INDIVIDUAL-BASED MODELLING OF

POPULATION DYNAMICS AND EVOLUTION

OF RESISTANCE TO PHOSPHINE

IN LESSER GRAIN BORER

by

Mingren SHI (史明仁)

This thesis is presented for the degree of

Doctor of Philosophy

The University of Western Australia

Faculty of Natural and Agricultural Sciences

School of Plant Biology

November 2012

TABLE OF CONTENTS

Statement of original contribution

i

Abstract ii

Publications arising from the thesis iv

Acknowledgements vi

Chapter 1 Introduction 1

Chapter 2 Numerical algorithms for estimation and calculation of parameters

in modelling pest population dynamics and evolution of

resistance. [with M. Renton, Math Biosci 233(2): 77-89 (2011)]

17

Chapter 3 Modelling mortality of a stored grain insect pest with fumigation:

probit, logistic or Cauchy model? [with M. Renton, Math Biosci

243: 137–146 (2013)]

31

Chapter 4 Constructing a new individual-based model of phosphine resistance

in lesser grain borer (Rhyzopertha dominica): do we need to

include two loci rather than one? [with M. Renton, T.J. Ridsdill-

Smith and P.J. Collins, Pest Science 85(4): 451-468 (2012).

41

Chapter 5 Individual-based modelling of the efficacy of fumigation tactics to

control lesser grain borer (Rhyzopertha dominica) in stored grain.

[with P.J. Collins, T.J. Ridsdill-Smith and M. Renton, J Stored

Prod Res, 51: 23-32 (2012)]

59

Chapter 6 Dosage consistency is the key factor in avoiding evolution of

resistance to phosphine and population increase in stored grain.

[with pests Renton M, Collins PJ, T James Ridsdill-Smith J and

Emery RN, Pest Manage. Sci., DOI: 10.1002/ps.3457 (2012)]

69

Chapter 7 Summary and discussion 81

Appendix Mortality estimation for individual-based simulations of phosphine

resistance in lesser grain borer (Rhyzopertha dominica). [with M.

Renton and P.J. Collins, Modelling and Simulation Society of

Australia and New Zealand, December 2011. pp. 352-358]

http://www.mssanz.org.au/modsim2011/A3/shi.pdf

87

i

STATEMENT OF ORIGINAL CONTRIBUTION

The research presented in this thesis is an original contribution to the field of simulation

modelling in plant biosecurity.

My estimated percentage contribution to each of the published papers is at least 90%

(see “Publications arising from this thesis” below). Others who made significant

contributions to the research are acknowledged in chapters 2-6.

The supervisors of this project, A/Prof. Michael Renton, Prof. T James Ridsdill-Smith,

Prof. Yonglin Ren and Mr. Robert N Emery, guided and supported me through the

process of searching literature, constructing models, setting model assumptions, coding

in Python and writing up and editing manuscripts for submission to journals and

conferences. My supervisors also helped in providing biological data, understanding the

needs for such modelling, ensuring the models were relevant to the real world, and

provided constructive criticism, feedback and suggestions on drafts of the thesis

chapters. Michael introduced me to individual based modelling, coding in Python and

provided advice during building the models. However, all model construction, analysis,

interpretation and writing was my own original work, and the modelling methods and

approaches employed were based on my own original ideas.

I declare that this thesis has been completed during the course of enrolment in a PhD

degree at the University of Western Australia, and the thesis is my own account of my

research and contains as its main content work which has not previously been submitted

for a degree at any tertiary educational institution.

ii

Abstract

The aim of this study is to use individual-based two-locus simulation modelling to

predict population dynamics and the evolution of resistance to phosphine (PH3)

fumigation in the lesser grain borer, Rhyzopertha dominica, and thus significantly

contribute to evaluating resistance management strategy options.

Individual-based modelling is a cutting-edge approach that explicitly represents the fact

that R. dominica populations consist of individual beetles, each of a particular genotype

and a particular life stage.

Four numerical algorithms for generating or estimating key parameters within such

models are described first in Chapter 2. The results of numerical experiments

demonstrated that the developed algorithms are valid and efficient.

In Chapter 3, two- and four-parameter probit models for phosphine mortality estimation

within a two-locus resistance simulation model (which comprises nine possible

genotypes) are developed. These probit models fit extensive experimental data very well

and include mortality predictions that vary with concentration, exposure time, and

genotype.

In Chapter 4, the differences between the predictions of one- and two-locus individual-

based models are compared in three cases: in the absence of phosphine fumigation, and

under high and low dose phosphine treatments. Simulation results show the importance

of basing resistance evolution models on realistic genetics and that using over-

simplified one-locus models to develop pest control strategies runs the risk of not

correctly identifying tactics to minimise the incidence of pest infestation.

In Chapter 5, the individual-based two-locus model is used to judge the management

tactics of single phosphine fumigation by investigating some biological and operational

factors that influence the development of phosphine resistance in R. dominica.

Simulation results indicate that extending exposure duration is a much more efficient

control tactic than increasing the phosphine concentration, and suggest that if the

original frequency of resistant insects is increased n times, then the fumigation needs to

iii be extended, at most, n days to achieve the same level of insect control. It yields from

simulation results that to control initial populations of insects that are n times larger, it

is necessary to increase the fumigation time by about n days.

Finally, in Chapter 6, the individual-based two-locus model is used to investigate the

impact of two important issues, the consistency of pesticide dosage through the storage

facility and the immigration rate of the adult pest, on overall population control and

avoidance of evolution of resistance to phosphine in lesser grain borer. Simulation

results indicate that achieving a consistent fumigant dosage is a key factor in preventing

evolution of resistance to phosphine and maintaining control of populations of R.

dominica.

iv

PUBLICATIONS ARISING FROM THIS THESIS

Primary authored manuscripts published in international journals

1. Shi M and Renton M, Numerical algorithms for estimation and calculation of

parameters in modelling pest population dynamics and evolution of resistance.

Mathematical Biosciences 233(2): 77-89 (2011). (The estimated percentage

contribution of the candidate is 95%.)

2. Shi M, Renton M, Ridsdill-Smith J and Collins PJ, Constructing a new

individual-based model of phosphine resistance in lesser grain borer

(Rhyzopertha dominica): do we need to include two loci rather than one? Pest

Science 85(4): 451-468 (2012). (The estimated percentage contribution of the

candidate is 90%.)

3. Shi M, Collins PJ, Ridsdill-Smith J and Renton M, Individual-based modelling

of the efficacy of fumigation tactics to control lesser grain borer (Rhyzopertha

dominica) in stored grain. Journal of Stored Product Research 51: 23-32 (2012).

(The estimated percentage contribution of the candidate is 90%.)

4. Shi M, Renton M, Collins PJ, T James Ridsdill-Smith J and Emery RN, Dosage

consistency is the key factor in avoiding evolution of resistance to phosphine

and population increase in stored grain pests, Pest Management Science

(wileyonlinelibrary.com) DOI: 10.1002/ps.3457 (2012). (The estimated

percentage contribution of the candidate is 90%.)

5. Shi M and Renton M, Modelling mortality of a stored grain insect pest with

fumigation: probit, logistic or Cauchy model? Mathematical Biosciences 243:

137–146 (2013). (The estimated percentage contribution of the candidate is

90%.)

Primary authored manuscripts published in refereed conference proceedings:

1. Shi M, Renton M and Collins PJ, Mortality estimation for individual-based

simulations of phosphine resistance in lesser grain borer (Rhyzopertha

dominica). In: Chan, F., Marinova, D. and Anderssen, R.S. (eds) MODSIM2011,

19th International Congress on Modelling and Simulation. Modelling and

Simulation Society of Australia and New Zealand, December 2011. pp. 352-358.

http://www.mssanz.org.au/modsim2011/A3/shi.pdf (2011).

v

(The estimated percentage contribution of the candidate is 90%.)

2. Shi M and Renton M, Phosphine mortality estimation for simulation of

controlling pest of stored grain: lesser grain borer (Rhyzopertha dominica), In:

Proceedings of International Conference on Modelling and Simulation, July 5-6,

2012, Zurich, Switzerland. World Academy of Science, Engineering and

Technology, Issue 67, July 2012, pp. 113-116.

https://www.waset.org/journals/waset/v67.php (2012). (The estimated

percentage contribution of the candidate is 90%.)

3. Shi M and Renton M, An individual-based two-locus modelling of pest control

in a spatial heterogeneous storage facility with pest immigration, In: S. Navarro,

H.J. Banks, D.S. Jayas, C.H. Bell, R.T. Noyes, A.G. Ferizli, M. Emekci, A.A.

Isikber, K. Alagusundaram (eds.), Proceedings of an International Conference

on Controlled Atmosphere and Fumigation in Stored Products. 15-19 October

2012. Antalya, Turkey. ARBER Professional Congress Services, Turkey, pp.

638-643. (2012). (The estimated percentage contribution of the candidate is

90%.)

vi

ACKNOWLEDGEMENTS

The research for this thesis was undertaken in the School of Plant Biology, The

University of Western Australia working with the Cooperative Research Centre for

National Plant Biosecurity (CRCNPB). Thanks to my coordinating supervisor, A/Prof.

Michael Renton’s encouragement and direction, and CRCNPB’s financial assistance I

had the chance to study for my second PhD degree in modelling of Plant Biosecurity

(The first PhD in Applied Mathematics was received at Murdoch University, Australia,

in 1997).

I would like to express my sincere gratitude to all those who helped me in completing

this thesis. Special thanks go to A/Prof. Michael Renton for being a great coordinating

supervisor and mentor, for his patience and guidance throughout my period of study. I

am very grateful to him for suggesting this challenging research topic in the first place.

He always was available for scientific discussion. He helped me in initiating

programming in Python and in improving my scientific writing skills. Deep thanks for

the great benefits from his supervision along the way.

I would also offer my sincere thanks to my co-supervisors. My thanks to Prof. T James

Ridsdill-Smith for his valuable suggestions for questions addressed in models, for his

joining discussions for construction of models, for his great help in entomology and

genetics, and for his checking and revising manuscripts. I am grateful to A/Prof.

Yonglin Ren and Mr. Robert N Emery, for their arrangement of visiting famers and

Cooperative Bulk Handling to obtain my fist hand biosecurity knowledge, and for their

great help in providing key information about beetle life cycles, silo fumigation and

biochemical interaction in the fumigation facilities.

My deep gratitude goes also to Dr. P.J. Collins. His name was not officially listed in my

supervisors, but indeed he is one of my mentors playing an indispensable role in

directing me to complete my thesis and a co-author of my four published

journal/conference paper arising from this thesis. He gave me great support and help in

the genetics and provision of raw data, in conceiving questions addressed and setting

assumptions in models, and in checking and revising papers.

vii The assistance of many members of staff at the School of Plant Biology is highly

appreciated.

viii

1

Chapter 1

Introduction

This thesis presents a number of new studies, all aiming to use simulation modelling to

help predict population dynamics and the evolution of resistance to phosphine pesticide

in the lesser grain borer stored grain pest, in order to evaluate monitoring strategies and

management options.

1. Background

1.1 Lesser grain borer

The lesser grain borer, Rhyzopertha dominica (Fabricius) (Coleoptera: Bostrichidae), is

an introduced, cosmopolitan, major pest of stored grains, which is difficult to control in

Australia and many other countries.

It attacks a wide variety of stored cereal grains, such as corn, rice and millet (Hagstrum

and Subramanyam, 2009). Both the larvae and adults bore irregularly shaped holes into

whole, undamaged kernels of grain, feeding inside the grain (Collins, 2008a, 2008b,

2008c; Emery, 2006). R. dominica impacts on food safety, trade, market access, market

development and, ultimately, the profitability and sustainability of plant industries.

Lesser grain borer is particularly destructive of stored cereals in warm temperate to

tropical climates where it is a voracious feeder on whole grains. Lesser grain borer is an

active flyer and rapid coloniser, completing its life cycle in 4-5 weeks under favourable

conditions. Adults lay 200-400 eggs through their life of 2-3 months (Arbogast, 1991).

2

1.2 Phosphine fumigation

Owing to its international market acceptance and a lack of acceptable, cost-effective

alternatives, disinfestation with phosphine (PH3) fumigant is a key component (Emery

and Nayak, 2007) and a fundamental tool used world-wide in the management of stored

grain pests including lesser grain borer.

Since the 1980’s, the Australian grain industry has come to rely heavily on the use of

phosphine to disinfest stored commodities, particularly grain to meet domestic and

international market demand for high quality grain, free of insects. Currently,

phosphine is the preferred fumigant used to protect bulk grains and oil seeds (more than

85% grains were treated / re-treated with phosphine) in each of the linkages from on-

farm storage to the grain terminal (Ren et al., 2012).

Reliance on phosphine developed because grain protectants, the previous commonly

used method of control, became vulnerable on two fronts: firstly, resistance strong

enough to cause control failures had emerged in many species and secondly, there was

growing adverse consumer reaction to the presence of chemical residues. In addition,

techniques for applying phosphine had improved and other chemical fumigants were

being criticised. Phosphine, in contrast to grain protectants, is unique; there is no

alternative that possesses the combined advantages that have made it so attractive for

use in the Australian grain industry. Phosphine is relatively easy to apply (compared

with other fumigants), versatile and cost-effective (out of patent and produced in India

and China) with international acceptance for being an environmentally benign treatment

that results in much less chemical residue. In addition, alternative chemical treatments

are generally not acceptable for environmental, health and safety reasons, and physical

treatments are significantly more expensive, not effective enough to meet market

standards and often do not comply with grain handling logistics. The main disadvantage

of phosphine is that it is a slow-acting poison and requires extended contact time at

appropriate concentrations if adult and immature insect life stages are to be killed. The

use of phosphine with poor delivery systems and in unsealed silos is resulting in

insufficient concentrations and exposure time, leading to poor kill rates and the

evolution of resistance to phosphine in stored grain pests (Collins et al., 2002)

3

1.3 Evolution of resistance to phosphine

However, heavy reliance on phosphine has resulted in the development of strong

resistance in several major pest species including R. dominica in many countries

including Australia. This is widely recognized as a serious concern to both primary

producers and grain handling organisations and a crucial issue for the long-term

sustainability of Australian agricultural systems. The appearance of resistance is

particularly critical when the pest management strategy is so reliant on a single

pesticide (phosphine), as is the case for the lesser grain borer (Collins et al., 2002).

The threat that insects may also develop resistance to phosphine was first noted in the

Food and Agriculture Organization’s (FAO) 1972/1973 global survey on pesticide

resistance (Champ and Dyte, 1976). In Australia, a mild level of resistance in R.

dominica was first recorded in 1990 (White and Lambkin, 1990). This led to resistance

monitoring projects being initiated across all cereal growing regions of Australia. With

industry support, these projects have now amalgamated to form a national phosphine

resistance monitoring and management program. In addition, research is being

undertaken to fully characterise the resistance and to develop improved control options

such as changes to fumigation concentrations and exposure periods.

Since 1997 strong resistance has been recorded in four of the five major insect pest

species, including lesser grain borer. Evolution of this new strong resistance is a major

challenge to the grain industry (Schlipalius et al., 2002; Ebert et al., 2003). As the

growth of resistance to phosphine in populations of the lesser grain borer seriously

threatens effective insect pest management (Collins, 2006; Emekci, 2010), there is a

world-wide need for preventing the selection of populations of resistant insects which

are difficult to control.

1.4 Note on the words ‘evolution’ and ‘selection’

In this thesis I model the way that the frequencies of certain alleles and genotypes

change within a population over generations under a selection pressure imposed by the

application of a fumigant that affects different genotypes in different ways. It could

possibly be argued that this is not ‘really’ modelling evolution, as I do not explicitly

4

model the occurrence of chance mutations in the population, leading to the appearance

of alleles conferring resistance to the fumigant, but instead assume a ‘background’ level

of these resistance alleles. However, if evolution is defined simply as the change in

genetic composition of a population over successive generations, then it would seem

that what I am modelling called be termed ‘evolution’. In any case, I acknowledge that

there are many subtleties in the debate of what is and isn’t really evolution, but in this

thesis I have chosen to use the terms ‘evolution of resistance’ and ‘selection of

resistance’ quite interchangeably, along with others such as ‘development of resistance’

and ‘emergence of resistance’, to mean the increase in frequencies of resistance-

conferring alleles and more resistant genotypes in a population over successive

generations. Certainly, in no case are we assuming or suggesting that the fumigant

causes mutations or alters mutation rates.

1.5 Factors impacting on evolution of resistance

The development of resistance to pesticides in insects is affected by a variety of

interacting influences, including genetic factors, biological/ecological factors and

management (operational) factors (National Research Council, 1986). Furthermore,

many factors can affect whether an insect population will survive under phosphine

fumigation treatments. Some of the more important factors include gas concentration,

duration of fumigation, temperature and developmental stages of the insects present.

Genetic factors likely to influence the development of populations with resistance to

insecticides include the initial gene frequencies of R (resistance) alleles in the

population and the degrees of dominance (interaction between alleles at the same locus)

and epistasis (interaction between genes) of the resistance genes.

Biological/ecological factors include the amount of time spent in each life stage (egg,

larvae, pupae, adult), fecundity, (i.e. number of eggs laid by a female), mortality, (i.e.

death rates of different genotypes and life stages), and insect movement into and out of

storage facility (e.g. migration rates).

Management factors include phosphine fumigation frequency, duration of fumigation

used, dose/concentration of phosphine, consistency of dose/concentration across time

5

and/or space, and the type, timing and frequency of hygiene practices in the storage

facilities and transportation areas (such as trains, trucks, ships).

The purpose of this PhD project is to use individual-based modelling to investigate how

these genetic factors, biological/ecological factors and management factors interact to

affect the evolution of resistance to the grain fumigant phosphine in the stored-grain

pest, the lesser grain borer, and thus identify optimal management strategies for

delaying or avoiding the evolution of resistance.

2. Computer simulation models

2.1 Importance of simulation models

Computer simulation models have been important and very useful tools for scientific

investigation in nearly all areas of science (Keller, 2002; Peck, 2004; Winsberg, 2003).

Computer simulation models can provide a relatively fast, safe and inexpensive means

to project the consequences of different assumptions about resistance, to understand the

causative processes of effective management, and to judge and weigh the merits of

various management options. In the study of complex systems, modelling can be used to

identify important gaps in knowledge, assess risks, and perform virtual experiments that

are impossible to perform in reality because of cost, logistics, or ethics. Also models can

synthesise and integrate our understanding of the different genetic, ecological and

management factors underlying resistance. For these reasons, modelling can play a

critical role in making predictions about the evolution of resistance, insect population

dynamics and the effects of different pest management strategies.

2.2 Examples of simulation models in pest management

Methods for simulation modelling in general have been well studied (Caprio et al., 2008

and references therein). Longstaff (1991) and Throne (1994) have provided overviews

of the role of modelling in evaluating alternative pest management strategies for stored

products. Several studies have focused on the genetics behind phosphine resistance in R.

dominica. Concise knowledge of how resistance is inherited and how it is controlled at

a genetic level enables modelling of the response of specific genotypes to different

treatments.

6

A number of other population dynamics models have been used to consider

management issues in stored grain systems. Hagstrum and Heid (1988) constructed a

simple model to predict Rhyzopertha numbers in grain stores in the USA. Flinn and

Hagstrum (1990) built upon the basic population dynamics model of Hagstrum and

Throne (1989) to predict effects of aeration, fumigation and protectants upon

Rhyzopertha, and Hagstrum and Flinn (1990), in turn, expanded this to consider another

four species. Kawamoto et al. (1989, 1991) developed population dynamics models for

Cryptolestes ferrugineus and Acarus siro and these are being used as the biological

basis for a composite granary-ecosystem model, into which is also built a physical

model (Metzger and Muir, 1983; Flinn et al., 1992) for changes in grain temperature

and moisture content with and without aeration. Sinclair and Alder (1985) developed a

model to simulate the management of stored grain pests on farms and the implications

of this for the central grain-handling system.

Apart from the above available modelling resources and methods, Renton (2009), Neve

(2008), and many others (e.g. see references cited in Renton (2009) and Neve (2008))

have developed simulation models representing population dynamics and evolution of

resistance to herbicides in weeds. Simulation models have also been used to predict the

development of resistance to pesticides in populations of insects (Comins, 1986;

Crowder et al., 2006; Groeters and Tabashnik, 2000; Roush and McKenzie, 1987). My

simulation modelling can draw on their results since modelling principles and methods

for population dynamics and resistance evolution to herbicides in weeds and to

insecticides in insect pests are similar in many aspects.

2.3 Individual-based modelling

In this thesis, a cutting-edge modelling approach, stochastic discrete individual-based

modelling, will be applied to the problem of simulating insect resistance. Individual-

based models simulate the insects as individual agents, which explicitly represents the

fact that R. dominica populations consist of individual beetles, each of a particular

genotype and a particular life stage.

Individual-based resistance models allow more aspects of the individual variability and

biological reality to be included, and allow the modeller to relatively easily incorporate

7

new attributes that I wish to investigate, such as different genetics, different initial

frequencies of genotypes and/or different proportions of life stages, spatial location,

movement, interactions between individual insects etc (Renton 2012).

Note that the aim and value of individual-based models is to gain insights into

qualitative relationships and interactions, rather than simple precise quantitative

prediction (Grimm and Railsback 2004, Renton 2012).

2.4 One-locus modelling

In previously published resistance modelling research “survivorship (or mortality) was

not explicitly included in the model because adequate data were not available”

(Hagstrum and Flinn, 1990), and thus a simple one-locus model was used. That is,

previous modelling of the evolution of major-gene resistance to pesticides in insects

generally assumes resistance is conferred by a single gene (Sinclair and Alder, 1985;

Tabashnik and Croff, 1982).

The recent discovery that two genes at loci on distinct chromosomes are responsible for

strong phosphine resistance has motivated the development of a simulation model of

two-locus genetic inheritance in these studies. Two-locus models will simulate the pest

population dynamics and the evolution of resistance of insect pests in stored grains, and

will be used for predicting optimum control strategies in this study.

3. Outline of this thesis

3.1 Estimating model parameters The usefulness of simulation models relies on the accurate estimation of important

model parameters. Various parameters must be calculated or estimated before these

models are used to predict the effects of different possible management strategies.

Estimating parameters based on measured empirical data is a critical issue in simulation

models of population dynamics and evolution of resistance in stored-grain insect pests

(Collins et al., 2002).

8

In Chapter 2, four numerical algorithms are described for four independent issues

generating or estimating key parameters within such models:

(1) A novel method to generate an offspring genotype table, where by an offspring

genotype table meaning a table that lists all possible combinations of parental

genotypes, and, for each possible parental combination, gives the expected

proportions of offspring genotypes (Hedrick, 2005, p76). Such a table is

indispensable for a genetic model simulating evolution of resistance, or other

traits.

(2) A generalized inverse matrix to find a least-squares solution to an over-

determined linear system for estimation of parameters in probit models to

predict mortalities.

(3) A simple algorithm to randomly select initial frequencies of genotypes.

(4) Converting the problem of estimating the intrinsic rate of natural increase of a

population to a root-finding problem and then the bisection algorithm can be

used to find the rate.

The mortality estimation in this chapter focusses on the numerical algorithm used for

model fitting, with only a limited use of experimental data for illustration.

Mortality is a very important parameter in modelling of resistance evolution and

population dynamics. In previous modelling research, when different mortalities for

different genotypes were included, they were only roughly divided into a few levels (e.g.

Tabashnik, B.E., 1989; Longstaff, 1988) and a simplified single gene model was used.

In other cases, mortalities were varied with temperature and moisture in some detail but

differences due to concentration, exposure time, or genotype were not included (e.g.

Flinn et al., 1992). The ability to estimate mortality for the different genotypes at a

range of concentrations and exposure times based on experimental data is critical for the

accuracy of the new two-locus individual-based simulation models.

Therefore, in Chaper 3 I conduct a more comprehensive study on the best way to model

phosphine mortality for nine different genotypes in two-locus models. Three models

were considered, probit, logistic and Cauchy models to fit the available data sets

employing either C (concentration or dose) and t (exposure time) themselves or log(C)

and log(t) as the independent model variables. These models were used to fit data sets

for five strains, each of which corresponds one of the nine genotypes, and the resistance

factors for these five genotypes (strains) are estimated based on the fitted models. The

9

study showed how the resistance factors for the other four genotypes can be estimated

by making some basic assumptions regarding genetic interactions according to the

strength and the dominance of the 1st and 2nd genes, and the synergism between the two

genes. Finally, the mortality estimation for the remaining four genotypes can be done

with a two-parameter probit model (as after comparing the relative accuracy of the

probit, logistic and Cauchy models, the probit is the best).

3.2 Comparison of one- and two-locus models Resistance to phosphine is an inherited trait and simple one-locus models have been

used in previous modelling research. However, recent research has indicated the

existence of two resistance phenotypes in R. dominica, weak and strong (Collins, 1998),

and revealed that the presence of homozygous resistance alleles at two loci confers

strong resistance (Collins et al., 2002; Schlipalius et al., 2002).

In Chapter 4, I describe and compare one- and two-locus individual-based models

constructed to investigate how genetic factors influence the development of phosphine

resistance in the lesser grain borer. Thus the importance of including two genes in

resistance model were evaluated, and whether following previous studies and

simplifying by assuming resistance to be conferred by a single gene would make little

difference to the model’s predictions were investigated.

The predictions of the one- and two-locus models under three simulated scenarios were

compared: in the absence of fumigation; under a high concentration treatment; and

under a low concentration treatment. Whether differences between the models

predictions could be overcome with a simple adaptation to the one-locus model, or

whether differences in model predictions were more significant were also investigated.

The simulation results show the importance of basing models of the development of

resistant populations on realistic genetics and the importance of including the full

complexity of the polygenic resistance in the model, rather than using an over-

simplified one-locus model. Hence, later simulations are carried out using two-locus

models.

10

3.3 Test of the efficacy of different short-term fumigation tactics

In practice, the same grain store may be fumigated multiple times, but usually for the

same exposure period and concentration. Simulating a single fumigation allows us to

look more closely at the effects of this standard treatment.

In Chapter 5, my individual-based, two-locus model is used to test the efficacy of

different short-term fumigation tactics in maintaining control by investigating some

biological and operational factors that influence the development of phosphine

resistance in R. dominica. This has been done by addressing three key questions about

the use of phosphine fumigant in relation to the development of PH3 resistance. First, is

long exposure time with a low concentration or short exposure period with a high

concentration more effective for insect control? Second, how long should the

fumigation period be extended to deal with higher frequencies of resistant insects in the

grain? A third question is how does the presence of varying numbers of insects inside

grain storages impact the effectiveness of phosphine fumigation?

3.4 Evaluation and identification of viable, long-term strategies

In previous chapters, however, my model has not yet been used to evaluate and identify

viable, long-term strategies to support the management of phosphine resistance and pest

infestation, which is the ultimate aim of my research.

In Chapter 6, the individual-based two-locus model was extended to include spatial

variability in dosage, and immigration into the storage of adult insects to investigate the

importance and impact of two important factors in managing pest population number

and the evolution of resistance in stored grain pests. These are: 1) achieving a

consistent fumigant dosage (or concentration) within the storage facility, and 2)

controlling immigration of insects from outside the facility (Opit et al., 2012).

11

References

(1) Arbogast R.T. (1991), Beetles: Coleoptera. In: Gorham JR (ed) Ecology and

management of Food-Industry Pests, Association of Official Analytical Chemists,

Arlington, Virginia, USA, pp 131-176.

(2) Caprio M.A., Storer N.P., Sisterson M.S., Peck S.L. and Maia A.H.N. (2008),

Assessing the Risk of the Evolution of Resistance to Pesticides Using Spatially

Complex Simulation Models, Chapter 4 of the book “Global Pesticide Resistance

in Arthropods”, Edited by Whalon M.E., Mota-Sanchez D., and Hollingworth

R.M., CAB international.

(3) Champ B.R. and Dyte C. (1976), Report on the FAO global survery of pesticide

susceptibility of stored grain pests. In FAO Plant Production and Protection Series,

pp. 90–99.

(4) Collins P.J. (1998), Resistance to grain protectants and fumigants in insect pests of

stored products in Australia. in: Proceeding of First Australian Postharvest

Technical Conference. ed. by Banks HJ, et al Canberra, Australia, pp 55–57.

(5) Collins P.J. (2006), Resistance to chemical treatments in insect pests of stored grain

and its management. In: Lorini, I., Bacaltchuk, B., Beckel, H., Deckers, D.,

Sundfeld, E., Santos, J.P., Biagi, J.D., Celaro, J.C., Faroni, L.R., Bortolini, L.O.F.,

Sartori, M.R., Elias, M.C., Guedes, R.N.C., Fonseca, R.G. and Scussel, V.M. (Eds).

Proceedings of the 9th International Working Conference on Stored Product

Protection, 15-18 October 2006, Campinas, Brazil, pp. 277-282.

(6) Collins P.J. (2008a), Coordinated resistance monitoring, GroundCover, Sep.-Oct.,

2008, p16.

(7) Collins P.J. (2008b), Phosphine resistance narrow options, GroundCover, Sep.-Oct.,

p14-15.

(8) Collins P.J. (2008c), The emergence of strong resistance, GroundCover, Sep.-Oct.,

2008, p14-15.

(9) Collins P.J., Daglish G.J., Bengston M., Lambkin T.M. and Pavic H. (2002),

Genetics of resistance to phosphine in Rhyzopertha dominica (Coleoptera:

Bostrichidae). Journal of Economic Entomology 95(4): 862–869

(10) Collins P.J., Emery R.N. and Wallbank B.E. (2002), Two decades of monitoring

and managing phosphine resistance in Australia. Advances in Stored Product

Protection. Proceedings of the 8th International Working Conference on Stored

12

Product Protection, pp. 570-575. Credland P.F., Armitage D.M., Bell C.H., Cogan,

P.M. and Highley E. (Edt.), York, UK, 22-26 July, 2002.

(11) Comins H.N. (1986), Tactics for resistance management using multiple pesticides.

Agriculture, Ecosystems & Environment 16, pp. 129–148.

(12) Crowder D.W., Carriere Y., Tabashnik B.E., Ellsworth P.C. and Dennehy T.J.

(2006), Modeling evolution of resistance to pyriproxyfen by the sweetpotato

whitefly (Hemiptera: Aleyrodidae). Journal of Economic Entomology 99, pp.

1396–1406.

(13) Emekci, M. (2010), Quo vadis the fumigants? In: Carvalho, O.M, Fields, P.G.,

Adler, C.S., Arthur, F.H., Athanassiou, C.G., Campbell, J.F., Fleurat-Lessard, F.,

Flinn, P.W., Hodges, R.J., Isikber, A.A. Navarro, S., Noyes, R.T., Riudavets, J.,

Sinha, K.K., Thorpe, G.R., Timlick, B.H., Trematerra, P., White, N.D.G. (Eds.),

Proceedings of the 10th International Working Conference on Stored Product

Protection, 27 June-2 July 2010, Estoril, Portugal, pp. 303-313.

(14) Emery R.N. (2006), Practical Experience on Integrated Pest Management on

Stored Grain: “The Clean Pipeline in Western Australia”, Proceedings of the 9th

International Working Conference on Stored Product Protection, 2006.

(15) Emery R.N. and Nayak M.K. (2007), Chapter 2 Cereals – Pests of Stored Grains.

in: Pests of field crops and pastures: identification and control. ed. by Peter Bailey,

CSIRO Publishing. http://www.publish.csiro.au/pid/3165.htm.

(16) Ebert P.R., Reilly P.E.B., Mau Y., Schlipalius D.I. and Collins P.J. (2003), The

molecular genetics of phosphine resistance in the lesser grain borer and

implications for management, Proceedings of the Australian Postharvest Technical

Conference, Canberra, 25–27 June 2003. CSIRO Stored Grain Research

Laboratory, Canberra, Wright E.J., Webb M.C. and Highley E. , ed., Stored grain

in Australia 2003.

(17) Flinn P.W. and Hagstrum D.W. (1990), Simulations comparing the effectiveness of

various stored-grain management practices used to control Rhyzopertha dominica

(Coleoptera: Bostrichidae). Environmental Entomology 19, 725–729.

(18) Flinn P.W., Hagstrum D.W., Muir W.E. and Sudayappa K., (1992), Spatial model

for simulating changes in temperature and insect population dynamics in stored

grain. Environmental Entomology 21, 1351-1356.

(19) Grimm V. and Railsback S.F. (2004), Individual-based Modeling and Ecology.

Princeton University Press.

13

(20) Groeters F.R. and Tabashnik B.E. (2000), Roles of selection intensity, major genes,

and minor genes in evolution of insecticide resistance. Journal of Economic

Entomology 93, pp. 1580–1587.

(21) Hagstrum D.W. and Flinn, P.W. (1990), Simulations comparing insect species

differences in response to wheat storage conditions and management practices.

Journal of Economic Entomology 83, 2469-2475.

(22) Hagstrum D.W. and Heid W.G. (1988), U.S. wheat marketing system: An insect

ecosystem. Bulletin of the Entomological Society of America 34, 33-36.

(23) Hagstrum D.W. and Subramanyam B. (2009), Stored-product Insect Source.

AACC International St Paul, Minnesota, USA

(24) Hagstrum D.W. and Throne J.E. (1989), Predictability of stored-wheat insect

population trends from life-history traits. Environmental Entomology 18, 660-664.

(25) Hedrick P.W. (2005), Genetics of Population, (3rd ed.), Jones and Bartlett

Publishers.

(26) Kawamoto H., Sinha R.N., Muir W.E. and Woods S.M. (1991), Simulation model

of Acarus siro (Atari: Acaridae) in stored wheat. Environmental Entomology 20,

1381-1386.

(27) Kawamoto H., Woods S.M., Sinha R.N. and Muir W E. (1989), A simulation

model of population dynamics of the rusty grain beetle, Cryptolestes ferrugineus in

stored wheat. Ecological Modelling 48, 137-157.

(28) Keller E.F. (2002), Making Sense of Life: Explaining Biological Development

with Models, Metaphors, and Machines, Harvard University Press.

(29) Longstaff B.C. (1988), A modeling study of the effects of temperature

manipulation upon the control of Sitophilus oryzae (Coleoptera: Curculionidae) by

insecticide. Journal of Applied Ecology 25, 163-175.

(30) Longstaff B.C. (1991), The role of modelling in the management of stored-product

pests. In: Proceedings of the Fifth International Working Conference on Stored-

Product Protection III (Edited by Fleurat-Lessard, F. and Ducom, P.), pp. 1995-

2007, Bordeaux, France, 9-14 September 1990.

(31) Metzger J.F. and Muir W.E. (1983), Aeration of stored wheat in the Canadian

Prairies. Canadian Agricultural Engineering 25, 127-137.

(32) National Research Council, Committee on Strategies for the Management of

Pesticide Resistant Pest Populations, 1986. Pesticide Resistance: Strategies and

Tactics for Management. National Academy Press, Washington, D.C.

14

(33) Neve P. (2008), Simulation modelling to understand the evolution and

management of glyphosate resistance in weeds, Pest Management Science 64:392–

401.

(34) Opit G., Collins P.J. and Daglish G.J. (2012), Chapter 13, Resistance management.

In: Hagstrum DW, Thomas W, Phillips TW and Cuperus G (eds), Stored Product

Protection, Kansas State Research and Extension Publication S156.

(35) Peck S.L. (2004), Simulation as experiment: a philosophical reassessment for

biological modelling. Trends in Ecology & Evolution 19(10), 530-534.

(36) Ren Y.L., Newman J., Agarwal M. and Cheng H. (2012), Nitrogen application

offers for both control of insect and grain quality. In: S. Navarro, H.J. Banks, D.S.

Jayas, C.H. Bell, R.T. Noyes, A.G. Ferizli, M. Emekci, A.A. Isikber, K.

Alagusundaram (eds.), Proceedings of an International Conference on Controlled

Atmosphere and Fumigation in Stored Products. 15-19 October 2012. Antalya,

Turkey. ARBER Professional Congress Services, Turkey, pp. 471-477.

(37) Renton M. (2009), The weeds fight back: Individual-based simulation of evolution

of polygenic resistance to herbicides. In: Anderssen RS, Braddock RD, Newham

LTH (eds) Proceeding of 18th World IMACS Congress and MODSIM09

International Congress on Modelling and Simulation. MSSANZ and IMACS,

Cairns, Australia, pp 574-580.

(38) Renton M. (2012), Shifting focus from the population to the individual as a way

forward in understanding, predicting and managing the complexities of evolution of

resistance to pesticides. Pest Management Science (2012) (accepted 7/4/2012, DOI

10.1002/ps.3341).

(39) Roush R.T. and McKenzie J.A. (1987), Ecological genetics of insecticide and

acaricide resistance. Annual Review of Entomology 32, pp. 361–380.

(40) Schlipalius D.I., Cheng Q., Reilly P.E.B., Collins P.J. and Ebert P.R. (2002),

Genetic Linkage Analysis of the Lesser Grain Borer Rhyzopertha dominica

Identifies Two Loci That Confer High-Level Resistance to the Fumigant Phosphine,

Genetics 161, 773–782.

(41) Sinclair E.R. and Alder J. (1985), Development of a computer simulation model of

stored product insect populations on grain farms. Agricultural Systems 18, 95-113.

(42) Tabashnik B.E. (1989), Managing resistance with multiple pesticide tactics: theory,

evidence, and recommendations. Journal of Economic Entomology 82, 1263-1269.

15

(43) Tabashnik B.E., Croft B.A. (1982), Managing pesticide resistance in crop-

arthropod complexes: interactions between biological and operational factors.

Environment Entomology 11: 1137-1144.

(44) Throne J.E. (1994), Computer modeling of the population dynamics of stored-

product pests. In Stored Grain Ecosystems (Edited by Jayas, D.S., White, N.D.G.

and Muir, W.E.), pp. 169195. Marcel Dekker, New York, USA. ISBN 0-8247-

8983-0.

(45) White, G.G. and Lambkin T.A. (1990), Baseline responses to phosphine and

resistance status of stored-grain beetle pests in Queensland, Australia. Journal of

Economic Entomology 83, 1738–1744.

(46) Winsberg E. (2003), Simulated experiments: methodology for a virtual world.

Philosophy of Science, 70: 105-125.

16

Numerical algorithms for estimation and calculation of parameters in modeling

pest population dynamics and evolution of resistance

Mingren Shi a,b,⇑, Michael Renton a,b,c

a School of Plant Biology, University of Western Australia, 35, Stirling Highway, Crawley, WA 6009, Australiab Cooperative Research Centre for National Plant Biosecurity, Australiac CSIRO Ecosystem Sciences, Underwood Avenue, Floreat, WA 6014, Australia

a r t i c l e i n f o

Article history:

Received 8 March 2011

Received in revised form 19 May 2011

Accepted 20 June 2011

Available online 13 July 2011

Keywords:

Parameter estimation

Offspring genotype table

Probit models

Mortality estimation

Population dynamics

Resistance evolution

a b s t r a c t

Computational simulation models can provide a way of understanding and predicting insect population

dynamics and evolution of resistance, but the usefulness of such models depends on generating or esti-

mating the values of key parameters. In this paper, we describe four numerical algorithms generating or

estimating key parameters for simulating four different processes within such models. First, we describe

a novel method to generate an offspring genotype table for one- or two-locus genetic models for simu-

lating evolution of resistance, and how this method can be extended to create offspring genotype tables

for models with more than two loci. Second, we describe how we use a generalized inverse matrix to find

a least-squares solution to an over-determined linear system for estimation of parameters in probit mod-

els of kill rates. This algorithm can also be used for the estimation of parameters of Freundlich adsorption

isotherms. Third, we describe a simple algorithm to randomly select initial frequencies of genotypes

either without any special constraints or with some pre-selected frequencies. Also we give a simple

method to calculate the ‘‘stable’’ Hardy–Weinberg equilibrium proportions that would result from these

initial frequencies. Fourth we describe how the problem of estimating the intrinsic rate of natural

increase of a population can be converted to a root-finding problem and how the bisection algorithm

can then be used to find the rate. We implemented all these algorithms using MATLAB and Python code;

the key statements in both codes consist of only a few commands and are given in the appendices. The

results of numerical experiments are also provided to demonstrate that our algorithms are valid and

efficient.

Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction

Estimating parameters based on measured empirical data is a

critical issue in biosecurity models, such as simulation models of

population dynamics and evolution of resistance in stored-grain

insect pests [12]. These simulation models are based on integrating

sub-models representing different key biological processes, such as

genetic recombination and mortality due to pesticides. Various

parameters for different sub-models must be calculated or esti-

mated before these models are used to predict the effects of differ-

ent possible management strategies. These parameters include: the

chance of certain genotypes being produced as the result of the

mating of certain parent genotypes (which we call offspring geno-

type tables), initial frequencies of genotypes, mortalities of insect

pests under various pesticide doses, and the intrinsic rate of natural

increase of an insect population. These are important parameters

within the sub-models for simulating genetic recombination and

thus determining the genotype of offspring, initialisation of the

population, simulating the effects of pesticide applications and cal-

culating the number of eggs produced by each insect, respectively.

By an offspring genotype table we mean a table that lists all pos-

sible combinations of parental genotypes, and, for each possible

parental combination, gives the expected proportions of offspring

genotypes (see Hedrick’ book [19, p. 76] for an example of this kind

of table, although no formal name is provided in this or other liter-

ature). Such a table is indispensable for a genetic model simulating

evolution of resistance, or other traits. We develop a novel method

to generate the offspring genotype table for a one-locus genetic

model: quantifying all possible genotypes of parents and offspring

and then using a block-matrix multiplication approach to generate

the full table describing the chance of certain genotypes being

produced as the result of the mating of each and every possible

combination of parent genotypes. The offspring genotype tables

for more than one locus are then produced recursively, with the

table for a model with a higher number of loci produced from the

tables for lower numbers of loci. This algorithm for the one- and

0025-5564/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved.

doi:10.1016/j.mbs.2011.06.005

⇑ Corresponding author at: School of Plant Biology, University of Western

Australia, 35, Stirling Highway, Crawley, WA 6009, Australia. Tel.: +61 8 6488

1992; fax: +61 8 6488 1108.

E-mail addresses: Shi.mingren@gmail.com (M. Shi), mrenton@cyllene.uwa.

edu.au (M. Renton).

Mathematical Biosciences 233 (2011) 77–89

Contents lists available at ScienceDirect

Mathematical Biosciences

journal homepage: www.elsevier .com/locate /mbs

Chapter 2

17

two-locus cases is given in Section 2.1. We also explain how this

algorithm can be extended for models with more than two loci.

Many problems of quantitative inference in biological and tech-

nological research concern the relation between a stimulus (e.g.

phosphine fumigation dose) and a binomial response (e.g. mortality

of insect pests). A binomial generalized linear model, with a link

function such as the probit function (the inverse of cumulative dis-

tribution function), is usually used to analyse the empirical data.

Normally, maximum likelihood estimation or chi-square approxi-

mation is applied to fitting the parameters of such probit models.

In fact, however, in such probit models the probit is a linear function

of parameters or metameter (e.g. log) of parameters and the corre-

sponding equations with respect to the parameters form an over-

determined linear system. We used a generalized inverse matrix

method to find the least-squares solution of the regularization equa-

tions. We describe the method in Section 2.2. This method has

advantages over other methods [4] if we only need to estimate

parameters without other statistical information such as signifi-

cance or confidence intervals for the estimates: it is simple with only

one key command, provides a more accurate estimate of parameters,

and even if the coefficient matrix of the over-determined linear sys-

tem is not numerically (column) full ranked it will still work and

yield a solution with minimum error in the L2 norm sense [4].

In some situations, we may wish to randomly select some or all

of the initial frequencies of genotypes for a biological or genetic

model. These frequencies must satisfy two simple constraints:

each frequency is in the range [0,1] and the sum is equal to 1. In

Section 2.3, we describe how we select the initial frequencies

either without any extra conditions, or with some pre-selected fre-

quencies, or with linear equality and inequality constraints. Also

we give a simple block-matrix multiplication method to calculate

the equilibrium proportions that should result from these initial

frequencies according to the Hardy–Weinberg Principle [29].

The intrinsic rate of natural increase (or development rate) is an

important parameter in modeling the dynamics of an insect popu-

lation. In Section 2.4, we describe how we converted the problem

of estimating this parameter into a root-finding problem and used

a bisection method to find the rate to any desired accuracy.

All the above algorithms are implemented using MATLAB

(www.mathworks.com) and Python (www.python.org) code, using

the Scientific Python library (www.scipy.org), and the key state-

ments and results of numerical experiments are given in Section

3 demonstrating that our algorithms are valid and efficient.

2. Methods

2.1. Quantification of genotypes through block-matrix multiplication

algorithm for creation of offspring genotype table

We developed a novel quantification of genotypes through block-

matrix multiplication algorithm to generate the offspring genotype

tables for a one-locus genetic model. In this section we describe

how this algorithm can be used to induce the two-locus table from

the one-locus table by block-matrix multiplication, and then how

this algorithm can recursively be extended to generate the off-

spring genotype tables for models with more than two loci. Based

on assumptions of random mating and no dependence of inheri-

tance on gender, this algorithm now makes it relatively straightfor-

ward to express genotype frequencies of an insect population as

the proportion of offspring from all possible parental unions that

belong to each genotype. Note that we developed the method in

this paper only for diploid species, i.e. where each locus has two al-

leles, but the idea for developing this algorithm is also suitable for

constructing algorithms for species where each locus has more

than two alleles.

2.1.1. One-locus case

To use computational methods for generating the one-locus off-

spring genotype table, we need to quantify the parental and off-

spring’s genotypes first. In the one-locus case, the two alleles,

dominant ‘‘A’’ and recessive ‘‘a’’, are distributed among offspring

in the usual, binomial ratios. Each mating of ‘‘female parent � male

parent’’ will produce four possible combinations: [each of 2 alleles

of female parent (F1,F2)] � [each of 2 alleles of male parent

(M1,M2)]. For example, the mating Aa � Aa, will produce F1 � M1

: AA, F2 � M1 : aA (=Aa), F1 � M2 : Aa and F2 � M2 : aa. This process

can be obtained by a schematic or a diagrammatic method, known

as the Punnett square, or by constructing a tree diagram [31]. The

Punnett square, named after the geneticist Reginald C. Punnett,

for the above case is shown in Table 1.

Hence the proportions of offspring are equal to 2/4 = 0.5 for

genotype Aa, 1/4 = 0.25 for aa and also 1/4 = 0.25 for AA. It is

important to note that Punnett squares give probabilities only for

genotypes, not phenotypes. The way in which the A and a alleles

interact with each other to affect the phenotype of the offspring

depends on how the gene products (proteins) interact. For classical

dominant/recessive genes, like that which determines whether a

rat has black hair (A) or white hair (a), the dominant allele will

mask the recessive one. Thus in the example above 75% of the off-

spring will be black (AA or Aa) while only 25% will be white (aa).

The ratio of the phenotypes is 3:1.

The proportion of each genotype in the offspring can be calcu-

lated by hand by counting the number of this genotype in the Pun-

nett square or by calculating the probability using a multiplication

rule in the tree diagram [31]. Our more efficient computer-based

method to do this works as follows. First we use numbers to denote

the genotypes of parents: ‘‘1’’ for the allele A and ‘‘2’’ for the allele

a. Then the Aa genotype of female and male parents can be ex-

pressed by the following matrices respectively:

FAa ¼1

2

� �; MAa ¼ ½1;2�: ð2:1Þ

The genotypes and numbers of four possible combinations of their

offspring can be generated by matrix multiplication:

FAaMAa ¼ 1

2

� �½1;2� ¼ 1 2

2 4

� �: ð2:2Þ

In the product, which can be regarded as a digitized or quantified

Punnett square, ‘‘1’’ stands for the genotype AA (as 1 � 1 = 1), ‘‘2’’

for Aa (1 � 2 = 2 � 1 = 2) and ‘‘4’’ for aa (2 � 2 = 4). We do not need

to produce all of the ‘‘products’’ of the different genotypes one by

one, instead, the whole offspring genotype table can be obtained

at once by the following process:

(i) Let M be a 1 � 6 matrix (or a 1 � 3 block-matrix) represent-

ing the three possible genotypes of the male parent:

ð2:3Þ

and F = MT (transpose of M) be a 3 � 1 block-matrix representing

the three genotypes of the female parent.

(ii) Then the block-matrix product FM is a 3 � 3 block-matrix

with each block being a 2 � 2 sub-matrix where

ð2:4Þ

78 M. Shi, M. Renton / Mathematical Biosciences 233 (2011) 77–89

18

Each 2 � 2 sub-matrix then corresponds to the result of a possible

mating; for example, the sub-matrix in the middle is the one shown

in Eq. (2.2).

(iii) Calculate the proportions of the three different genotypes of

offspring for each possible mating (sub-matrix) by

ðthe number of\1"s or \2"s or \4"sÞ=4: ð2:5Þ

We can use ‘‘s’’ to denote ‘‘AA’’, ‘‘h’’ for ‘‘Aa’’ and ‘‘r’’ for ‘‘aa’’, and

Py�z is used to denote the ‘‘proportion list’’, which is a list (or row

vector) of proportions of offspring reproduced by the cross of

female parent having genotype y with male parent having geno-

type z. Then for the example represented by the sub-matrix in

Eq. (2.2),

ð2:6Þ

Note that each of Py�z is a 1 � 3 matrix or a row vector.

If each sub-matrix in FM is replaced by its corresponding pro-

portion list in Eq. (2.6) we have a block-matrix P where

P ¼

Ps�s Ps�h Ps�r

Ph�s Ph�h Ph�r

Pr�s Pr�h Pr�r

2664

3775¼

ð1;0;0Þ ð0:5;0;5;0Þ ð0;1;0Þ

ð0:5;0:5;0Þ ð0:25;0:5;0:25Þ ð0;0:5;0:5Þ

ð0;1;0Þ ð0;0:5;0:5Þ ð0;0;1Þ

2664

3775:

ð2:7Þ

2.1.2. Two-locus case

The two-locus offspring genotype table can be obtained from

the one-locus offspring genotype table directly; fortunately we

do not need to construct the two-locus Punnett squares as this

would be very time-consuming. For example, the 4 � 4 Punnett

square shown in Table 2 (from [31] but in our notation) is for the

mating AajBb � AajBb (or hjh � hjh: note ‘‘xjy’’ means the genotype

x from the 1st locus and y from the 2nd locus).

Therefore the proportions of the offspring’s genotypes from this

mating are

ð2:8Þ

where Phjh�hjh is a 1 � 9 matrix or a row vector.

Note that if we assume classical dominant/recessive genes the

corresponding phenotype ratios are

A�B�: aaB�

: A�bb : aabb ¼ ðss þ sh þ hs þ hhÞ : ðrs þ rhÞ : ðsr þ hrÞ: rr ¼ 9 : 3 : 3 : 1;

where the ‘⁄’ indicates that the corresponding allele could be any of

A, B, a, or b. For example if ‘⁄ = b’ then ‘aaB⁄’ becomes ‘aaBb’ or ‘rh’

genotype.

To produce the offspring genotype table for the two-locus case,

we would need to make 81 such squares and calculate the propor-

tions. How time-consuming it would be!

Now we describe our efficient method to calculate the probabil-

ities of offspring’s genotypes. For the above mating AajBb � AajBb,

from the 1st locus cross Aa � Aa (or h � h), there are 1/4 = 25% of

genotype AA (s) and aa (r) and 50% Aa (h) in the offspring (see

the ‘‘Aa � Aa’’ row in Table 3). For each possible combination of

the 1st locus crosses, there are 1/4 = 0.25 of genotype BB (s) and

bb (r) and 2/4 = 0.5 of Bb (h) from the 2nd locus cross Bb � Bb (or

h � h). Hence the proportions of genotypes (in the order shown

in Eq. (2.8)) of offspring resulting from mating AAjBb �AajBb(sjh � hjh) can be obtained by

ð2:9Þ

Define XC as the column vector obtained by arranging the col-

umns of a matrix X one by one below each other in the original or-

der and simply denote the row vector (XC)T by XCT. Now we have

PTh�hPh�h

� �CT

¼ Phjh�hjh (see Eq. (2.8)), which is then placed in the

‘‘AajBb � AajBb (or hjh � jhjh)’’ row of the two-locus offspring geno-

type table (see Appendix B.2). Note that the 1 � 9 row vector

Phjh�hjh is equivalent to the ‘‘proportion lists’’ discussed in the

one-locus case. This definition comes from the fact that the order

of genotypes in Eq. (2.8) can be obtained by (YTY)C where

Y = (s,h,r) and the multiplication of letters is defined by combining

the two letters in the original order.

The whole offspring genotype table can thus be obtained by the

following steps:

(i) Find each of the following product of two sub-matrices:

X ijkl ¼ PTi�jPk�l; for i; j; k; l 2 fs;h; rg: ð2:10Þ

In computer code, the two loops for i and j are associated with the

mating in the 1st gene and the other two loops (k and l) with the

mating in the 2nd gene (see Appendix A.1).

(ii) The corresponding 1 � 9 vector XCTijkl or ‘‘proportion list’’

forms a row of the offspring genotype table.

2.1.3. Cases with more than two loci

Furthermore, this method can be extended recursively to form

offspring genotype tables for the N-locus case where NP 3. As

each locus has 3 possibilities: s, h and r, there are 3N different geno-

types for the parents and 3N � 3N = 32N mating combinations. That

is, the offspring table has 32N rows and 3N columns. Let

M1 ¼ M2 ¼ k; if N ¼ 2k

M1 ¼ k ÿ 1; M2 ¼ k; if N ¼ 2k ÿ 1for k ¼ 2;3; . . . ð2:11Þ

If

� I and J are any of the possible M1-locus genotypes, representing

the first M1 loci of the female and male genotype respectively,

� K and L are any of the possible M2-locus genotypes, representing

the last M2 loci of the female and male genotype respectively,

then the row of the offspring genotype table for the cross

between female genotype IjK and male genotype JjL can be

obtained from the matrix product PðM1ÞI�J

� �T

PðM2ÞK�L

� �CT

, where

Table 1

The Punnett square for the mating Aa � Aa.

Maternal

A a

Paternal A AA Aa

a Aa aa

Table 2

The 4 � 4 Punnett square for the mating AajBb � AajB.

AB Ab aB ab

AB AABB (ss) AABb (sh) AaBB (hs) AaBb (hh)

Ab AABb (sh) AAbb (sr) AaBb (hh) Aabb (hr)

aB AaBB (hs) AaBb (hh) aaBB (rs) aaBb (rh)

ab AaBb (hh) aaBb (rh) aaBb (rh) Aabb (rr)

M. Shi, M. Renton / Mathematical Biosciences 233 (2011) 77–89 79

19

PðM1ÞI�J for the cross I � J and P

ðM2ÞK�L for the cross K � L are row vec-

tors in the M1-locus and M2-locus offspring genotype table

respectively.

For example, the 11-locus offspring table would be derived by

multiplying elements of the 5 and 6-locus genotype tables.

2.2. Generalized inverse matrix for fitting the parameters of probit

models

2.2.1. Probit models

In statistics, the generalized linear model (GLM) in the form of

Y ¼ a þ b1x1 þ b2x2 þ � � � þ bkxk;

is a flexible generalization of ordinary least squares regression that

allows the linear model to be related to the response variable via

a link function for Y and the magnitude of the variance of each mea-

surement to be a function of its predicted value [14,18]. GLM

includes ordinary linear regression, Poisson regression, logistic

regression (with the canonical logit link) and probit regression.

The probit (=‘‘probability unit’’) link function is the inverse

cumulative distribution function (CDF) associated with the stan-

dard normal distribution [7,16]. Many problems of quantitative

inference in biological and technological research concern the rela-

tion between a stimulus (e.g. phosphine fumigation) and a re-

sponse (e.g. mortality of insects). Bliss [7] used all observations

of mortality response to each of a range of exposure times for each

of a range of fumigation concentrations, i.e. ‘‘all the information in

such a family of curves and not just that from a single point on each

component’’. Using this approach, a probit plane

Y ¼ a þ b1 logðtÞ þ b2 logðCÞ ð2:12Þ

may be fitted to the data, where t and C are respectively exposure

time and concentration, and Y is the probit mortality, which means

the probability of mortality P is related to Y by the following CDF

expression:

P ¼ 1ffiffiffiffiffiffiffi2p

pZ Yÿ5

ÿ1exp ÿ1

2u2

� �du: ð2:13Þ

In the case that the available independent data consist only of

the products Ct, rather than C and t separately, the parameters b1

and b2 can be merged into a single parameter, b:

Y ¼ a þ b logðCtÞ: ð2:14Þ

Whether common logarithms (base 10) or natural logarithms (ln or

loge,base e) are used in model (2.12) or (2.14) is immaterial, since

results obtained using either base are easily converted to the other

base: log10 x = (log10 e) lnx = 0.43429 lnx.

It is an implicit assumption in Eq. (2.12) that concentration and

time act independently. Alternatively, an extra term b3log(t) log(C)

can be added to describe the interaction of the dosage variables t

and C, which may be seen, for example, as a systematic change

in the slope of individual regressions of probit mortality on dosage

with change in exposure time:

Y ¼ a þ b1 logðtÞ þ b2 logðCÞ þ b3 logðtÞ logðCÞ: ð2:15Þ

Bell [3] applied a more conventional model to mortality data:

log t ¼ log k ÿ n log C or log C ¼ log a þ b log t;

ða ¼ k1=n

; b ¼ ÿ1=n; or k ¼ an; n ¼ ÿ1=bÞ:ð2:16Þ

This equation yields the familiar Haber-type model:

Cn ¼ k or C ¼ tb; ð2:17Þ

where C is the dosage that when applied for a time t achieves a par-

ticular specified response level (e.g. 50% or 99% mortality) and n, k,

a, and b are parameters that define the specific characteristics of the

response relationship.

Note that the integrand function in formula (2.13) is for the

standard normal distribution N(0,1). Probits may sometimes be

transformed by subtracting 5 from them i.e. Z = Y ÿ 5 where Z is

the normal equivalent deviate or N.E.D. [16].

It should be pointed out that in model (2.12) or (2.14) probit

mortality (Y), but not mortality percentage (P), is a linear function

of log time (log t) and log concentration (logC) or log(Ct), but not of

time and concentration themselves. Similarly, log t is a linear func-

tion of logk and n (or logC is a linear function of loga and b) in

model (2.16).

2.2.2. The generalized inverse matrix method

A number of approaches can be used to estimate the parameters

of mortality models such as those above. Maximum-likelihood esti-

mation remains popular and is the default method in many statis-

tical computing packages. Other approaches, including Bayesian

approaches and least squares have been developed [14,18]. Algebra-

ically, when any one of the above models (2.12), (2.14), (2.15) and

(2.16) is fitted to a data set, we have an over-determined system of

linear equations with respect to the parameters to be estimated. For

example, for the model (2.12), the N-equations with 3 variables

(a,b1,b2) corresponding to the data set fY i; ti;CigNi¼1 are as follows:

Y i ¼ 1 � a þ ðlogðtiÞ � b1 þ ðlog CiÞ � b2; ði ¼ 1;2; . . . ;NÞ: ð2:18Þ

The matrix form of the above equations is Ax = b where

x = (a,b1,b2)T,

A ¼

1 log t1 log C1

1 log t2 log C2

..

. ... ..

.

1 log tN log CN

266664

377775

and b ¼

Y1

Y2

..

.

YN

266664

377775: ð2:19Þ

Then the maximum-likelihood method maximizes their joint log-

likelihood function provided that the expected value E[ATA] exists

and is not singular [14,18].

The method of least squares is often used to generate estimators

and other statistics in regression analysis [33]. If a solution

minimizes

XN

i¼1

ðeY i ÿ Y iÞ2 ¼XN

i¼1

ð½a þ b1 logðtiÞ þ b2 logðCiÞ� ÿ Y iÞ2; ð2:20Þ

where eY i ¼ a þ b1 logðtiÞ þ b2 logðCiÞ is the ith predicted value, then

the solution is called a least squares solution [17]. Normally, the

least squares method can be used to solve the regularized equations

of Ax = b : ATAx = ATb, provided that ATA is non-singular. Actually, if

A+ is the generalized inverse (or Moore–Penrose pseudo-inverse) of

matrix A, then A+b is such a solution [4,23,26]. Note that if A is a

non-singular square matrix then A+ = Aÿ1. If A is column full-ranked,

then ATA is non-singular and A+ = (ATA)ÿ1AT. But while this equation

could theoretically be used to calculate A+, it is of limited practical

use for calculating A+ numerically, because using QR decomposition

or singular value decomposition (SVC) to obtain A+ will give much

smaller numerical errors than direct calculation of (ATA)ÿ1AT

[20,27].

2.3. Selection of initial frequencies and calculation of the equilibrium

frequencies of genotypes

2.3.1. Selection of initial frequencies of genotypes without special

constraints

Generating random initial frequencies of genotypes without

special conditions is simple. In general, if pi denotes the frequency

of genotype i then the following constraints apply

80 M. Shi, M. Renton / Mathematical Biosciences 233 (2011) 77–89

20

ðiÞ 0 6 pi 6 1 and ðiiÞXk

i¼1

pi ¼ 1 ðk

¼ total number of genotypes;i ¼ 1;2; . . . ; kÞ: ð2:21Þ

We can randomly generate k uniformly distributed numbers be-

tween zero and one to satisfy the 1st constraint, calculate the

sum of these k values, and then divide each value by this sum of

the k values, thus ensuring the 2nd condition is satisfied, while also

ensuring the 1st constraint is maintained.

2.3.2. Selection of initial frequencies of genotypes with some

preselected values

In some cases, we may want some initial frequencies to have

special values. For example, when we want to simulate the impact

of the initial proportion of resistant rr beetles (prr) on the evolution

of phosphine resistance, we may want to double or triple the value

of prr that was previously used, while maintaining the 2nd con-

straint. Now suppose m(<k) proportions: p1,p2, . . . ,pm have been

preselected and their sum Sm ¼ Pmi¼1pi < 1. Firstly, we randomly

generate m ÿ k uniformly distributed numbers x1,x2, . . .,xkÿm, and

calculate the sum of those values Skÿm ¼ Pkÿmi¼1 xi. Secondly, each

of xi, for i = 1, . . . ,k ÿ m is divided by S where S = Skÿm/(1 ÿ Sm).

Thus those k ÿ m values together with the preselected m values

will satisfy the second condition in constraints (2.21) since

Xkÿm

i¼1

xi=S þXm

i¼1

pi ¼Skÿm

Sþ Sm ¼ ð1 ÿ SmÞ þ Sm ¼ 1: ð2:22Þ

More generally, if constraints for the set of frequencies form a linear

system of equations and/or inequalities (note that the 1st condition

in formula (2.21) is a set of inequalities and the 2nd one is an equa-

tion), we can use any technique (e.g. [5,25,28]) for finding a feasible

solution of a linear programming problem to find a possible set of

frequencies.

2.3.3. Calculation of the Hardy–Weinberg equilibrium genotype

proportions for two-locus case using allelic proportion matrix

Now we describe how to calculate the Hardy–Weinberg equilib-

rium genotype proportions that should result from any particular

initial genotype proportions. For the one-locus case, let alleles A

and a be in proportions p1 and q1(=1 ÿ p1) respectively. Then, be-

cause half the alleles in genotype Aa or h are ‘‘A’’, p1 = ps + 0.5ph,

where px is the initial frequency of genotype, x 2 {s,h,r}. According

to the Hardy–Weinberg principle, with neutral selection pressure

over time the frequency of the three genotypes s, h and r within

the population will tend towards the equilibrium proportions

p21 : 2p1q1 : q2

1 [29]. Similarly, for a two-locus model, suppose al-

leles A and a on the 1st locus are in proportions p and q respec-

tively and alleles B and b on the 2nd locus are in proportions u

and v respectively. Let pxy be the initial frequency of genotype xy,

x, y 2 {s,h,r}. Then

p ¼ pss þ psh þ psr þ 0:5ðphs þ phh þ phrÞ; q ¼ 1 ÿ p;

u ¼ pss þ phs þ prs þ 0:5ðpsh þ phh þ prhÞ; v ¼ 1 ÿ u:ð2:23Þ

The equilibrium proportions for this two locus case, PE2, can be ob-

tained using the matrix product rather than element-wise calcula-

tion, by letting

A ¼s

h

r

p2

2pq

q2

264

375; B ¼ u2 2uv v

2� �s h r

ð2:24Þ

Then the nine Hardy–Weinberg equilibrium genotype proportions

can be obtained from the product AB:

P ¼ AB ¼s

h

r

p2u2 2p2uv p2v

2

2pqu2 4pquv 2pqv2

q2u2 2q2uv q2v

2

264

375

s h r

ð2:25Þ

2.4. Bisection method to estimate the intrinsic rate of natural increase

of an insect population

In any study of the biology of insect pests, one of the first ques-

tions is: How fast can the insect population multiply? The intrinsic

rate of natural increase, r, is defined as the rate of increase per

head under specified physical conditions, in an unlimited environ-

ment [1,6]. This rate plays a key role in fields as diverse as ecology,

genetics, demography and evolution.

Given the age-specific survival rates (lx) and the age-specific

fecundity rates (mx) at age x, an approximation of the value of r

may be calculated from the Lotka equation [15]:X

x

eÿrxlxmx ¼ 1: ð2:26Þ

In Birch’s approximation of the value of r [6], he neglected the contri-

bution of the older age groups (similarly in our Example 3.4 in Section

3.4, the summation of the expression (2.26) is not carried beyond the

age-group centered at x = 13.5). He then substituted a number of trial

values of r into the expression (2.26) using 4-figure tables for calcu-

lating eÿrx to find the value of r which would make the summation

approximately equal to 1. Carey [10] determined the r value by using

a procedure based on Newton’s method rn+1 = rn ÿ f(rn)/f0(rn). Maia et

al. [21] used a jackknife technique and Meyer et al. [22] compared

jackknife and bootstrap techniques for estimating r. Also the simplex

method has been used to obtain a numerical solution of r [30]. Here

we describe the use of an alternative and very simple iteration

approach, the bisection method, to find the value of r.

2.4.1. Bisection method

The bisection method can be found in most textbooks of numer-

ical analysis (e.g. [9]). Some authors [11,24,32,34] mentioned the

name ‘‘bisection’’ in the context of estimating the development

rate, but their papers did not describe any details, which biologists

may be interested in.

The bisection method is a root-finding algorithm, where ‘root’

means the value of x at which a function f(x) is equal to zero. The

method repeatedly bisects an interval then selects the subinterval

in which a root must lie for further iteration. If f(x) is a continuous

function on the interval [a,b], then the bisection method is very sim-

ple and guaranteed to converge to a root of f, x⁄, provided f(a) and f(b)

have different signs, i.e. f(a)f(b) < 0. The process of iteration is termi-

nated when the length of the iterated interval <2e and then the mid-

point of the interval is then chosen as the estimate of x⁄. Hence the

accuracy of the estimate of x⁄ is less than the desired accuracy e.

To find r using our bisection method, we first define a continu-

ous function with respect to r:

gðrÞ ¼X

x

eÿrxlxmx ÿ 1: ð2:27Þ

Then the problem is reduced to finding the root of g(r), i.e. find the

value of r which will make g(r) = 0, and the bisection method can be

applied in the normal way.

3. Results

3.1. The offspring genotype table

3.1.1. For the one- and two-locus cases

The Python function and main program for creating the off-

spring genotype tables for the one- and two-locus models are given

M. Shi, M. Renton / Mathematical Biosciences 233 (2011) 77–89 81

21

in Appendices A.1 and B.1 respectively. The table for the two-locus

model is given in Appendix B.2. Creating this large table with 729

entries takes less than 0.01 s using the functions in the appendices.

The resulting offspring genotype table for the one-locus model is

shown in Table 3.

3.1.2. An example for the three-locus case

For N = 3, in which case k = 2 (see formula (2.11)), the 3 � 9

matrix form of the offspring’s genotype proportions resulting from

the mating AajBBjCc � AAjbbjCc (or hjsjh � sjrjh) can be obtained by

calculating the product PTh�sPsjh�rjh where Ph�s = (0.5,0.5,0) from

Table 3. and Psjh�rjh = (0,0.25,0,0,0.5,0,0,0.25,0) from ‘‘ sjh � rjhor AABb � aaBb’’ row of Table B.2 (two-locus). Then convert it into

a 1 � 27 vector. The order of 27 genotypes for the three-locus case

can be obtained by (YTZ)CT where Y = (s,h,r) and Z is the 1 � 9

‘‘vector’’ shown in Eq. (2.8), which is

ðYT ZÞCT ¼ ðsss; hss; rss; shs;hhs; rhs; srs;hrs; rrs; ssh;hsh; rsh; shh; hhh;

rhh; srh; hrh; rrh; ssr;hsr; rsr; shr;hhr; rhr; srr;hrr; rrrÞ:ð3:1Þ

Hence the six non-zero proportions of PTh�sPsjh�rjh

� �CT

are:

shs;hhs; shh; hhh; shr;hhr

ð0:125; 0:125;0:25;0:25; 0:125;0:125Þ ð3:2Þ

This result can be confirmed by using a multiplication rule in a tree

diagram. For example, for hhs (and similarly for other genotypes):

3.2. Examples of using the generalized inverse matrix method to fit

parameters of probit models

3.2.1. MATLAB and Python codes

In MATLAB, commands Y = norminv (P) + 5 and P = normcdf

(Y ÿ 5) can be used to convert between the probit value Y (see Sec-

tion 2.2.1) and mortality percentage P. In Python, we can use sim-

ilar commands ‘‘Y = norm.ppf (P ) + 5 and P = norm.cdf (Y ÿ 5)’’

with ‘‘from scipy.stats import ⁄’’. The probit value Y can then

be converted to and from the N.E.D value Z by adding or subtract-

ing five.

In either MATLAB or Python, the command for finding the gen-

eralized inverse of matrix A is pinv (A).

3.2.2. Examples

Example 3.1 (Collins, 2010, unpublished data). The dose of phos-

phine (PH3) and response of strain QRD14 of the insect R. dominica

to phosphine fumigation are listed in the first three columns of

Table 4. Here t is a constant (48 h) and ‘‘dose’’ (mg/l) means

‘‘concentration’’.

For the purpose of analysis, the 15 observations should be di-

vided into 5 groups (so N = 5) each having 3 observations with

the same dose. The response rate (or mortality), listed in the last

column of Table 4, is the aggregated rate of the 3 observations;

for example, for the dose 0.0010 the aggregated rate is

(2 + 1 + 0)/(49 + 50 + 50) = 0.0201. Note that if the response rate

is p = 100% (e.g. to the dose 0.0040 in Table 4) then we change it

from 1 to 0.9999. Otherwise the corresponding probit value is infi-

nite, which cannot be used to fit the parameters. Similarly we

should change p = 0 (with examples in the other two data sets)

to something very close to zero, such as p = 0.0001, otherwise the

corresponding probit value will be negative infinity.

The probit model for this example is shown in formula (2.24)

and the coefficients matrix of the corresponding over-determined

linear system, A, has 5 rows and only 2 columns with the elements

in the 1st column being all ‘‘1’’ (see Eq. (2.19)). The fitted parame-

ters (obtained using A+) are a = 15.0324 and b = 9.2291 respec-

tively. Note that the fitted parameters using Maximum Likelihood

(ML) are a⁄ = 14.2743 and b⁄ = 8.5963 respectively.

Fig. 1 (a) and (b) shows the probit lines (probit values against log

(dose)) and mortality (%) (against dose) curves obtained using the

two methods. It can be seen that the two probit lines and mortality

curves for QRD14 are close to each other (similar for the other two

strains). But it can be seen from comparing the least squares (LS)

errors that our method has smaller numerical error in the sense

of formula (2.20): 0.2214 compared with 0.3850. For the data sets

of strain QRD569 and Comb F1, the numerical errors for our method

are 3.1589 and 0.3034 respectively, also smaller than the values of

4.7249 and 0.8035 (obtained using ML) respectively.

Example 3.2. Daglish [13] observed a range of concentrations in

combination with exposure times of 20, 48, 72 and 144 h, to

Table 3

The offspring genotype table for the one-locus model showing the expected

proportions of each possible offspring genotype resulting from each possible parental

genotype mating combination.

Female parent Male parent Offspring

AA (s) Aa (h) aa (r)

AA (s)� AA (s) 1.0 0 0

Aa (h) 0.5 0.5 0

aa (r) 0 1.0 0

Aa (h)� AA (s) 0.5 0.5 0

Aa (h) 0.25 0.5 0.25

aa (r) 0 0.5 0.5

aa (r)� AA (s) 0 1.0 0

Aa (h) 0 0.5 0.5

aa (r) 0 0 1.0

Table 4

Results of phosphine dose–response trials for the QRD14 strain of R. dominica

showing the phosphine dose used for 48 h exposure, the number of insects used, the

number of insects dying and the aggregate response mortality rate for each trial.

Observations Response rate

Dose (C) No. used No. response

0.0010 49 2 0.0201

0.0010 50 1

0.0010 50 0

0.0015 50 20 0.3200

0.0015 49 13

0.0015 51 15

0.0020 50 38 0.7047

0.0020 50 34

0.0020 49 33

0.0030 50 47 0.9733

0.0030 50 49

0.0030 50 50

0.0040 50 50 0.9999

0.0040 48 48

0.0040 50 50

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � ! � � " � � # � " $ � � ! # � % % & "� � � ' � � � � � � � � � ' � � � � � �� � � ' � � � � � � � � � ( � � � � � �

82 M. Shi, M. Renton / Mathematical Biosciences 233 (2011) 77–89

22

determine the time-concentration combinations required to

achieve mortality rates of 50% and 99% for strain QRD14 (Suscep-

tible-S), QRD369 (Resistant-R) and QRD369 � QRD14 (Hybrid-H).

These concentrations are denoted LC50 and LC99 (Lethal Concen-

tration value for 50% and 99% mortalities). Their observed data and

prediction equations (Haber-type model (2.17)) associated with

LC50 are listed in Table 5 (see Table 1 in [13]).

Note that the fitted value for the index n in model (2.17) ob-

tained by Daglish [13] was different for the LC50 data and the

LC99 data, which means it is not possible to develop a Haber-type

rule with which to successfully extrapolate predicted mortalities

between exposure scenarios [8].

For the purposes of predicting the mortalities at different con-

centrations and different expose times, we employ the probit mod-

el (2.15) to refit all the data of Daglish [13], including both the LC50

and LC99 data sets together, to fit the four parameters for each of

the three strains (C: mg/l, t: h). Note that the four t values should

be repeated twice in the 2nd column of coefficient matrix A (see

Eq. (2.19)). The fitted equations are as follows (the logarithmic

base is 10):

QRD369 : Y ¼ÿ10:8398þ16:1356logðtÞþ1:9145logðCRÞþ4:0846logðtÞ logðCRÞ;QRD14 : Y ¼3:9749þ12:3267logðtÞþ3:8700logðCSÞþ1:9247logðtÞ logðCSÞ;369�14 : Y ¼11:2847þ3:7764logðtÞþ6:9650logðCHÞÿ1:0105logðtÞ logðCHÞ:

ð3:3Þ

3.3. Calculation of equilibrium proportions

The Python function and main program for selection of uni-

formly distributed random numbers that the sum = 1 with some

or without any preselected numbers are given in Appendices A.2

and C.1 respectively. The Python function and main program for

calculation of equilibrium frequencies of genotypes are given in

Appendices A.3 and C.2 respectively.

Example 3.3. If the initial frequencies of genotypes PI2 for the two-

locus case are

PI2 ¼ss hs rs sh hh rh sr hr rr

0:2040; 0:1203; 0:0875; 0:1064; 0:0690; 0:0894; 0:0467; 0:1197; 0:1570

ð3:4Þ

then p = 0.5116, q = 0.4884, u = 0.5442, v = 0.4558, according to for-

mula (2.23), and

PE2 ¼ss hs rs sh hh rh sr hr rr

0:0775; 0:1480; 0:0706; 0:1298; 0:2479; 0:1183; 0:0544; 0:1038; 0:0496

ð3:5Þ

according to formula (2.25), after converting the matrix form to a

row vector form (see Eqs. (2.9) and (2.8)).

Note that if we choose the above proportions PE2 in Eq. (3.5) as

the initial ones then the equilibrium proportions are PE2 them-

selves. However many other different sets of initial frequencies

could also result in the same set of equilibrium proportions; PE2

is a special solution.

3.4. Bisection method for finding the development rate r

Example 3.4. The pivotal age in weeks (x), age-specific survival

rates (lx) and the age-specific fecundity rates (mx) are shown in

Table 6 (Table 2 in [6]).

The results for this example obtained using our bisection algo-

rithm for three tests (different initial intervals) are listed in Table 7.

The iteration process for Test 3 is shown in Fig. 2. Given the initial

interval [0.6,1.0] the iteration steps are as follows: the interval

Fig. 1. The probit lines (a) and percentage mortality (b) curves obtained using the least squares (LS) and maximum likelihood (ML) methods with the observed values for the

QRD14 strain.

Table 5

Concentration (LC50 & LC99) values (mg/l) required to achieve 50% and 99% mortality for different exposure times (t) for three strains of R. dominica, together with the model fitted

by Daglish [13] for the LC50 data.

Strain Mortality (%) 20 h 48 h 72 h 144 h Model (2.17)

QRD14 50 0.0052 0.0017 0.0011 0.00064 C0.8673t = 0.2088

99 0.0091 0.0037 0.0021 0.0014

QRD369 50 0.20 0.052 0.032 0.017 C0.8673t = 4.0908

99 0.40 0.091 0.060 0.028

QRD369� 50 0.010 0.0042 0.0023 0.0011 C0.8673 t = 0.3863

QRD14 99 0.026 0.013 0.0066 0.0025

M. Shi, M. Renton / Mathematical Biosciences 233 (2011) 77–89 83

23

resulting from the previous iteration is bisected, the function value

at the two ends of the interval is calculated, and then the half sub-

interval for which the function values at the two ends have differ-

ent signs is kept as the new interval. This process continues until

the desired accuracy is achieved.

4. Discussion

Our quantification and block-matrix multiplication approach to

generate the offspring genotype tables involves many fewer opera-

tions than classical methods such as that based on Punnett squares

(PS). For each mating in any N-locus cases, the PS method requires

three processes: constructing the PS, counting the number and

calculating the proportions of each genotype of progeny. In the

one-locus case, our method requires multiplying two matrices with

the ‘‘quantified’’ elements and the same counting and calculating

processes. It can be seen from Table 1 and (2.2) that both have the

same number of operations (four) if we regard defining the genotype

of one cell in Table 1 as one operation (although this operation is

more complex than multiplication of two numbers). For each mating

in the N-locus case (NP 2), constructing such a PS requires

2N � 2N = 22N operations as there are N loci each having 2 alleles

for both the female or male parent. In addition, there are 22N counts

and 22N divisions for calculating the proportions. If our method is

used, the only need is to find the product of two matrices which re-

quires 2N multiplications (If N = 2k, then 2k � 2k = 22k = 2N. If

N = 2k ÿ 1, then 2k � 2kÿ1 = 22kÿ1 = 2N). This is because our method

calculates the N-locus case recursively using the results obtained

for k or k ÿ 1 loci (N = 2k ÿ 1 or 2k). It could be argued that in some

matings, e.g. AAjBB � Aajbb for the two-locus case, the parents have

only 2 different genotypes AABb and AaBb and so the PS has only 4

cells. However, the other cells appearing in Table 1 correspond to

zero elements in this case and our algorithm would not perform

any multiplications by zeros in the computer codes.

Theoretically the two methods (maximum likelihood and least

squares) to fit the probit models should obtain the same parame-

ters. Numerically, however, they yield small differences in results.

The generalized inverse matrix approach described here provides

an efficient method for fitting probit models. The advantages of

using this approach are

(1) It provides a more numerically accurate estimate of

parameters.

(2) Even if A is not (column) full-ranked and thus the coefficient

matrix of the regularized equations, (ATA), is singular, there

still exits a matrix A+ where the linear system has a solution

A+b with minimum L2 norm:

kAþbk2 ¼ minfkxk2 : AT Ax ¼ ATbg:

The generalized inverse matrix with the least squares technique

can also be used to fit the parameters of a model when the model

or modified model is a linear function with respect to parameters

or metameter of parameters. For example, it can be used to fit

the parameters a and b in Freundlich adsorption isotherm models

[2] defined by the equation Y = aebt, where t is time of exposure (h)

and Y is the ratio (C/Co) of concentration C at time t to the applied

concentration Co at time t = 0. This follows since the equivalent

log–log model is ln(Y) = ln(a) + bt where ln(Y) is a linear function

of parameters ln(a) and b.

Our algorithm for randomly generating initial genotype frequen-

cies based on matrix products is very simple and efficient. For find-

ing the Hardy–Weinberg equilibrium genotype proportions, using

matrix products requires 23 multiplications; forming the elements

of matrix A or B in Eq. (2.24) requires 7 multiplications for each

and calculating the product of AB in Eq. (2.25) requires 9 multiplica-

tions. On the other hand, element-wise calculation for the nine ele-

ments of matrix P in Eq. (2.25) requires 41 multiplications;

calculating the elements in the 4 corners requires 16 multiplications

(4 for each element) and in the other 5 positions requires 25 multi-

plications (5 for each). The matrix product method we propose is

thus efficient and avoids repeated calculations.

The advantage of the bisection method we propose for deter-

mining the intrinsic rate of increase is that it is also very

simple and normally only a few iterations are needed to find the

development rate as the desired accuracy is normally to two deci-

mal places.

As stated previously, accurately and efficiently determining

parameter values of key sub-models within biological simulation

Table 6

Raw data: age x (weeks), age-specific survival rate lx per week and age-specific fecundity rate mx per week.

x 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5

lx 0.87 0.83 0.81 0.8 0.79 0.77 0.74 0.66 0.59 0.52 0.45 0.36 0.29 0.25 0.19

mx 20.0 23.0 15.0 12.5 12.5 14.0 12.5 14.5 11.0 9.5 2.5 2.5 2.5 4.0 1.0

Table 7

The estimate of rate r and accuracy for three initial intervals.

Test Initial interval [a,b] Iteration number Estimate of r g(r) Accuracy

1 [0,1] 6 0.76 0.02 <0.01

2 [0,1] 13 0.7620 ÿ0.0002 <0.0001

3 [0.6,1.0] 5 0.76 0.03 <0.01

Fig. 2. Illustration of the iteration process for Test 3, showing the initial interval

[0.6,1.0], the function g(r) curve, together with the function values at the left and

right endpoints and the iterated intervals after each of the 5 iterations.

84 M. Shi, M. Renton / Mathematical Biosciences 233 (2011) 77–89

24

models, such as models simulating population dynamics and evo-

lution of resistance in stored-grain insect pests, is a critical issue

[12]. We conclude that the methods presented in this paper pro-

vide a toolkit for estimating a number of important parameter val-

ues for such resistance simulation models, which will allow these

models to be used to predict the effects of different possible man-

agement strategies.

Acknowledgements

The authors would like to acknowledge the support of the Aus-

tralian Government’s Cooperative Research Centres Program. We

also thank P.J. Collins for his great help in the genetics and provi-

sion of raw data.

Appendix A. Three Python functions

A.1. Python function for creation of offspring genotype table

from numpy import ⁄ # If the three functions are separated the left

from pylab import ⁄ # five commands should be put in the beginning of

from random import ⁄ # the file for each of other two functions

from math import ⁄

from scipy import ⁄

def GenTable (nL):

‘‘‘‘‘‘Create offspring genotypes table for one- or two-locus cases by set nL = 1 or nL = 2’’’’’’

# nL: number of locus, nL = 1 or 2

# 1? A, 2? a:

F = matrix ([1,1,1,2,2,2]) # genotypes of Female parent

M = F # genotypes of Male parent

# 1? AA, 2? Aa, 4? aa in the product FM

FM = F.T � M # F.T: Transpose of F

Table1 = matrix (zeros ((9,3),float)) # The table for one-locus

for i in range (3): # Count the numbers and

for j in range (3): # calculate the proportions

N1 = 0. # for calculation of numbers 1, 2, or 4

N2 = 0. # in each sub-matrix

N4 = 0.

X=(FM[2 � i,2 � j],FM[2 � i + 1,2 � j],FM[2 � i,2 � j + 1],FM[2 � i + 1,2 � j + 1])

YS = choose (greater(X,1),(X,0))

k = 3 � i + j

Table1[k,0] = sum (YS)/4.0

YHR = choose (equal (X,1),(X,0))

YH = choose (greater (YHR,2),(YHR,0))

Table1[k,1] = sum (YH)/8.0

YR = choose (equal (YHR,2),(YHR,0))

Table1[k,2]=sum (YR)/16.0

if nL==1:

return Table1

if nL==2:

n = 0

Table2 = zeros ((9 � 9,9),float)for f2 in range (3): # 2nd gene of Female

for f1 in range (3): # 1st gene of Female

for m2 in range (3): # 2nd gene of Male

for m1 in range (3): # 1st gene of Male

k1 = 3 � f1 + m1 # The index in one-locus table

# for 1st cross

k2 = 3 � f2 + m2 # The index in one-locus table

# for 2nd cross

C1 = Table1[k1,:]

C2 = Table1[k2,:]

C1C2 = (C1).T � C2FM2 = ((C1C2[:,0]).T,(C1C2[:,1]).T,(C1C2[:,2]).T)

Table2[n,] = reshape (FM2,(1,9))

n = n + 1

return Table2

M. Shi, M. Renton / Mathematical Biosciences 233 (2011) 77–89 85

25

A.2. Python function for selection of uniformly distributed random proportions

def UniformRandom (Str,Pm,IDXm):

‘‘‘‘‘‘Creating K random (uniformly distributed) numbers with sum = 1.0 with or without m preselected ones for

creating initial proportions of genotypes or life stages or others’’’’’’

# Str: A vector of strings indicating the random variable

# Pm: A 1xm row vector of preselected uniformed distributed

# numbers with sum Sm < 1.0. If m = 0, input Pm as []

# INXm: Indices of Pm in the returned array Pk.

# If m = 0, input IDXm as []

K = len (Str) # Number of the random digits

m = len (Pm) # Number of preselected uniformed distributed digits

print ‘n n m=’, m, ‘IDXm:’, IDXm, ‘n n Pm:’,Pm

Pk = zeros (K,float)

x = zeros (K-m,float)

Sm = sum (Pm) # Sum of the preselected random numbers

if Sm > 1.0:

print Sm,’ !!! The sum of preselected numbers > 1.0’

Pk=[]

return Pk

from random import ⁄

for i in range (K ÿ m):

x[i]=random ()

Sk_m = sum (x)

print ‘nn x:’, x, ‘nn sum=’, Sk_m

S = Sk_m/(1.-Sm)

Pk_m = x/S

print ‘nn Pk_m:’, Pk_m

if m==0:

Pk = Pk_m

else:

Pk[IDXm]=Pm

IDXk_m = delete (range (K),IDXm)

Pk[IDXk_m]=Pk_m

return Pk

A.3. Python function for calculation of equilibrium frequencies of genotypes

def EqiFre_pquv (nL,IniF):

‘‘‘‘‘‘Given initial frequency of genotypes calculate allelic proportions p& q/u& v and the frequencies in

equilibrium for 1- or 2-locus model"""

# nL: number of locus

# IniF: 1 � 3 or 1 � 9 list - initial frequency of genotypes

if nL==2:

x1 = take (IniF,[0,3,6])

x2 = take (IniF,[1,4,7])

x3 = take (IniF,[0,1,2])

x4 = take (IniF,[3,4,5])

p = sum (x1) + sum (x2)/2. # proportion of allele ‘A’

u = sum (x3) + sum (x4)/2. # proportion of allele ‘B’

q = 1.-p

v = 1.-u

A = matrix ([[p ⁄⁄ 2],[2.0 � p � q],[q ⁄⁄ 2]])

B = matrix ([u ⁄⁄ 2,2.0 � u � v,v ⁄⁄ 2])

P=(A � B).TEP2 = reshape (P.flat,(1,9))[0]

return (p,q,u,v,EP2)

if nL==1:

p = IniF[0] + 0.5 � IniF[1]q = IniF[2] + 0.5 � IniF[1]EP1=[p ⁄⁄ 2,2. � p � q,q ⁄⁄ 2]

return (p,q,EP1)

86 M. Shi, M. Renton / Mathematical Biosciences 233 (2011) 77–89

26

Appendix B. Offspring genotype table

B.1. Python code for creating offspring genotype table for 1- and 2-locus models

def PrintTable (GenTypes,Table):

(row,col) = shape (Table)

for i in range (row):

k = i/col

j = i ÿ k � colif i==k � col:print GenTypes[k],‘X’,GenTypes[j],Table[i]

else:

print ’ ’,GenTypes[j],Table[i]

return

print ‘nnnn### Offspring Genotypes Table for one-locus ###nnnn’T1 = GenTable (1)

GenType1=[‘AA (s)’,‘Aa (h)’,‘aa (r)’]

PrintTable (GenType1,T1)

T2 = GenTable (2)

GenotypesA=[‘AABB’, ‘AaBB’, ‘aaBB’, ‘AABb’, ‘AaBb’, ‘aaBb’,‘AAbb’, ‘Aabb’, ‘aabb’]

Genotypes=[‘ss[0]’,‘hs[1]’,‘rs[2]’,‘sh[3]’,‘hh[4],’,‘rh[5]’,‘sr[6]’,‘hr[7]’,‘rr[8]’]

print ‘nnnn### Offspring Genotypes Table for two-locus ###nnnn’print GenotypesA,‘nn’,Genotypes,‘nn’PrintTable (GenotypesA,T2)

B.2. Offspring genotype table (two loci)

Female parent Male parent AABB AaBB aaBB AABb AaBb aaBb AAbb Aabb aabb

AABB� AABB 1. 0. 0. 0. 0. 0. 0. 0. 0.

AaBB 0.5 0.5 0. 0. 0. 0. 0. 0. 0.

aaBB 0. 1. 0. 0. 0. 0. 0. 0. 0.

AABb 0.5 0. 0. 0.5 0. 0. 0. 0. 0.

AaBb 0.25 0.25 0. 0.25 0.25 0. 0. 0. 0.

aaBb 0. 0.5 0. 0. 0.5 0. 0. 0. 0.

AAbb 0. 0. 0. 1. 0. 0. 0. 0. 0.

Aabb 0. 0. 0. 0.5 0.5 0. 0. 0. 0.

aabb 0. 0. 0. 0. 1. 0. 0. 0. 0.

AaBB� AABB 0.5 0.5 0. 0. 0. 0. 0. 0. 0.

AaBB 0.25 0.5 0.25 0. 0. 0. 0. 0. 0.

aaBB 0. 0.5 0.5 0. 0. 0. 0. 0. 0.

AABb 0.25 0.25 0. 0.25 0.25 0. 0. 0. 0.

AaBb 0.125 0.25 0.125 0.125 0.25 0.125 0. 0. 0.

aaBb 0. 0.25 0.25 0. 0.25 0.25 0. 0. 0.

AAbb 0. 0. 0. 0.5 0.5 0. 0. 0. 0.

Aabb 0. 0. 0. 0.25 0.5 0.25 0. 0. 0.

aabb 0. 0. 0. 0. 0.5 0.5 0. 0. 0.

aaBB� AABB 0. 1. 0. 0. 0. 0. 0. 0. 0.

AaBB 0. 0.5 0.5 0. 0. 0. 0. 0. 0.

aaBB 0. 0. 1. 0. 0. 0. 0. 0. 0.

AABb 0. 0.5 0. 0. 0.5 0. 0. 0. 0.

AaBb 0. 0.25 0.25 0. 0.25 0.25 0. 0. 0.

aaBb 0. 0. 0.5 0. 0. 0.5 0. 0. 0.

AAbb 0. 0. 0. 0. 1. 0. 0. 0. 0.

Aabb 0. 0. 0. 0. 0.5 0.5 0. 0. 0.

aabb 0. 0. 0. 0. 0. 1. 0. 0. 0.

AABb� AABB 0.5 0. 0. 0.5 0. 0. 0. 0. 0.

AaBB 0.25 0.25 0. 0.25 0.25 0. 0. 0. 0.

aaBB 0. 0.5 0. 0. 0.5 0. 0. 0. 0.

AABb 0.25 0. 0. 0.5 0. 0. 0.25 0. 0.

(continued on next page)

M. Shi, M. Renton / Mathematical Biosciences 233 (2011) 77–89 87

27

Appendix B (continued)

Female parent Male parent AABB AaBB aaBB AABb AaBb aaBb AAbb Aabb aabb

AaBb 0.125 0.125 0. 0.25 0.25 0. 0.125 0.125 0.

aaBb 0. 0.25 0. 0. 0.5 0. 0. 0.25 0.

AAbb 0. 0. 0. 0.5 0. 0. 0.5 0. 0.

Aabb 0. 0. 0. 0.25 0.25 0. 0.25 0.25 0.

aabb 0. 0. 0. 0. 0.5 0. 0. 0.5 0.

AaBb� AABB 0.25 0.25 0. 0.25 0.25 0. 0. 0. 0.

AaBB 0.125 0.25 0.125 0.125 0.25 0.125 0. 0. 0.

aaBB 0. 0.25 0.25 0. 0.25 0.25 0. 0. 0.

AABb 0.125 0.125 0. 0.25 0.25 0. 0.125 0.125 0.

AaBb 0.0625 0.125 0.0625 0.125 0.25 0.125 0.0625 0.125 0.0625

aaBb 0. 0.125 0.125 0. 0.25 0.25 0. 0.125 0.125

AAbb 0. 0. 0. 0.25 0.25 0. 0.25 0.25 0.

Aabb 0. 0. 0. 0.125 0.25 0.125 0.125 0.25 0.125

aabb 0. 0. 0. 0. 0.25 0.25 0. 0.25 0.25

aaBb� AABB 0. 0.5 0. 0. 0.5 0. 0. 0. 0.

AaBB 0. 0.25 0.25 0. 0.25 0.25 0. 0. 0.

aaBB 0. 0. 0.5 0. 0. 0.5 0. 0. 0.

AABb 0. 0.25 0. 0. 0.5 0. 0. 0.25 0.

AaBb 0. 0.125 0.125 0. 0.25 0.25 0. 0.125 0.125

aaBb 0. 0. 0.25 0. 0. 0.5 0. 0. 0.25

AAbb 0. 0. 0. 0. 0.5 0. 0. 0.5 0.

Aabb 0. 0. 0. 0. 0.25 0.25 0. 0.25 0.25

aabb 0. 0. 0. 0. 0. 0.5 0. 0. 0.5

AAbb� AABB 0. 0. 0. 1. 0. 0. 0. 0. 0.

AaBB 0. 0. 0. 0.5 0.5 0. 0. 0. 0.

aaBB 0. 0. 0. 0. 1. 0. 0. 0. 0.

AABb 0. 0. 0. 0.5 0. 0. 0.5 0. 0.

AaBb 0. 0. 0. 0.25 0.25 0. 0.25 0.25 0.

aaBb 0. 0. 0. 0. 0.5 0. 0. 0.5 0.

AAbb 0. 0. 0. 0. 0. 0. 1. 0. 0.

Aabb 0. 0. 0. 0. 0. 0. 0.5 0.5 0.

aabb 0. 0. 0. 0. 0. 0. 0. 1. 0.

Aabb� AABB 0. 0. 0. 0.5 0.5 0. 0. 0. 0.

AaBB 0. 0. 0. 0.25 0.5 0.25 0. 0. 0.

aaBB 0. 0. 0. 0. 0.5 0.5 0. 0. 0.

AABb 0. 0. 0. 0.25 0.25 0. 0.25 0.25 0.

AaBb 0. 0. 0. 0.125 0.25 0.125 0.125 0.25 0.125

aaBb 0. 0. 0. 0. 0.25 0.25 0. 0.25 0.25

AAbb 0. 0. 0. 0. 0. 0. 0.5 0.5 0.

Aabb 0. 0. 0. 0. 0. 0. 0.25 0.5 0.25

aabb 0. 0. 0. 0. 0. 0. 0. 0.5 0.5

aabb� AABB 0. 0. 0. 0. 1. 0. 0. 0. 0.

AaBB 0. 0. 0. 0. 0.5 0.5 0. 0. 0.

aaBB 0. 0. 0. 0. 0. 1. 0. 0. 0.

AABb 0. 0. 0. 0. 0.5 0. 0. 0.5 0.

AaBb 0. 0. 0. 0. 0.25 0.25 0. 0.25 0.25

aaBb 0. 0. 0. 0. 0. 0.5 0. 0. 0.5

AAbb 0. 0. 0. 0. 0. 0. 0. 1. 0.

Aabb 0. 0. 0. 0. 0. 0. 0. 0.5 0.5

aabb 0. 0. 0. 0. 0. 0. 0. 0. 1.

88 M. Shi, M. Renton / Mathematical Biosciences 233 (2011) 77–89

28

References

[1] H.G. Andrewartha, L.G. Birch, Selections from the Distribution and Abundanceof Animals, The University of Chicago, 1982.

[2] H.J. Banks, Behaviour of gases in grain storages, in: Fumigation and ControlledAtmosphere Storage of Grain, Proceedings of an Iinternational Conference Heldat Singapore, 1989, pp. 96–107.

[3] C.H. Bell, The efficiency of phosphine against diapausing larvae of Ephestiuelutellu (Lepidoptera) over a wide range of concentrations and exposure times,J. Stored Prod. Res. 15 (1979) 53.

[4] A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications,second ed., Springer, New York, NY, 2003.

[5] M.J. Best, K. Ritter, Linear Programming: Active Set Analysis and ComputerPrograms, Prentice-Hall, Englewood Cliffs, New Jersey, 1985.

[6] C. Birch, The intrinsic rate of increase of an insect population, J. Animal Ecol. 17(1948) 15.

[7] C.I. Bliss, The relation between exposure time, concentration and toxicity inexperiments on insecticides, Ann. Em. Sot. Am. 33 (1940) 721.

[8] N.J. Bunce, R.B. Remillard, Haber’s rule: the search for quantitativerelationships in toxicology, Human Ecol. Risk Assess. 9 (4) (2003) 973.

[9] R.L. Burden, J.D. Faires, Numerical Analysis, eighth ed., Thomson, Belmont, CA.,2005.

[10] J.R. Carey, Applied Demography for Biologists, Oxford University Press, 1993.[11] H. Chi, H. Su, Age-stage, two-sex life tables of Aphidius gifuensis (Ashmead)

(Hymenoptera: Braconidae) and its host myzus persicae (Sulzer) (Homoptera:Aphididae) with mathematical proof of the relationship between femalefecundity and the net reproductive rate, Environ. Entomol. 35 (1) (2006) 10.

[12] P.J. Collins, G.J. Daglish, M. Bengston, T.M. Lambkin, H. Pavic, Genetics ofresistance to phosphine in Rhyzopertha dominica (Coleoptera: Bostrichidae), J.Econ. Entomol. 95 (2002) 862.

[13] GJ. Daglish, Effect of exposure period on degree of dominance of phosphineresistance in adults of Rhyzopertha dominica (Coleoptera: Bostrychidae) andSitophilus oryzae (Coleoptera: Curculionidae), Pest Manage. Sci. 60 (8) (2004)822.

[14] A.J. Dobson, A.G. Barnett, Introduction to Generalized Linear Models, third ed.,Chapman and Hall/CRC, Boca Raton, FL, 2008.

[15] L.I. Dublin, A.J. Lotka, On the true rate of natural increase, J. Am. Stat. Assoc. 20(1925) 305.

[16] D.J. Finney, Probit Analysis, third ed., Cambridge University, 1971.[17] S. Gilbert, Introduction to Linear Algebra, third ed., Wellesley-Cambridge Press,

Wellesley, Massachusetts, 2003.

[18] J. Hardin, J. Hilbe, Generalized Linear Models and Extensions, second ed., StataPress, College Station, 2007.

[19] P.W. Hedrick, Genetics of Population, third ed., Jones and Bartlett Publishers,2005.

[20] S. Josef, B. Roland, Introduction to Numerical Analysis, third ed., Springer-Verlag, Berlin, New York, 2002.

[21] A De H.N. Maia, A.J.B. Luiz, C. Ampanhola, Statistical inference on associatedfertility life table parameters using jackknife technique: computationalaspects, J. Econ. Entomol. 93 (2) (2000) 511.

[22] J.S. Meyer, C.G. Ingersoll, L.L. McDonald, S. Marks Boyce, Estimatinguncertainty in population growth rates: jackknife vs. bootstrap techniques,Ecology 67 (5) (1986) 1156.

[23] E.H. Moore, On the reciprocal of the general algebraic matrix, Bull. Am. Math.Soc. 26 (1920) 394.

[24] L.D. Mueller, M.R. Rose, Evolution Evolutionary theory predicts late-lifemortality plateaus, Proc. Natl. Acad. Sci. USA 93 (1996) 15249.

[25] M.R. Osborne, Finite Algorithms in Optimization and Data Analysis, Wiley,Chichester, 1985.

[26] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51(1955) 406.

[27] M. Shi, L. Gao, Z. Chen, Z. Yang, Matrix Computation in Engineering: Theories,Algorithms and FORTRAN Programs, BUT Publishing House, 1990.

[28] M. Shi, M.A. Lukas, An L1 estimation algorithm with degeneracy and linearconstraints, Comp. Statist. Data Anal. 39 (2002) 35.

[29] W.D. Stansfield, Theory and Problems of Genetics, second ed., McGraw-HillBook Company, Sturtevant, 1983.

[30] A. Taberner, P. Castaera, E. Silvestre, J. Dopazo, Estimation of the intrinsic rateof natural increase and its error by both algebraic and resampling approaches,Bioinformatics 9 (5) (1993) 535.

[31] R.H. Tamarin, Principles of Genetics, fourth ed., Boston University, Wm. C.Brown Publisher, 1989.

[32] M.F.J. Taylor, Field measurement of the dependence of life history on plantnitrogen and temperature for a herbivorous moth, J. Animal Ecol. 57 (1988)873.

[33] J. Wolberg, Data Analysis Using the Method of Least Squares: Extracting theMost Information from Experiments, Springer, 2005.

[34] T. Yang, H. Chi, Life tables and development of Bemisia argentifolii(Homoptera: Aleyrodidae) at different temperatures, J. Econ. Entomol. 99 (3)(2006) 691.

Appendix C. Python code for selection of uniformly distributed random proportions and calculation of equilibrium frequencies of

genotypes

C.1. Python code for selection of uniformly distributed random proportions

# IniFrq = UniformRandom (Genotypes,[],[]) # without preselected numbers

#Pm = [0.1248,0.0,0.9000] # Test if the sum > 1.0

Pm = [0.1248,0.0,0.4352]

IDXm = [0,2,7]

IniFrq = UniformRandom (Genotypes,Pm,IDXm)

print ‘nn### Initial frequencies for two-locus ###nnnn’print Genotypes,‘nn’,array (IniFrq)

print ‘Check sum = 1?’,sum (IniFrq)

C.2. Python code for calculation of equilibrium frequencies of genotypes

IniF = [0.2040,0.1203,0.0875,0.1064,0.0690,0.0894,0.0467,0.1197,0.1570] (p,q,u,v,EP2)=EqiFre_pquv (2,IniF)

print ‘nnnn### The allelic proportions p,q,u,v:nn’print ‘p=’,p,‘q=’,q,‘u=’,u,‘v=’,v

print ‘nn### The equilibrium frequencies of genotypes:nn’,Genotypesprint ‘nn’,EP2,‘nn Check sum = 1?’,sum (EP2)

M. Shi, M. Renton / Mathematical Biosciences 233 (2011) 77–89 89

29

30

Modelling mortality of a stored grain insect pest with fumigation: Probit, logistic

or Cauchy model?

Mingren Shi a,b,⇑, Michael Renton a,b,c

a School of Plant Biology, University of Western Australia, 35, Stirling Highway, Crawley, WA 6009, Australia b Cooperative Research Centre for National Plant Biosecurity, Australia c CSIRO Ecosystem Sciences, Underwood Avenue, Floreat, WA 6014, Australia

a r t i c l e i n f o

Article history:

Received 16 June 2012

Received in revised form 27 January 2013

Accepted 6 February 2013

Available online 5 March 2013

Keywords:

Phosphine mortality estimation

Probit model

Logistic model

Cauchy model

Lesser grain borer

Pest control simulation

a b s t r a c t

Computer simulation models can provide a relatively fast, safe and inexpensive means to judge and

weigh the merits of various pest control management options. However, the usefuln ess of such simula-

tion models reli es on the accurate estimati on of important model parameters, such as the pest mortality

under different treatments and conditions. Recently, an individual-based simulation model of populatio n

dynamics and resistance evolution has been developed for the stored grain insect pest Rhyzopertha

dominica, based on experimental results showing that alleles at two different loci are involved in resis-

tance to the grain fumigant phosphine. In this paper, we describe how we used three generalized linear

models, probit, logistic and Cauchy models, each employing two- and four-parameter sub-models, to fit

experimental data sets for five genotype s for which detailed mortality data was already available. Instead

of the usual statistical iterative maximum likelihood estima tion , a direct algebraic approach, generalized

inverse matrix techniq ue , was used to estimate the mortality model parameters. As this technique needs

to perturb the observed mortality proportions if the proportions include 0 or 1, a golden section search

approach was used to find the optimal perturbation in terms of minimum least squares (L2) error. The

results show that the estimates using the probit model were the most accurate in terms of L2 errors

between observed and predicted mortality values. These errors with the probit model ranged from

0.049% to 5.3%, from 0.381% to 8.1% with the logistic model and from 8.3% to 48.2% with the Cauchy

model. Meanwhile, the generalized inverse matrix technique achieved similar results to the maximum

likelihood estimation ones, but is less time consuming and computationally demanding. We also describe

how we constructed a two-parameter model to estimate the mortalities for each of the remaining four

genotypes based on realistic genetic assumptions.

Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction

The lesser grain borer, Rhyzopertha dominica , is a very destruc-

tive primary pest of stored grains. Fumigation with phosphine

(PH3) is a key component in the managemen t of controlling pest

infestations worldwide. However heavy reliance on PH 3 has re-

sulted in the development of strong resistance in several major

pest species including R. dominica. Computer simulatio n models

can provide a relatively fast, safe and inexpensive means to judge

and weigh the merits of various management options for control-

ling populations and avoiding or delaying resistance evolution in

pests such as R. dominica. But the usefulness of simulation models

such as these relies on the accurate estimation of key model

paramete rs, which should be based on reliable experimental data

as much as possible.

In previously published simulation modelling research on this

important topic of phosphine resistance in stored grain insect

pests, accurate survivorship of different genotypes was not explic-

itly included in the model because adequate data were not avail-

able [1], and thus a simple single gene model was used.

However , recent fumigant response analyses of PH 3 resistance in

R. dominica in Australia have indicated the existence of two resis-

tance phenotypes , which are labelled Weak and Strong Resistance

[2–4]. The genetic linkage analysis undertak en by Schlipalius et al.

[5,6] revealed that two loci confer strong resistance, thus motivat-

ing us to construct a more detailed and realistic two-locus individ-

ual-based simulation model of resistance [7–10]. In our two-locus

model, for simplicit y we assume that there are two possible alleles

(resistance or susceptibility) at each of the two loci, meaning nine

genotypes in total. As phosphine concentr ation and time of

0025-5564/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.mbs.2013.02.005

⇑ Corresponding author at: School of Plant Biology, University of Western

Australia, 35, Stirling Highway, Crawley, WA 6009, Australia. Tel.: +61 8 6488

1992; fax: +61 8 6488 1108.

E-mail addresses: shi.mingren@gmail.com (M. Shi), mrenton@cyllene.uwa.

edu.au (M. Renton).

Mathematical Biosciences 243 (2013) 137–146

Contents lists available at SciVerse ScienceDi rect

Mathe matical Biosc iences

journal homepage: www.elsevier .com/locate /mbs

Chapter 3

31

exposure are both important in determini ng the intensity of

response to the fumigant, the ability to estimate mortality for the

different genotypes at a range of concentr ations and exposure

times based on experimental data is critical for the accuracy of this

new two-locus simulatio n model.

Experiments by Collins et al. [11,12] and Daglish [13] pro-

vided results that help to quantify the expected mortality of

these nine resistance genotypes. These were conducted on in-

sects that had been purified to produce strains. The data sets

from the experiments of Collins et al. contain three strains

QRD14, QRD569 and their Combined F1 (QRD14 � QRD569)

and those from Daglish’s experime nts contain QRD14 (the same

as Collins’), QRD369 and their hybrid (QRD14 � QRD369). Each

of the five strains corresponds to a single genotype of the nine

possible genotypes in our two-locus model (Table 1), whereas

in most previous studies (e.g. [14]), datasets were obtained from

population samples from the field likely to contain various mix-

tures of resistance genes. Hence these new results provide the

means to accurately estimate mortaliti es for the available five

strains, which is the first phase of our mortality modellin g.

Moreover, those estimates can in turn be used to construct a

model to estimate mortalities for the remaining four genotypes ,

based on reasonable genetic assumpti ons [8], which is the sec-

ond phase.

Our previous papers [7,8] presente d some discussion of phos-

phine mortality estimation. However, these focussed on the

numerical algorithm used for model fitting, with only a limited

use of experimental data for illustration [7], and presented preli-

minary modelling results based on simple probit models only [8].

Moreover, there was a limitation in the numerical treatment in

the two papers. The kill rates in some of the experiments included

one and zero [11]; to enable the probit least squares approach to be

used, these two values need to be changed from 1 to 1 ÿ e or from 0

to e where e is a small perturbation , otherwise, their link function

values (see below) are undefined. But in the previous two papers

[7,8] the choice of e was arbitrary and fixed and certainly not opti-

mal in terms of minimum least square error. This limitation moti-

vated us to conduct this current more comprehensive study on the

best way to model mortality phosphine mortality for different

genotypes.

In this paper we describe how we used three models, probit, lo-

gistic and Cauchy models to fit the available data sets using either C

(concentration or dose) and t (exposure time) themselv es or log(C)

and log(t) as the independen t model variables, and compared the

relative accuracy of probit, logistic and Cauchy models for mortal-

ity estimation. We also tested and compare d two-parameter and

four-paramete r sub-models for each of the three models. We also

show how we developed an approach for identifying an optimal

perturbation value when mortality was 0 or 1 based on the golden

section search method.

2. Materials and methods

2.1. Two-locus model with nine genotypes

To para met eris e the mo rtali ty compo nent of our sim ula tion mo d-

el, we need ed to dev elo p empi ric al mo rtal ity mo dels for each geno-

typ e. Sin ce ther e are two loc i in the mo del [7–10] , wit h two pos sibl e

all ele s on each of the loc i, ther e are nine gen oty pes in tota l (Tabl e 1).

2.2. Three generalise d linear models of mortality

In statistics , the generalised linear model (GLM) in the form of

Y ¼ a þ b1x1 þ b2x2 þ � � � þ bkxk ð1Þ

is a flexible general ization of ordinar y least squares regressi on that

allows the linear model to be related to the response variable via

a link function for Y and the magnit ude of the varianc e of each mea-

surement to be a function of its predicted value [15]. GLMs applica -

ble to binomial mortal ity data include probit regression (with a

probit link function), logistic regression (with the canonical logit

link) and Cauch y regressi on (with the tangent link) which we used

to fit the experime ntal data sets.

2.2.1. Probit model

The probit (=‘‘probability unit’’) link function U(P) (Y = U(P) + 5)

is the inverse cumulative distribution function (CDF) associate d

with the standard normal distribut ion [16,17]:

P ¼ Uÿ1ðY ÿ 5Þ ¼ 1ffiffiffiffiffiffiffi

2pp

Z Yÿ5

ÿ1exp ÿu2=2

� du ð2Þ

where P is the actual mortality (proportion that died, 0 6 P 6 1) and

Y is the probit transformed mortality . Note that adding five to U(P)

just ensures all Y values are positive in practice , and simply means

the paramete r a is transf ormed by a constant value of five compared

to an alternative link function where this is not applied.

Using a three-paramet er probit model [16], a probit plane

Y ¼ a þ b1mðtÞ þ b2mðCÞ ð3Þ

may be fitted to the data, where t and C are respective ly exposure

time and concentrat ion, and Y is the probit mortal ity. We consid-

ered two choices for the function m: the logarit hmic function, i.e.

m(t) = log(t) and m(C) = log(C) (whether common logarithm s (base

10) or natural logarithms (base e) are used is immaterial), or the

identit y function, i.e. m(t) = t and m(C) = C.

In the case that the available independent data consist only of

the products Ct (e.g. a range of C but a fixed time t), rather than

independen t values of C and t separately, the parameters b1 and

b2 can be merged into a single parameter, b, resulting in a two-

paramete r probit model:

Y ¼ a þ bmðCtÞ ð4Þ

Table 1

The identifiers of the nine genoty pes (ss,sh, . . .,rr) in the two-locus model, and the correspondence of genotypes and the five strains for which

experimental mortality data was available [s – homogeneous (‘‘homo’’) suscep tible (‘‘suscept’’); r – homogeneous resis tant; h – heterozygous

(‘‘hetero’’)].

138 M. Shi, M. Renton / Mathematical Biosciences 243 (2013) 137–146

32

An extra term ‘‘b3 m(t) m(C)’’ can be added to (3) to describe the

interaction of the variables t and C, thus extending to a four-

parameter probit model

Y ¼ a þ b1mðtÞ þ b2mðCÞ þ b3mðtÞmðCÞ ð5Þ

2.2.2. Logistic models

The most typical link function for logistic models is the canon-

ical logit link [18] (to distinguish we use z instead of the Y in Eq.

(1)):

z ¼ wðPÞ ¼ ln½P=ð1 ÿ PÞ� or P ¼ wÿ1ðzÞ ¼ 1=ð1 þ eÿzÞ ð6Þ

where P is the mortal ity as shown in Eq. (2) and Y ÿ 5 is replaced by

z, the logistic transform ed mortal ity. The two-par ameter and four-

paramete r logistic models correspondi ng to Eqs. (4) and (5) are

respective ly,

z ¼ a þ bmðCtÞ ð7Þ

z ¼ a þ b1mðtÞ þ b2mðCÞ þ b3mðtÞmðCÞ ð8Þ

2.2.3. Cauchy model

The cumulative distribut ion function (CDF) of the Cauchy distri-

bution, whose curve is sigmoid like the probit and logistic curves,

is as follows [19]:

P ¼ Fðx; x0; cÞ ¼1

parctan

x ÿ x0

c

� �þ 1

2ð9Þ

It can be reduced to

tan p P ÿ 1

2

� �� �¼ x ÿ x0

c

� �¼ a þ bx ð10Þ

Here x0 is a location paramete r and c is a scale param eter and

a ¼ ÿx0=c and b ¼ 1=c; or x0 ¼ ÿa=b and c ¼ 1=b ð11Þ

We can look at Eq. (10) as a two-param eter (a and b) GLM model

with a ‘‘link function’’ n(P) = tan[ p(P ÿ 0.5)], where P is also as

shown in Eq. (2) but changi ng Y ÿ 5 to n, the Cauchy transformed

mortality . When it applies to mortal ity estimation, we should set

x = m(Ct). This can be extended to a four-parame ter Cauchy model:

nðPÞ ¼ a þ b1mðtÞ þ b2mðCÞ þ b3mðtÞmðCÞ ð12Þ

However, the relations hips between x0, c and the paramete rs shown

in Eq. (11) no longer hold in Eq. (12).

2.3. Generalized inverse matrix approach to fitting mortality models

We employed a direct algebraic approach, generalized inverse

matrix technique to fit the mortality models. This alternative gener-

alized inverse matrix technique achieves similar results to the

more common statistical iterative approach, maximum likelihood

estimation, but is less time consuming and computati onally

demanding [7]. We briefly explain this approach for those inter-

ested in the details.

Algebraicall y, when any one of the above models is fitted to a

data set, we have an over-determined system of linear equations

with respect to the parameters to be estimated. Let the set of N ob-

served data points be fLi; ti;CigNi¼1 where Li, ti, and Ci are the ith link

function value, exposure time and concentratio n respectively and L

is one of U, W and n. Then the over-determi ned system is (only the

first two items for 2-paramete r models):

Li ¼ 1 � a þ ½mðtiÞ� � b1 þ ½mðCiÞ� � b2

þ ½mðtiÞmðCiÞ� � b3ði ¼ 1;2; . . . ;NÞ ð13Þ

Let the matrix form of the over-determi ned system be Ax = b

where x is model parameter vector, b ¼ ðLiÞNi¼1 and A is an N � p

(p = number of parameters) matrix whose 1st column has all

‘‘1’’ element, 2nd column has ti (or log(ti)) as the ith element

and so on.

The method of least squares is often used to generate estimators

and other statistics in regression analysis [20]. If a solution

minimize s

XN

i¼1

�Li ÿ Li

ÿ �2 ¼XN

i¼1

½a þ b1mðtið Þ þ b2mðCiÞ þ b3mðtiÞmðCiÞ� ÿ LiÞ2

ð14Þ

where �Li = a + b1m(ti) + b2m(Ci) + b3m(ti)m(Ci) is the ith predicted va-

lue, then the solution is called a Least Square s solution. Normall y,

the least squares method can be used to solve the regularized equa-

tions of Ax = b i.e. ATAx = ATb, provided that ATA is non-singula r.

Actually, if A+ is the generalized inverse (or Moore–Penrose pseu-

do-inver se) of matrix A, then A+b is such a solution [21]. Note that

if A itself is a non-singula r square matrix then A+ = Aÿ1. If A is col-

umn full-ranked , then ATA is non-singula r and A+ = (ATA)ÿ1AT. But

while this equation could theoretica lly be used to calculate A+, it

is of limite d practica l use for calculating A+ numerica lly, because

using QR decomp osition (a decomp osition of a matrix X into a prod-

uct X = QR of an orthogonal matrix Q and an upper triangular matrix

R) or singular value decomp osition to obtain A+ will give much

smaller numerical errors than direct calculatio n of (ATA)ÿ1AT [21].

Even if A is not (column) full-ranked and thus the coefficient matrix

of the regularized equations, (ATA), is singular, there still exits a ma-

trix A+ where the linear system has a solution A+b with minimum L2

norm [21]: ||A+b||2 = min{|| x||2: ATA x = ATb}. Note that this general-

ized inverse matrix method minimises the error on the link-func-

tion-tran sformed mortalities rather than the mortalities

themse lves.

2.4. Experimen tal data on mortality

Collins et al. [11] observed mortalities under a range of phos-

phine concentrations (C: mg/l) at a fixed exposure time (t) 48 h

for susceptible (strain QRD14 – correspondi ng to genotype ss)

and strong resistant (strain QRD569 - rr) phenotypes and their

combined F1 progeny ((569 � 14) + (14 � 569) – hh). The consoli-

dated raw data are listed in ‘‘observed’’ rows of Table 2) and the

LT99.9 values (lethal time to achieve 99.9% mortality) are listed in

‘‘observed ’’ rows of Table 3. These values are derived from the

experime ntally observed data for strain QRD569 [12] which was

exposed to a series of concentration from 0.1 to 1.0 mg/l at a range

of exposure periods. The raw data of Collins et al. was consolidated

by averaging replicates.

Daglish [13] determined LC50 (lethal concentr ation to achieve

50% mortality ) and LC99 (to achieve 99% mortality) for phosphine-

susceptibl e (QRD14 – ss) and weak-resist ance (QRD369 – rs) pheno-

types and their F1 progeny (QRD369 � QRD14 – hs) over a range of

concentr ations and exposure times (Table 4).

The two-para meter probit model (4), logistic model (7) and

Cauchy model (10) were used, each to fit the data sets for ss and

hh respectively (Table 2). The four-paramete r probit models (5),

(8), and (12) were used to fit the two data sets for rr genotype

(Tables 2 and 3) since the interaction of exposure time and concen-

tration was considered and more accurate results could thus be ob-

tained. Exposure time t (hrs) values used to fit this rr data are as

follows:

t ¼24 � ð2;2;2;2;2;14:02;12:74;8:509;7:144;6:55;5:628;

4:233;3:74Þ ðhrsÞ ð15Þ

M. Shi, M. Renton / Mathematical Biosciences 243 (2013) 137–146 139

33

2.5. Golden section search method for perturbed value

It can be seen from Table 2 that the mortalities include values of

1 and 0. For the generaliz ed inverse matrix fitting approach to

work, these values need to be changed from 1 to 1 ÿ e and from

0 to e, where e is a small perturba tion; otherwise their link function

value of each of the three models is undefined. It is not the case

that the smaller the perturba tion e the smaller the least squares

(L2) error. In fact it can be observed that there is a minimum func-

tion value of least squares errors between predicted and observed

mortalities (Eq. (14)) in terms of e. Fig. 1(a) shows two predicted

mortality curves for the ss genotype obtained using the probit

model; one is obtained by changing 1 to 1–10ÿ8 (e = 10 ÿ8, see

Table 2 at C = 0.004) and the other is obtained by choosing 1 ÿ e

as the value associated with the minimum L2 error. It is clear that

the latter fitted the observed data very well with (global) L2 er-

ror = 0.0004892 compare d to the former with L2 error = 0.0191.

At C = 0.002 mg/l the predicted mortalities by the former and later

are 0.7061, 0.8398 respectively and the observed one is 0.7047

(Table 2); the difference of two predicted mortalities is up to

0.134 (13.4%).

Fig. 1(b) shows that the least squares error between observed

and predicted mortaliti es (for the ss genotype in Table 2 using pro-

bit model) is a unimodal function of 1 ÿ e. The golden section search ,

introduce d by Kiefer [22], is a technique for finding the extreme

(minimum or maximum ) of a unimoda l function f(x) by succes-

sively narrowing the range of values inside which the minimum

(for our problem) is known to exist [23] as quickly as possible. In

any iteration step, the algorithm maintain s triples of points,

(x0,x1,x3) from left to right (see the 1st text row in Fig. 1(b)), such

that the distances of the triples form a golden ratio :

ðx3 ÿ x1Þ : ðx3 ÿ x0Þ ¼ ðffiffiffi5

pÿ 1Þ=2 � 0:618 or

ðx1 ÿ x0Þ : ðx3 ÿ x0Þ � 1 ÿ 0:618 ¼ 0:382: ð16Þ

For this algorithm, initially two end points, x0 = 0.998 and

x3 = 0.9999 (row 1, Table 5) are chosen. These points define a rea-

sonable interval to search: assuming we are choosing a perturba -

tion for a value of 1, in general the upper limit is very close to

one and the lower limit should be chosen so it is just a little greater

than any other observed mortality value; alternatively, if the num-

ber of individuals correspondi ng to the observed mortality of 1 is

Table 2

The consolidated data of phosphine dose and the aggregate response mortality rate for the genoty pes ss, hh, and rr at a fixed exposure time t = 48 h [11] and predicted mortalities

using 2-parameter probit models for ss and hh, and 4-para meter probit models for rr data sets listed here and in Table 3.

QRD14 (ss)

Dose C (mg/l) 0.001 0.0015 0.002 0.003 0.004

Mortality Observed 0.0201 0.32 0.7047 0.9733 1.0000

Predicted 0.0209 0.30 0.7061 0.9798 0.9991

Comb F1 (hh)

Dose C (mg/l) 0.0025 0.004 0.005 0.0075 0.01 0.02

Mortality Observed 0.0000 0.3445 0.3940 0.8047 0.8591 0.9868

Predicted 0.0951 0.3111 0.4583 0.7259 0.8645 0.9894

QRD569 (rr)

Dose C (mg/l) 0.1 0.25 0.5 1.0 1.25

Mortality Observed 0.0000 0.0200 0.2254 0.5203 0.5705

Predicted 1.33 � 10ÿ6 0.0305 0.1030 0.5844 0.7546

Table 3

Response to PH 3 of mixed-a ge cultures of strain QRD569 (rr) [12] and predicted morta lities using 4-parameter probit model for rr data sets listed here and in Table 2.

Dose: C (mg/l) 0.1 0.15 0.2 0.3 0.4 0.5 0.75 1.0

LT99.9 (day) Observed 14.02 12.74 8.509 7.144 6.55 5.628 4.233 3.74

Predicted 14.75 11.27 9.394 7.347 6.22 5.484 4.402 3.74 aMortality Predicted 0.9981 0.9998 0.9964 0.9985 0.9995 0.9995 0.9993 0.9982

a Recall that the observed mortalities are all 0.999 = 99.9% (LT99.9).

Table 4

Concentration (LC50 and LC99) values (mg/l) required to achieve 50% and 99% mortality for different exposure times (t) [13] and the L2 errors of predicted and observed LC50 and

LC99 for ss, hs and rs data using the three 4-paramet er models (5), (8), and (12). Note that the probit, logistic and Cauchy models all give the same predicted LC50 and LC99 values

and thus the same L2 errors; although for othe r values models give different predictions (see Fig. 3).

Strain (mortality) 20 h 48 h 72 h 144 h L2 error

ss (50%) Observed 0.0052 0.0017 0.0011 0.00064

Predicted 0.0044 0.0017 0.0012 0.00065 6.22 � 10ÿ7

(99%) Observed 0.0091 0.0037 0.0021 0.0014

Predicted 0.0102 0.0036 0.0024 0.0013 1.37 � 10ÿ6

hs (50%) Observed 0.010 0.0042 0.0023 0.0011

Predicted 0.010 0.0040 0.0024 0.0010 2.65 � 10ÿ7

(99%) Observed 0.026 0.013 0.0066 0.0025

Predicted 0.027 0.011 0.0070 0.0029 5.03 � 10ÿ6

rs (50%) Observed 0.20 0.052 0.032 0.017

Predicted 0.19 0.052 0.033 0.017 4.00 � 10ÿ5

(99%) Observed 0.40 0.091 0.060 0.028

Predicted 0.41 0.095 0.057 0.028 6.78 � 10ÿ5

140 M. Shi, M. Renton / Mathematical Biosciences 243 (2013) 137–146

34

known to be n (i.e. n out of n individua ls died) then a lower limit of

(n ÿ 1)/n would be reasonable. Similarly, if we are choosing a per-

turbation for a value of zero, in general the lower limit is very close

to zero and the upper limit should be chosen either so it is just a

little less than any other observed mortality value, or to be 1/ n if

n is known. Normally the least squares error would be expected

to be unimodal over this interval and to reach a local minimum

at some point within this interval.

Since f(x) is unimoda l in the Fig. 1(b) example, f1 = f(x1) is smal-

ler than either f(x0) or f(x3). We then calculate the function value

f2 = f(x2) at a new point x2 such that (x2 ÿ x0): (x3 ÿ x0) � 0.618

(x2 is to the right of x1, see the 1st text row in Fig. 1(b)). If the ‘‘right

value’’ f(x2) > ‘‘left value’’ f(x1) we discard the ‘‘right part’’ of the

interval, i.e. [x2,x3], otherwise we discard the ‘‘left part’’ [x0,x1]. In

the Fig. 1(b) example, the ‘‘left part’’ [x0,x1] is removed after the

1st iteration (Table 5). Thus, the remaining interval always con-

tains the minimum and its length is always 0.618 of the original

one. In the next iteration, if we discard the ‘‘left part’’, first we rela-

bel the remaining triple (x1,x2,x3) to (x0,x1,x3). This is the case illus-

trated in Fig. 1(b), but the new relabeled values are denoted by ‘s0,

s1, s3’ instead for clarity). Otherwise we relabel the remaining triple

(x0,x1,x2) to (x0,x1,x3). In both cases we relabel the relevant relative

function value (Table 5). The new triples still form a golden ratio

(16). Then calculate the new function value (at s2 in the example)

and compare the function values (at s1 and s2 in the example) and

narrow the interval again, and so on until the desired accuracy is

reached. In the process, except for the initial step, we only need

to calculate the new point x2 and its function value. See Fig. 1(b)

and Table 5 for details for our example; after seven iterations we

chose x2 = 0.999305 as the best perturbation since f(x1) > f(x2),

and the correspondi ng minimum L2 error (f(x2)) = 0.0004892.

We used the golden section search technique to estimate the

minimum L2 error between predicted and observed mortalitie s for

QRD14 (ss) and Combined F1 (hh) and successfully fit the two mod-

el parameters (see Eqs (4), (7), and (10), as the exposure time is a

fixed number) of the three alternativ e GLM models.

However , there is a different story for the strain QRD569 (rr)

since we have two different data sets: one is a set of different con-

centrations and fixed time (48 h) (Table 2) resulting in different

(observed) mortalities, the other is a set of different concentrations

and fixed mortality (99.9%) (Table 3) resulting in different

(observed) lethal times. We found that if the same criterion of

minimum L2 error with respect to mortaliti es (for the data set in

Table 2) was used to find the best perturbation e and the model

paramete rs were then fitted to the two data sets, the L2 error of

the predicted and observed LT 99.9 times (for the data set in Table 3)

is big. As rr beetles are strongly resistant pest beetles and we want

to accurately simulate mortality under fumigation treatments

likely to cause high but not perfect mortality for this genotype,

accuracy of LT99.9 estimates are of prime importance . Hence we

change the criterion for the best perturbation e to be that which

minimize s the L2 error of the LT99.9 predictio ns only; using this cri-

terion, the overall L2 error of mortality is small though it is not

minimal.

2.6. Mortality estimation for the remaining four genotypes

Note that when ss beetles mate with rr beetles (ss � rr), 100% of

the F1 offspring produced will be the hh genotype, and when ss

beetles mate with rs beetles (ss � rs), 100% of the F1 offspring pro-

duced will be the hs genotype [7]. However, any other pair of mat-

ing among the five strains will not reproduce 100% of a single

genotype offspring; for example, hs � rr results in 50% of rh and

50% of hh [7]. Hence it is impossible to obtain strains each corre-

sponding to a single one of the remaining four genotypes. There-

fore we develope d a method to estimate the mortalities for the

remaining sh, sr, hr and rh genotypes in the two-para meter model

based on reasonable genetic assumptions . The approach is summa-

rised briefly here, since full details are available in [8].

(a) (b)

Fig. 1. (a) The two predicted mortality curves associated with perturbing the observed mortality proportion from 1 to either 1–10ÿ8 or 0.998, the proportion corresponding to

the minimum L2 error, and (b) (The curve of the L2 errors as a function of the perturbed mortality proportions from 0.998 to 0.9999, and the iteration intervals under the

curve) resulting from the golden section search technique. In both cases the probit mortality model was applied to Collins’ QRD14 strain.

Table 5

The four points (triples (x0,x1,x3) and x2 in the six iterations of golden secti on search

shown in Fig. 1(b).

Iteration x0 x1 x2 x3 Discarded

interval

1 0.998000 0.998726 0.999175 0.999900 Left [x0,x1]

2 0.998726 0.999175 0.999451 0.999900 Right [x2,x3]

3 0.998726 0.999003 0.999174 0.999451 Left [x0,x1]

4 0.999003 0.999174 0.999280 0.999451 Left [x0,x1]

5 0.999174 0.999280 0.999346 0.999451 Left [x0,x1]

6 0.999174 0.999240 0.999280 0.999346 Right [x2,x3]

7 0.999240 0.999280 0.999305 0.999346

M. Shi, M. Renton / Mathematical Biosciences 243 (2013) 137–146 141

35

As a step towards achieving to this, we first estimated the resis-

tance factors for the first five genotypes (strains) based on our fit-

ted models. The resistance factor of a genotype x for a 48 h

fumigation duration is defined as the ratio between the PH 3 con-

centration that achieves 50% mortality in a sub-population of geno-

type x and the lower PH 3 concentratio n that achieves 50% mortality

in a susceptibl e sub-populati on [24]. We then estimated the resis-

tance factors for the other four genotypes by making some basic

assumptions regarding genetic interactio ns; log-transformed resis-

tance factors for the nine genotypes can be expresse d in terms of

five ‘genetic interaction’ paramete rs which represent respectively

the strength and the dominance of the 1st and 2nd genes, and

the synergism between the two genes. Since we had estimates

for five genotypes, we could calculate values for the five ‘genetic

interaction’ parameters, which then let us estimate the resistance

factors for the other four genotypes . Finally we modelled survival

rates for the other four genotypes using the two-para meter probit

model. We assumed that the parameter b for these genotypes was

the same as the b value for one, or the mean of two, of the five

strains. Finally, the value for parameter a for each of the four geno-

types could then be obtained by direct substitut ion of the esti-

mated resistance factors and their related values C, t and Y into

the two-parame ter probit model.

3. Results

For all three models, log-transform ing the explanatory variables

(choosing m as a logarithmic function rather than the identity

function) gave better results (less error). Therefore, only results

for the logarithmic function are presented here.

For the data of Collins et al. [11,12], the optimal perturbation

used for mortality values of 0 or 1 obtained using the golden sec-

tion search technique are listed in Table 6, together with the corre-

sponding L2 errors for mortality and/or LT99.9 value. For the data of

Daglish [13] predictions and L2 errors (with respect to the lethal

concentratio n values LC50 and LC99) are included in Table 4; note

that the observed mortalities for this data set only included values

of 0.5 or 0.99 so no perturba tions were required and we just di-

rectly used the data set to fit the parameters for the three models.

It is clear from Tables 4 (last column, the three models achieve

the same L2 errors) and Table 6 and Figs. 2 and 3 that the probit

model is the best entirely in terms of smallest L2 errors. Hence

we only list the predicted values in Tables 2–4 (‘‘Predicted’’ rows),

and the fitted parameters in Table 8 obtained using the probit

models in the first phase. Fig. 4(a) shows the plot of probit values

based on our fitted models against a range of concentratio n values

[0.01,1.0] (mg/l) at a fixed t = 24 h for the five strains.

Also the predicted and observed LC50 values (mg/l) at 48 h ob-

tained using the probit models are listed in Table 7. Note that

the observed and predicted LC50 values for Daglish’s data are also

listed in the ‘‘50%’’ rows and ‘‘48 h’’ column of Table 4. It is these

values that are used to calculate the resistance factors of the five

strains, which are then used in the second phase to estimate the

resistance factors of the remaining four genotypes which are also

listed in Table 7. Finally, these factors are used to find the two

paramete rs of the probit model for the four remaining genotypes

(see [8] for details). The fitted paramete rs are also listed in Table 8.

The probit lines at t = 24 h for all of the nine genotypes are shown

in Fig. 4(b).

4. Discussion and conclusi on

In this study we have shown how the probit, logistic and Cau-

chy models with two- or four-paramete r sub-model s fit a variety

of data set types: observed mortalities at a range of concentr ation

with a fixed (48 h) exposure time (Table 2); observed lethal times

achieving 99.9% mortality at a range of concentrations (Table 3)

and observed lethal concentratio ns reaching 50% or 99% mortality

at a set of exposure times. Our results show that in all cases it was

better to log-transformed concentr ation and time as explanatory

variables in models of mortality due to phosphine fumigation,

rather than use the untransfor med variables. Moreover, for all data

sets, a probit model provided a better or equally as good fit to the

data as alternative Cauchy or logistic models. The differences in L2

errors between choosing m as the identity function or the logarith-

mic function were small for the logistic and Cauchy models but

bigger for the probit models. Therefore we conclude that the probit

models based on log-transform ed concentr ation and time provide

the best predictio ns of mortality under a range of concentratio ns

and times for use in our two-locus simulation model [7–10].

Daglish [13] used a Haber-ty pe rule [25] to fit his experiment

data [13]. However, this kind of rule only predicts the relationship

between the concentration and time to achieve a certain fixed

mortality , and does not provide a means to predict variable mortal-

ities between exposure scenarios [25]. Our re-fitted four-param e-

ter models can predict mortalities at a range of concentratio ns

and different exposure times which is required in a two-locus sim-

ulation model intended to be used to explore a wide range of phos-

phine fumigatio n practices and strategies.

Previous ly [7,8], it was an arbitrary decision about what value

to use instead of 0 and 1 (e.g. changing 0 to 0.0001 or 1 to

0.9999) when using a least-squares approach to fit probit

(two- or four-param eter) models to mortality/su rvival data. None-

theless, mortality estimations were getting strongly affected by

this arbitrary decision, especiall y with such a small data set, as

Table 6

Fitting Collins’ data (Tables 2 and 3): The optimal perturbed mortality values e or 1 ÿ e and the corresponding minimum L2 errors (proportions) between observe d and predicted

(from the 2-parameter models) mortality value s for ss and hh and for rr (predictions from the 4-parameter mode ls) denoted by ‘‘M’’ together with L2 errors between observed and

predicted LT99.9 values (days). Fitting Daglish’s data (Table 4): L2 errors (proportions) between observed and predicted (from the 4-parameter models) mortality values for ss, hs

and rs.

Model QRD14 (ss) Comb F1 (hh) QRD569 (rr)

1 ÿ e L2 error e L2 error e L2 errors (LT99.9/M)

Collins

Probit 0.999305 0.0004892 0.082918 0.0115106 4.38 � 10ÿ8 3.67/0.0533

Logistic 0.999902 0.0180498 0.241277 0.0338106 3.88 � 10ÿ5 3.99/0.0810

Cauchy 0.980154 0.0828693 0.141277 0.2029557 0.000744 4.41/0.4525

QRD14 (ss) QRD14 � QRD369 (hs) QRD369 (rs)

Daglish

Probit 0.0354032 0.0188711 0.0025313

Logistic 0.0509557 0.0281705 0.0038142

Cauchy 0.3721201 0.4819185 0.1553470

142 M. Shi, M. Renton / Mathematical Biosciences 243 (2013) 137–146

36

shown clearly in Fig. 1(a). Using the approach we have developed,

based on the golden section search technique, to find the best per-

turbation of proportion 1 or 0 with respect to the minimum L2 er-

ror, makes the fitted models much more accurate.

It can be seen from Table 6 that the least squares (L2) errors be-

tween observed and predicted mortality values ranged from

0.049% (bold in Table 6) to 5.3% (italic) with the probit model, from

0.381% to 8.1% with the logistic model and from 8.3% to 48.2% with

the Cauchy model. Overall, both probit and logistic models fit all

data sets well but the probit models are better in terms of small

least squares (numerical) errors. Cauchy models perform worse

in this sense. Also as the transformat ion from the link function val-

ues to mortality is the arctangent function, the predicted mortality

curves resulting from Cauchy models are flat in the two end parts

(mortalities are relatively close to 0 or 1) but very steep in the

‘‘middle’’ part. Note that the Cauchy C–T curve in Fig. 2(d) has sim-

ilar behaviour to the other two C–T curves since all of the three

curves plot the predicted lethal time to achieve a very high mortal-

ity of 99.9% at a range of concentr ations. On the other hand, the

predicted mortality curves from the Cauchy model are far from

the observed mortaliti es in the ‘‘middle’’ part (Fig. 2(a)–(c)).

The strain QRD14 (ss) was involved in both Collins’ data

(Table 2) and Daglish’s data (Table 4) and two different sub-models

were used to fit; two-parameter probit model for the former and

four-paramete r probit model for the latter. However, the two

sub-models result in the same predicted LC50 value 0.0017 mg/l

which is also the same as Daglish’s observed value (Table 7) to four

significance places. This also means the probit models performed

very well.

The C–T curve for the rs beetles using the logistic model is very

similar to that using the probit model plotted in Fig. 3(c). Compar-

ing (d) with (c) of Fig. 3 the ‘‘middle’’ part of the mortality curve

using the Cauchy model are still too steep to fit the observed mor-

talities in this part.

The estimations using 3-parameter probit (see Eq. (3)), logistic

and Cauchy models result in bigger L2 errors than those using cor-

respondi ng 4-paramete r models. This follows because the latter

considers the interactions between concentration and exposure

time which occur in reality.

To our knowledge, no previous models have included mortality

predictio ns for a range of resistance genotypes that vary with con-

centration and exposure time, based on extensive experime ntal

data like the models we have presente d here. The well-fitting

two- and four-paramete r probit models described in this paper al-

low us to accurately predict mortality of the nine resistance geno-

types of the lesser grain borer, R. dominica. This provides an

essential component of our two-locus individual-bas ed simulation

model, which will help us more confidently predict the evolution

(a) (b)

(c) (d)

Fig. 2. Predicted mortalities at a fixed exposure time (t = 48 h) and a range of doses using the three models (2-parameter) comparing to the observed mortalities listed in

Table 2 (a) for genotype ss, (b) for hh and (c) for rr. And (d) predicted times to achieve 99.9% mortality (LT99.9 values) at a range of concentrations C = [0.1,1.0] for rr using the

three models (4-parameter) comparing to the observed values listed in Table 3.

M. Shi, M. Renton / Mathematical Biosciences 243 (2013) 137–146 143

37

of phosphine resistance in R. dominica, in order to weigh the merits

of various managemen t options for delaying or avoiding evolution

of resistance in this destructive primary pest of stored grains.

With these more accurate mortality estimations as a basis, we

have already investiga ted the importance of basing resistance evo-

lution models on realistic genetics and found that predictions of

our realistic two-locus individua l-based model vary significantly

from those of an equivalent model based on a simplifying assump-

tion that resistance is conferred by a single gene [9]. We have also

compared how fumigatio n tactics based on extending the duration

of fumigation or increasing the concentration of fumigation influ-

ence the control of resistant and non-resistant lesser grain borer,

and how these are the impacted by different initial gene frequen-

cies [10]. Furthermore, we have used the individua l-based model

to investiga te the impact of phosphine dose consisten cy and immi-

gration rate on the effect of fumigation and the consequences for

the developmen t of phosphine resistance, and found that dose con-

sistency is the key factor in managing population numbers and

resistance levels [26]. This current paper shows that we can be

confident in the mortality estimate s for different genotypes,

(a) (b)

(c) (d)

Fig. 3. C–T curves at a range of pairs (dose C, exposure time t) comparing the observed mortalities (50% and 99%) at the range of doses listed in Table 4(a) for genotype ss using

probit model (b) for hs using logistic model (c) for rs using probit model and (d) for rs using Cauchy model.

Table 7

Predicted (‘‘Pre’’), observed (‘‘Obs’’) and estimated LC50 values (mg/l) [C – Collins’ data (Tables 2 and 3); D – Daglish’s data (Table 4)] for 48 h exposure

and resistance factors f(x) = LC50(x)/LC50(ss) for genotype x of the five strains, and estimated resistance factors for the othe r four genotypes.

144 M. Shi, M. Renton / Mathematical Biosciences 243 (2013) 137–146

38

exposure times and fumigant concentr ations used in the individ-

ual-based model, and thus more confident in the predictions and

recommend ations developed with it. The results of these investiga-

tions and of those conducted with the individual-bas ed model in

future will therefore help us to continue to use the relatively safe

and effective phosphine (PH3) fumigant for the control of infesta-

tions of this serious pest, and thus help safeguard world-wide grain

supplies.

Acknowled gments

The authors would like to acknowledge the support of the Aus-

tralian Governmen t’s Cooperative Research Centres Program. We

also thank P.J. Collins for his great help in the genetics and provi-

sion of raw data and an anonymous reviewer of our previous paper

[10] for the suggestion which motivated us to conduct this study.

References

[1] D.W. Hagstrum, P.W. Flinn, Simulations comparing insect species differences in response to wheat storage conditions and management practices, J. Econ.Entomol. 83 (1990) 2469.

[2] P.J. Collins, Resistance to grain protectants and fumigants in insect pests of stored products in Australia, in: H.J. Banks et al. (Eds.), Proc of First Australian Postharvest Technical Conference Canberra, Australia, 1998, pp. 55–57.

[3] J. Thorne, G. Fulford, A. Ridley, D. Schlipalius, P. Collin, Life stage and resistance effects in modeling phosphine fumigation of Rhyzopertha dominica (F.), in:Proc. 10th Int. Working Conf. Stored Product Protection, Lisbon, Julius Kuhn Publ., Berlin, Germany, June–July2010, pp. 438–445.

[4] K. Lilford, G.R. Fulford, D. Schlipalius, A. Ridley. Fumigation of stored grain insects – a two locus model of phosphine resistance, in: Proc. 18th World Imacs/Modsim Congress, Cairns, Australia, July 2009, pp. 540–546, <http://mssanz.org.au/modsim09>.

[5] D.I. Schlipalius, P.J. Collins, Y. Mau, P.R. Ebert, New tools for management of phosphine resistance, Outlooks Pest Manag. 17 (2006) 52.

[6] D.I. Schlipalius, W. Chen, P.J. Collins, T. Nguyen, P.E.B. Reilly, P.R. Ebert, Gene interactions constrain the course of evolution of phosphine resistance in the lesser grain borer, Rhyzopertha dominica , Heredity 100 (2008) 506.

[7] M. Shi, M. Renton, Numerical algorithms for estimation and calculation of parameters in modelling pest population dynamics and evolution of resistance in modelling pest population dynamics and evolution of resistance, Math.Biosci. 233 (2011) 77.

[8] M. Shi, M. Renton, P.J. Collins, Mortality estimation for individual-based simulations of phosphine resistance in lesser grain borer (Rhyzopertha

dominica), in: F. Chan, D. Marinova, R.S. Anderssen (Eds.), MODSIM2011,19th International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand, December, 2012, pp. 352–358, <http://www.mssanz.org.au/modsim2011/A3/shi.pdf>.

[9] M. Shi, M. Renton, J. Ridsdill-Smith, P.J. Collins, Constructing a new individual- based model of phosphine resistance in lesser grain borer (Rhyzoperthadominica): do we need to include two loci rather than one?, Pest Sci 85 (2012) 451.

[10] M. Shi, P.J. Collins, J. Ridsdill-Smith, M. Renton, Individual-based modelling of the efficacy of fumigation tactics to control lesser grain borer (Rhyzopertha

dominica) in stored grain, J. Stored Prod. Res. 51 (2012) 23.

Table 8

The fitted param eters for the two-parameter probit models fitted to the data for strain QRD14 (ss) and Comb F1 (hh) and for the four-param eter probit model fitted to the data for

QRD569 (rr), hs and rs. For the genotype ss, the two-parameter mode l was fitted to Collins et al. data (Table 2) and the four-param eter model was fitted to Daglish’s data (Table 4).

The parameter s for the two-parameter models derived for the other four genotypes are also listed.

Genotype 2-parameter model 4-parameter model

a b a b1 b2 b3

ss 14.2549 8.5610 3.9749 12.3267 3.8700 1.9247

hs 11.2847 3.7764 6.9650 ÿ1.0105

hh 7.3773 4.0045

rs ÿ10.8398 16.1356 1.9145 4.0846

rr ÿ12.2019 10.3584 2.5803 1.3853

sh 13.4147 8.5610 The same b value as that for ss

sr 6.3546 4.0045 The same b value as that for hh

hr 3.9243 5.7784 b = the mean of the two slopes of probit lines for rs and hh

rh 0.7072 7.5522 b = the slope of probit line for rs

(a) (b)

Fig. 4. Probit lines at the range concentration values [0.01,1.0] mg/l and t = 24 h (a) for the five strains (b) for all nine genotypes. The three horizontal lines are Y = ÿ1.706, 5

and 11.706 respectively corresponding to mortality = 0%, 50% and 100% respectively. Note that no two probit lines intersect at a point in the ‘‘middle’’ zone between

Y = ÿ1.706 and 11.706 implying that no same mortality (not 0, nor 1) at the same concentration are reached for two different genotypes.

M. Shi, M. Renton / Mathematical Biosciences 243 (2013) 137–146 145

39

[11] P.J. Collins, G.J. Daglish, M. Bengston, T.M. Lambkin, H. Pavic, Genetics of resistance to phosphine in Rhyzopertha dominica (Coleoptera: Bostrichidae), J.Econ. Entomol. 95 (2002) 862.

[12] P.J. Collins, G.J. Daglish, H. Pavic, R.A. Kopittke, Response of mixed-age cultures of phosphine resistant and susceptible strains of lesser grain borer,Rhyzopertha dominica , to phosphine at a range of concentrations and exposure periods, J. Stored Prod. Res. 41 (2005) 373.

[13] G.J. Daglish, Effect of exposure period on degree of dominance of phosphine resistance in adults of Rhyzopertha dominica (Coleoptera: Bostrychidae) and Sitophilus oryzae (Coleoptera: Curculionidae), Pest Manag. Sci. 60 (8) (2004)822–826.

[14] M.A.G. Pimentel, L.A. Faroni, M.R. Totola, R.N.C. Guedes, Phosphine resistance,respiration rate and fitness consequences in stored-product insects, Pest Manag. Sci. 63 (2007) 876.

[15] A.J. Dobson, A.G. Barnett, Introduction to Generalized Linear Models, third ed.,Chapman and Hall/CRC, Boca Raton, FL, 2008.

[16] C.I. Bliss, The relation between exposure time, concentration and toxicity in experiments on insecticides, Ann. Em. Sot. Am. 33 (1940) 721.

[17] D.J. Finney, Probit Analysis, third ed., Cambridge University Press, 1971.[18] A. Agresti, An Introduction to Categorical Data Analysis, second ed., Wiley,

Hoboken, New Jersey, 2007.

[19] J. Zhang, A highly efficient L-estimator for the location parameter of the Cauchy distribution, Comput. Stat. 25 (2010) 97.

[20] J.R. Wolberg, Data Analysis using the Method of Least Squares: Extracting the Most Information from Experiments, Springer, 2005.

[21] A. Ben, T.N.E. Greville, Generalized Inverses: Theory and Applications, second ed., Springer, New York, 2003.

[22] J. Kiefer, Sequential minimax search for a maximum, Proc. Am. Math. Soc. 4(1953) 502.

[23] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Section 10.2 golden section search in one dimension, in: Numerical Recipes: The Art of ScientificComputing, third ed. Cambridge University Press, New York, 2007.

[24] FAO, Recommended methods for the detection and measurement of resistance of agricultural pests to pesticides: Tentative method for adults of some major species of stored cereals with methyl bromide and phosphine. FAO Method No 16, FAO Plant Protection Bulletin 23 (1975) 12–25.

[25] N.J. Bunce, R.B.J. Remillard, Haber’s rule: the search for quantitative relationships in toxicology, Human Ecol. Risk Assess. 9 (6) (2003) 1547.

[26] M. Shi, M. Renton, P.J. Collins, T.J. Ridsdill-Smith, R.N. Emery, Dosage consistency is the key factor in avoiding evolution of resistance to phosphine and population increase in stored grain pests, Pest Manag. Sci. in press, http://dx.doi.org/10.1002/ps.3457.

146 M. Shi, M. Renton / Mathematical Biosciences 243 (2013) 137–146

40

ORIGINAL PAPER

Constructing a new individual-based model of phosphine

resistance in lesser grain borer (Rhyzopertha dominica): do we

need to include two loci rather than one?

Mingren Shi • Michael Renton •

James Ridsdill-Smith • Patrick J. Collins

Received: 10 November 2011 / Accepted: 16 February 2012 / Published online: 3 March 2012

Ó Springer-Verlag 2012

Abstract In this article, we describe and compare two

individual-based models constructed to investigate how

genetic factors influence the development of phosphine

resistance in lesser grain borer (R. dominica). One model is

based on the simplifying assumption that resistance is

conferred by alleles at a single locus, while the other

is based on the more realistic assumption that resistance is

conferred by alleles at two separate loci. We simulated the

population dynamic of R. dominica in the absence of

phosphine fumigation, and under high and low dose

phosphine treatments, and found important differences

between the predictions of the two models in all three

cases. In the absence of fumigation, starting from the same

initial frequencies of genotypes, the two models tended to

different stable frequencies, although both reached Hardy–

Weinberg equilibrium. The one-locus model exaggerated

the equilibrium proportion of strongly resistant beetles by

3.6 times, compared to the aggregated predictions of the

two-locus model. Under a low dose treatment the one-locus

model overestimated the proportion of strongly resistant

individuals within the population and underestimated the

total population numbers compared to the two-locus model.

These results show the importance of basing resistance

evolution models on realistic genetics and that using

oversimplified one-locus models to develop pest control

strategies runs the risk of not correctly identifying tactics to

minimise the incidence of pest infestation.

Keywords Individual-based model � One-locus �Two-locus � Phosphine resistance � Lesser grain borer

Introduction

The lesser grain borer, Rhyzopertha dominica, is a very

serious cosmopolitan pest of stored durable food com-

modities. It attacks a wide variety of stored foods (Hag-

strum and Subramanyam 2009) and is particularly

destructive of stored cereals in warm temperate to tropical

climates where it is a voracious feeder on whole grains. R

dominica is an active flyer and rapid coloniser, completing

its life cycle in 4–5 weeks under favourable conditions.

Adults lay 200–400 eggs through their life of 2–3 months

(Arbogast 1991).

Grain industries world-wide rely on phosphine (PH3)

fumigant as their key tool for the control of infestations of

R. dominica, because it is inexpensive, environmentally

Communicated by C. G. Athanassiou.

M. Shi (&) � M. Renton

School of Plant Biology M084, FNAS, The University

of Western Australia, 35 Stirling Highway, Crawley,

WA 6009, Australia

e-mail: shi.mingren@gmail.com

M. Shi � M. Renton � J. Ridsdill-Smith � P. J. CollinsCooperative Research Centre for National Plant Biosecurity,

Canberra, Australia

M. Renton � J. Ridsdill-Smith

CSIRO Ecosystem Sciences, Underwood Avenue, Floreat,

WA 6014, Australia

J. Ridsdill-Smith

School of Animal Biology M092, FNAS, The University of

Western Australia, 35 Stirling Highway, Crawley, WA 6009,

Australia

P. J. Collins

Department of Employment, Economic Development and

Innovation, Ecosciences Precinct, GPO Box 267, Brisbane,

QLD 4001, Australia

123

J Pest Sci (2012) 85:451–468

DOI 10.1007/s10340-012-0421-6

personal copyChapter 4

41

benign and accepted by world markets. In addition, alter-

native chemical treatments are generally not acceptable for

environmental, health and safety reasons and physical

treatments are significantly more expensive, not effective

enough to meet market standards and often do not comply

with grain handling logistics. However, heavy reliance on

PH3 has led to the development of resistance in R. domi-

nica in many countries, including Australia, and this is of

serious concern to both primary producers and grain han-

dling organisations.

Initial fumigant response analyses of PH3 resistance in

R. dominica in Australia indicated two resistance pheno-

types, which are labelled weak and strong resistance

(Collins 1998). Subsequent classical and molecular genetic

analyses revealed that weak resistance is controlled by a

single major gene, rph1, and that strong resistance is

mediated by the same major gene, rph1, in combination

with another major gene, rph2. Both rph2 and rph1 are

close to recessive so that only homozygous insects fully

express resistance to phosphine. In addition, both rph1 and

rph2 individually express a relatively low level of resis-

tance but when they occur in the same insect the resistance

mechanisms appear to synergise producing a much higher

level of resistance (Collins et al. 2002; Schlipalius et al.

2002).

The initial industry response to PH3 resistance has been

to increase PH3 application rates and fumigation frequency;

however, this is not viewed as a sustainable strategy. Our

research is aimed at contributing to the development of

viable, long-term strategies to support the management of

phosphine resistance by the grain industry. Our approach

was to use stochastic individual-based modelling to simu-

late and test a range of resistance management scenarios.

Models aimed at accurate prediction are usually validated

against independent data, but direct validation of resistance

evolution models is not generally possible due to the sto-

chasticity, variability and uncertainty of the processes

involved, and the fact that controlled and replicable studies

with the large numbers of insects and long time periods

involved in reality are impossible. Nonetheless, such

models may be one of the few tools for evaluating different

management options for delaying evolution of resistance.

Confidence in the predictions of such a model thus depends

strongly on testing which model assumptions make a sig-

nificant difference to its predictions and then ensuring that

these assumptions match reality as closely as possible.

In this article, we present the development of one- and

two-locus individual-based models and make some com-

parisons with a previous population-based model of resis-

tance selection in R. dominica (Lilford 2009; Lilford et al.

2009). We use our model to address the key question of the

importance of the number of loci involved in resistance.

Previous modelling of the evolution of major-gene

resistance to pesticides in insects generally assumes resis-

tance is conferred by a single gene (Sinclair and Alder

1985; Tabashnik and Croft 1982), but, as explained above,

in this case strong resistance appears to be conferred by

two genes. We wanted to investigate the importance of

including two genes in our resistance model, or whether

following previous studies and simplifying by assuming

resistance to be conferred by a single gene would make

little difference to the model’s predictions. We thus com-

pared the predictions of our one- and two-locus models

under three simulated scenarios: in the absence of fumi-

gation; under a high concentration treatment and under a

low concentration treatment. We also investigated whether

differences between the models predictions could be

overcome with a simple adaptation to the one-locus model,

or whether differences in model predictions were more

significant.

Models and methods

We describe the overall model dynamics, the model

assumptions, the simulation procedure and the genotypes

and offspring genotype tables for the one- and two-locus

models. We then explain how we aggregated the nine

genotypes in the two-locus model into the three genotypes

in the one-locus model for comparison purposes. We then

describe the simulations we conducted for this study, which

were chosen specifically to address our aim of comparing

the one- and two-locus models, rather than to simulate any

particular real situations.

Overview of individual-based model

Our approach is to use stochastic individual-based model-

ling that explicitly takes into account the fact that R.

dominica populations consist of individual beetles, each of

a particular genotype and a particular life stage. The overall

model dynamics with assumptions and simulation pro-

cesses are discussed first, followed by explanation of the

way that the model represents genotype and resistance

status and how this determines the death or survival of

individual beetles in the absence or in the presence of

phosphine fumigation.

The life stages of beetles consist of egg, larva, pupa and

adult. For simulation purposes, the model actually uses five

life stages; the adult stage is separated into two: a pre-

oviposition period when the female beetle is unable to lay

eggs and a subsequent mature egg-laying period. A 1:1 sex

ratio was assumed. This allowed us to set the start number

and count the population of female beetles, with the

assumption that total number and the allelic frequency for

the male beetles was the same as for the females.

452 J Pest Sci (2012) 85:451–468

123

Author's personal copy

42

The rate of development and survival of various life

stages of the lesser grain borer depend on several envi-

ronmental factors, particularly temperature, moisture con-

tent of the grain (or equilibrium relative humidity) and food

type. For this model we assumed ‘typical’ conditions based

on the FAO (1975) standard, i.e. grain-type wheat stored at

a temperature (T) of 25°C and relative humidity (r.h.) of

70%, under which many researchers have undertaken their

experiments (e.g. Collins et al. 2000, 2002, 2005; Daglish

2004; Herron 1990; Pimentel et al. 2007; Schlipalius et al.

(2008)). Life history parameters regarding the time spent

within each life stage were estimated from published

experimental data including Baldassari et al. (2005), Rees

(2004), Andrewartha and Birch (1982), Birch (1945, 1953),

Beckett et al. (1994) and Longstaff (1999). We set the

female’s pre-oviposition period as 15 days according to

Schwardt (1933) and Matin and Hooper (1974). Two-locus

model parameters with a brief description and the default

parameter value used in the model are listed in Tables 1, 2

and 3. Details of parameter estimation are included in

section ‘Mortality under phosphine fumigation’ and Shi

and Renton (2011) and Shi et al. (2011). We describe

differences in parameters for our one-locus model later.

The simulated dynamics for an individual are illustrated

in Fig. 1. We provide the following explanation for the

steps that are numbered in the figure.

(1) The initial number of each genotype within each life

stage is determined by the specified initial propor-

tions. The ‘time remaining within current life stage’

(TRICLS) for each individual beetle is then drawn

randomly from a log-normal distribution, with mean

and standard deviation depending on the life stage

(Table 3). The log-normal is used because it has the

appropriate shape and is simple to parameterise based

on available information (see Appendix 2 for details).

(2) The simulation runs on a daily timestep, with each

individual beetle updated separately each day.

(3) After each iteration, TRICLS is reduced by one for

every beetle. When TRICLS B 0 the current stage

ends and the beetle enters the next life stage and the

duration for the new stage (TRICLS) is determined by

drawing from the appropriate log-normal distribution.

When TRICLS B 0 for an adult beetle, the individual

dies and is removed from the population.

(4) It is assumed that no eggs are laid during periods of

fumigation. Whether the beetle survives through the

day during a period of fumigation is determined by

Table 1 The finite daily survival rate (FDSR) for the nine genotypes

of the two-locus model for phosphine treatment 0.01 mg/l 9 14 days

(exposure for 14 days at the concentration 0.01 mg/l)

1st Gene 2nd Gene

s h r

s 8.08 9 10-6 0.01050841 0.32155794

h 0.12774219 0.22071475 0.75579152

r 0.94699984 0.96198801 0.99995317

s Homozygous susceptible, r homozygous resistant, h heterozygous

Table 2 The FDSR for the nine genotypes of the two-locus model

for the high phosphine treatment 0.2 mg/l 9 8 days

1st Gene 2nd Gene

s h r

s \1.0 9 10-11\1.0 9 10-11 6.33 9 10-10

h \1.0 9 10-11\1.0 9 10-11 0.00238214

r 0.00030361 0.00719626 0.54591027

Table 3 General parameters for the two-locus model

Parameters Description Value

Mi Mean number of days

at life stage i

Megg = 11.9, Mlarva = 36.5,

Mpupa = 9.6,

Madult = 117

(Mtotal = 175 days

= 25 weeks)

Si Standard deviation of

number of days

at life stage i

Segg = 1.5, Slarva = 4.6,

Spupa = 1.2, Sadult = 15

bD Daily (finite) birth

rate

0.2242 (eggs per day

per female parent)

N0 Starting number of

(female) beetles

10,000

PI1 and PE1

I2 and PE2

Initial and

equilibrium

frequencies

For the one-locus model

see section ‘Theory and

calculation’

For the two-locus model,

see section ‘Theory

and calculation’

TRICLS Time remaining in

current life stage

Variable

Fig. 1 The overall model dynamics for each individual beetle

J Pest Sci (2012) 85:451–468 453

123

Author's personal copy

43

whether a generated uniformly distributed random

number B the survival probability SVxy, where xy is

the genotype of the beetle. This survival probability

depends on the concentration and exposure time of

the fumigant, as explained below.

(5) The number of eggs is determined by drawing

randomly from a Poisson distribution with mean

equal to the daily birth rate bD (Table 3), and then for

each egg separately and independently determined:

(i) the paternal genotype by drawing randomly from

a multinomial distribution based on the current

genotype frequencies of the population,

(ii) the genotype of the egg by drawing randomly

from a multinomial distribution based on the

maternal and paternal genotype and the (one- or

two-locus) offspring genotype table, as

explained below,

(iii) the sex of the egg; if a randomly generated

number[0.5, the sex of the egg is determined

as male, in which case this egg is removed from

the simulated population.

Assumptions regarding genotypes and resistance

in the one- and two-locus models

In the simplest case of a single locus, we consider two

alleles that govern the trait of interest; i.e. an individual’s

resistance to phosphine. The dominant allele is denoted by

A and the recessive allele by a. Due to the incompletely

recessive nature of the genes at each locus, PH3 resistance

is taken to be a qualitative trait that is present in three

forms:

• The recessive homozygous, or full expression of

resistance (aa or R);

• Heterozygous, or partial expression of resistance (Aa or

H) and

• The dominant homozygous, or fully susceptible (AA or

S).

The extension of the one-locus model includes two loci

for phosphine resistance. Dominant (susceptibility) alleles

are denoted by A and B, and recessive (resistance) alleles

by a and b. Instead of the three genotypes possible in the

one-locus model, the two-locus model includes nine pos-

sible genotypes:

• ss: with both loci homozygous susceptible (AA and BB)

• sh: with the first locus homozygous susceptible (AA)

and the second locus heterozygous (Bb)

• sr: with the first locus homozygous susceptible (AA)

and the second locus homozygous resistant (bb)

• and similarly hs, hh, hr, rs, rh, through to

• rr: with both loci homozygous resistant (aa and bb).

The model is also based on the following assumptions:

• Mating occurs randomly. For example, phosphine

resistance does not affect choice of mate.

• The grain food supply is non-limiting and its quality

does not affect the natural birth rate of the beetle.

• Fumigant concentrations do not vary with time or

location within the storage facility or silo. Variation

with time or space due to uptake or release of gas from

or into grain (sorption–desorption), diffusion, or leak-

age is negligible (Banks 1989).

• Temperature and relative humidity within the store

remains at 25°C and 70%, respectively, so that life

stage durations are constant.

Offspring genotype table

A novel computer-based method (Shi and Renton 2011)

was used to generate the offspring genotype table for both

the one- and two-locus cases. The one-locus table (Table 4)

has nine rows corresponding to the nine possible mating

combinations and three columns for the three possible

genotypes of offspring. The 4th row (Aa (h) 9 AA (s)) of

Table 4, for example, means that mating of a female parent

of genotype Aa with a male parent of genotype AA will

produce offspring of genotype AA with 50% probability,

genotype Aa with 50% probability and no offspring of

genotype aa.

For the two-locus model, the offspring genotype table is

an 81 9 9 table (Shi and Renton 2011). For example,

mating of AaBb 9 AaBb will produce the following pro-

portions of genotypes in the offspring,

AABB AaBB aaBB AABb AaBb aaBb AAbb Aabb aabb

ss hs rs sh hh rh sr hr rr

0.0625 0.125 0.0625 0.125 0.25 0.125 0.0625 0.125 0.0625

454 J Pest Sci (2012) 85:451–468

123

Author's personal copy

44

which corresponds to one row of the full offspring geno-

type table.

It should be pointed out that in our individual-based

model, the particular genotype of an individual female

beetle, together with the particular genotype of the ran-

domly selected male beetle with which it is assumed to

mate, determines which row in the table is used to calculate

the number of eggs of each genotype. However, a popu-

lation-based model is concerned with using the columns of

such a table to build a system of ordinary differential

equations. For example, the 1st column of Table 4 corre-

sponds to the offspring genotype AA. In a one-gene pop-

ulation-based model, assuming a constant birth rate b that

is independent of genotype and a time-dependent death

rate, aS(t) of the susceptible genotype this column would be

associated with the ordinary differential equation (Lilford

et al. 2009):

dNS=dt ¼ b

�NS 1:0NS=N þ 0:5NH=Nð ÞþNH 0:5NS=N þ 0:25NH=Nð Þ

�ÿ aSNS

ð1Þ

where Nx denotes the number of genotype x beetles and

N denotes the total number of beetles. Note that the four

numbers, 1.0, 0.5, 0.5 and 0.25, that appear in the above

equation, correspond to the four non-zero proportions in

the AA offspring genotype column in Table 4.

Aggregation

For the purpose of setting up a meaningful comparison

between the one- and two-locus models, we needed to

aggregate the nine genotypes from a two-locus model into

just three groups corresponding in some way to the three

genotypes in a one-locus model. The simple assumption for

constructing the one-locus model was that each genotype

corresponded to one of the observed phenotypic levels of

resistance: fully susceptible (homozygous susceptible),

weak (heterozygote) or strong (homozygous resistant). The

genetic linkage analysis undertaken by Schlipalius et al.

(2006, 2008) indicated that two loci, rph1 and rph2, confer

strong resistance and that rph1 is responsible for the weak

resistance phenotype. We therefore assumed that the strong

resistance phenotype (R or homozygous resistant in the

one-locus model) corresponds to the genotype in the two-

locus model that is homozygous resistant at both loci, i.e.

the genotype rr in this model. Full susceptibility to phos-

phine (S or homozygous susceptible in the one-locus

model) was assumed to correspond to when individuals in

the two-locus model are homozygous susceptible (s), at the

rph1 gene and either homozygous susceptible (s) or het-

erozygous (h) at the rph2 gene, i.e. the genotypes ss and sh

with the 1st and 2nd lowest survival rates. We then assume

the weak resistance (H or heterozygous in the one-locus

model) corresponds to all remaining genotypes. This

resulted in the nine genotypes model being classified into

three groups, denoted S2, H2 and R2:

• Susceptible group S2 = group ss and group sh

• Strongly resistant group R2 = group rr

• Weakly resistant group H2 = all other groups (sr, hs,

hh, hr, rs and rh).

These then corresponded directly to both the three

observed levels of resistance, fully susceptible, strong

and weak, and to the three genotypes in the one-locus

model, S, R and H. We next needed to set the survival rates

for the three genotypes in the one-locus model based on the

survival rates for the nine genotypes in the two-locus

model. Setting survival rates (SV) for R was straight

forward: SVR = SVrr. To set the survival rates for S and H

we used the mean value of the survival rates of the group

elements. To sum up,

SVS ¼ SVSS þ SVshð Þ=2;SVH ¼ SVsr þ SVhs þ SVhh þ SVhr þ SVrs þ SVrhð Þ=6;SVR ¼ SVrr

ð2Þ

Simulations for comparison

We compared the two models in three cases: under no

fumigation, under a high dose treatment (0.2 mg/l 9

8 days) and under a longer low dose treatment (0.01 mg/

l 9 14 days).

For the simulation in the absence of fumigation we

ran both of the two models for 350 days or 50 weeks,

each for six different sets of initial frequencies, to see if

the theoretical Hardy–Weinberg equilibrium frequen-

cies would be reached (see section ‘Hardy–Weinberg

principle and theoretical equilibrium proportions of

Table 4 The offspring genotype table giving the probabilities or

expected proportions of each offspring genotype resulting from each

possible mating combination of male and female parental genotypes

for the one-locus model

Female parent Male parent Offspring

AA (s) Aa (h) aa (r)

AA (s)9 AA (s) 1.0 0 0

Aa (h) 0.5 0.5 0

aa (r) 0 1.0 0

Aa (h)9 AA (s) 0.5 0.5 0

Aa (h) 0.25 0.5 0.25

aa (r) 0 0.5 0.5

aa (r)9 AA (s) 0 1.0 0

Aa (h) 0 0.5 0.5

aa (r) 0 0 1.0

J Pest Sci (2012) 85:451–468 455

123

Author's personal copy

45

genotypes’). The initial proportions of life stages were

assumed to be distributed evenly, that is 0.25 for each of

the four main life stages. We also tested other randomly

selected initial proportions of life stages and found that it

made no difference to results (these results not shown).

In preliminary test runs of the models for 350 days

starting from 10,000 female beetles, we found that our

computer would stop with a memory error when the female

population number reached about 8.6 million. To overcome

this difficulty, we used a strategy of re-setting the numbers

for each genotype group after every 175 days to make the

total number equal to 10,000, while keeping the current

genotype proportions unchanged.

For the simulations under fumigation, we set the initial

population to be 10,000 (female) beetles in total, with 25%

in each of the four main life stages. We assumed fumiga-

tion started immediately, simulated fumigation for the

appropriate time (8 or 14 days) and ran each of the two

models for 140 days (20 weeks) in total, enough time to

enable comparison.

Theory and calculation

In this section, we briefly describe the Hardy–Weinberg

equilibrium and how we compute the theoretical equilib-

rium proportions of genotypes for the simulations in the

absence of fumigation.

Hardy–Weinberg principle and theoretical equilibrium

proportions of genotypes

In the absence of fumigation and assuming no fitness dif-

ferences between genotypes, the simulation results should

tend towards the Hardy–Weinberg equilibrium for both of

the models. The Hardy–Weinberg principle states that both

allele and genotype frequencies in a population remain

constant. That is, they are in equilibrium from generation to

generation unless specific disturbing influences are intro-

duced. Disturbing influences include non-random mating,

selection, limited population size, ‘overlapping genera-

tions’, genetic drift and other factors (Stansfield 1991). It

should be pointed out that outside the lab, one or more of

these ‘disturbing influences’ are nearly always in effect,

which means an exact Hardy–Weinberg equilibrium is

unlikely to be found in nature. Genetic equilibrium is an

ideal that provides a baseline against which genetic change

can be measured.

We use the following randomly selected initial fre-

quencies of genotypes (denoted by PI2) for all simulations

of the two-locus model in this article:

The allelic proportions p, q, u, v and equilibrium fre-

quencies for the two-locus model, which we denote as PE2,

can be obtained using matrix multiplication (see Shi and

Renton 2011, for details):

p ¼ 0:5116; q ¼ 0:4884; u ¼ 0:5442; v ¼ 0:4558

ð4aÞPE2: ½0:0775; 0:1480; 0:0706; 0:1298; 0:2479;

0:1183; 0:0544; 0:1038; 0:0496�ð4bÞ

Note that the element for rr beetles, PE2(rr) is equal to

q2v2 = 0.0496.

According to the aggregation method, the initial fre-

quencies for the aggregated two-locus model correspond-

ing to those in (3) (denoted by PIA2) and its equilibrium

proportions (denoted by PEA2) are:

S2 H2 R2

PIA2: ½0:3104; 0:5326; 0:1570�S2 H2 R2

PEA2: ½0:2074; 0:7431; 0:0496� :ð5Þ

For the purposes of comparisons, the initial frequencies for

the one-locus model (denoted by PI1) should correspond to

PIA2 in (5). Then the proportions of alleles ‘A’ and ‘a’ are

p1 = 0.5767 and q1 = 0.4233, respectively, and the

equilibrium proportions, PE1 are:

S H R S H R

PE1: p21;�

2p1q1; q21�

¼ ½0:3326; 0:4882; 0:1792�ð6Þ

Note that the element for R beetles, PE1(R) is equal to

q21 ¼ 0:1792. In theory, the ratio between the equilibrium

frequency of strongly resistant beetles predicted by the

one-locus model, to that from the two-locus model, should

be PE1 Rð Þ=PE2 rrð Þ ¼ q21=q2v2. In this case, this would be

equal to 3.61, which is very far from one.

Natural birth rate

The maximum (instantaneous) intrinsic capacity for

increase (rm) of the lesser grain borer, measured as adult

ss hs rs sh hh rh sr hr rr

PI2 : ½0:2040; 0:1203; 0:0875; 0:1064; 0:0690; 0:0894; 0:0467; 0:1197; 0:1570� ð3Þ

456 J Pest Sci (2012) 85:451–468

123

Author's personal copy

46

female progeny/female/week, at 25°C and 70% r.h. is

estimated to be 0.4109 (Driscoll et al. 2000) and the mean

life duration of the beetle (Table 3) is 58 (immature

stages) ? 117 (adult stages) = 175 days or 25 weeks.

Therefore, the weekly mean death rate is d = 1/25 = 0.04

and the weekly (instantaneous) birth rate (Birch 1948) is

b = rm ? d = 0.4109 ? 0.04 = 0.4509. Finally, the

(natural) daily finite birth rate (eggs per day per female

parent) is

bD ¼ e0:4509=7 ¼ 1:5697=7 ¼ 0:2242 ð7Þ

Mortality under phosphine fumigation

Our individual-based simulation model required predic-

tions of finite daily survival at different concentrations for

all nine genotypes. Using data from a number of experi-

mental studies, we were able to construct empirical models

relating daily mortality/survival probability to genotype

and fumigant concentration. A brief description is provided

here; for full details see Shi et al. (2011).

Daglish (2004) determined mortality rates for phos-

phine-susceptible (strain QRD14—corresponding to geno-

type ss) and weak resistance (strain QRD369—rs)

phenotypes and their F1 progeny (QRD369 9 QRD14—

hs) over a range of concentrations at exposure times of 20,

48, 72 and 144 h. He used a Haber-type model, Cnt = k

(Bunce and Remillard 2003), where C (mg/l) is the con-

centration of phosphine and t (hour) is the exposure time,

to predict response to phosphine and the concentrations

needed to achieve 50% (LC50) and 99% (LC99) mortality at

certain times. But as both n and k in his equations vary with

both genotype and mortality level, it is not possible to

develop a Haber-type rule with which to successfully

extrapolate predicted mortalities between exposure sce-

narios (Bunce and Remillard 2003). Hence, we employed

the four-parameter probit model using a least squares

technique with a generalised inverse matrix approach (Shi

and Renton 2011) to refit his data:

Y ¼ aþ b1 logðtÞ þ b2 logðCÞ þ b3 logðtÞ logðCÞ; ð8Þ

where Y is the probit (the inverse cumulative distribution

function) value of mortality. We then obtained a model

predicting finite daily survival rate (FDSR) at different

concentrations for the ss, hs and rs genotypes.

Collins et al. (2002) observed mortalities under a range

of concentrations of phosphine at 48 h for susceptible

(strain QRD14—ss) and strongly resistant (strain

QRD569—rr) phenotypes and their combined F1 progeny

((569 9 14) ? (14 9 569)—hh). We fitted the two-

parameter probit model (Shi and Renton 2011)

Y ¼ aþ b logðCtÞ ð9Þ

to predict mortalities for each of ss and hh genotypes.

The experiments in Collins et al. (2005) were conducted

over a long period of time and the results were confirmed in

field trials and are the basis for the current rates used to

control resistant insects. Values derived from their

observed data included LT99.9—lethal time to achieve

99.9% mortality—for strain QRD569 (rr) exposed to a

series of fixed concentrations from 0.1 to 2.0 mg/l. We

tried the four-parameter (8), the three-parameter (dropping

the last term of 8), and the two-parameter (9) models to fit

several combinations of different parts of the two data sets

and obtained different predicted LT99.9 values. We com-

pared the predictions of the different models and found that

the deviations between the observed and predicted LT99.9

values were smallest when we used both data sets at C from

0.1 to 1.0 mg/l to fit the four-parameter probit model. The

close match between the observed and the predicted LT99.9

values (Table 5) provided verification for these fitted four-

parameter probit models of mortality, as did the fact that

the reported and predicted LC50 values were also very

close to one another (Table 6).

We still needed to construct a model predicting finite

daily survival at different concentrations for the remaining

sh, sr, hr and rh genotypes. As a step towards achieving

this, we first estimated the resistance factor for the first five

genotypes. The resistance factor of a genotype x for a given

fumigation duration is defined as the ratio of the PH3

concentration that achieves 50% mortality (LC50(x)) in a

sub-population of genotype x and the PH3 exposure con-

centration that achieves 50% mortality in a susceptible ss

(LC50(ss)) sub-population (FAO 1975). In other words, the

resistance factor for genotype x is yielded by

f ðxÞ ¼ LC50ðxÞ=LC50ðssÞ: ð10Þ

We estimated LC50 for 48 h exposure for the first five

genotypes using our fitted models and checked that they

were close to the previous estimates of Daglish (2004) and

Collins et al. (2002). The estimated LC50 values (Table 6)

were then used to estimate resistance factors for the five

genotypes.

Table 5 The LT99.9 values (days) reported by Collins et al. (2005) with the corresponding values predicted by the four-parameter probit models

for various phosphine doses (mg/l)

Dose 0.1 0.15 0.2 0.3 0.4 0.5 0.75 1.0

Reported 14.99 11.24 9.302 7.260 6.16 5.461 4.442 3.87

Predicted 14.02 12.74 8.509 7.144 6.55 5.628 4.233 3.74

J Pest Sci (2012) 85:451–468 457

123

Author's personal copy

47

We then estimated the resistance factors for the other

four genotypes (sh, sr, hr and rh) by making some basic

assumptions regarding genetic interactions. We assumed

that log-transformed resistance factors for the nine geno-

types can be expressed in terms of five parameters k, d1, d2,

s1 and s2 as per Table 7, where s1 and s2 represent the

strength of the rph1 and rph2 genes, respectively, d1 and d2represent the dominance of the rph1 and rph2 genes,

respectively, and k represents the synergism between the

two genes. We already had estimates for the factors f(ss),

f (hs), f(rs), f (hh) and f (rr). From experimental data

(Collins et al. 2002, 2005; Daglish 2004) we also knew

f(sr) is less than f(rs), and so s2\ s1. We therefore assumed

that s2 = 0.5s1 which gave an estimate for f(sr), and then

solved for the remaining unknowns in Table 7 to give

estimates for f(sh), f(rh) and f(hr). The resistance factors

for the nine genotypes are listed in Table 8 (Appendix 1).

Finally, we modelled survival rates for the other four

genotypes (sh, sr, hr and rh) using Eq. 9. We assumed that

the parameter b for these genotypes was the same as the

estimated parameter b for the hh genotype, b(hh). We

already had estimates for the resistance factors for t = 48 h

for each of the four genotypes. By multiplying these esti-

mates by LC50(ss) we obtained estimates for LC50 (see

Eq. 10) for each of the four genotypes. The value for

parameter a for each of the four genotypes could then be

obtained by direct substitution into Eq. 9.

At this point we had a model predicting survival rates at

a range of concentrations and times for each of the nine

genotypes (Fig. 2). For the hs, rs and rr genotypes this

model was of the form of Eq. 8 and for the other genotypes

it was of the form of Eq. 9. Note that in Fig. 2 the probit

value for the rr genotype at C = 1.0 mg/l is 2.102915 and

the survival rate is 0.998117 or about 99.81%, which

matches the observations from Collins et al. (2005).

There are two ways to estimate the FDSR under a PH3

treatment C 9 T at a fixed concentration C (mg/l) and a

range of times (1, 2, …, T days). Directly substituting the

fixed C value and the series of t values into the prediction

Eqs. 8 and 9 results in the cumulative survival rates (CSR).

The CSR for the final Tth day is the total survival rate

SVT(x) for genotype x for the full C 9 T treatment. One

way to estimate FDSR is based on the assumption that the

survival rate is the same each day during a fumigation

treatment, in which case the FDSR is SVs1(x) = [SVT(x)]1/

T. The estimated finite daily and total survival rates under

the two treatments of 0.01 mg/l 9 14 days and 0.2 mg/

l 9 8 days are listed in Tables 1 and 2. As the computed

values are accurate to 11 decimal places it is reasonable to

regard the survival rate\ (10-11) as zero or all dead. This

level of accuracy is more than enough for the numbers of

beetles that we simulated. The other way to estimate dif-

ferent FDSRs for each day is as follows: convert the

CSRi?1 and CSRi into the ith day’s daily survival rate

SVdi(x) by setting SVd1(x) = CSR1(x) and then letting

SVdiðxÞ ¼ CSRiþ1 xð Þ=CSRi xð Þ; i ¼ 1; 2; . . .; T ÿ 1: Note

that the total survival rate obtained by either of the two

ways will be the same for each genotype. However, since

our concern is with the results after fumigation, then we

chose to use the former approach with equal daily survival

rates, as it is simpler and faster to simulate. If we had also

been concerned with accurately representing on a daily

basis what happens during the fumigation process then we

could have used the latter approach instead.

Results and comparisons

No fumigation

All simulations using a range of initial frequencies resulted

in the statistically discrete analogue of the Hardy–

Weinberg equilibrium. As our main aim was to compare

the difference between the two models, we show the details

Table 6 LC50 values (mg/l) for 48 h exposure reported (R) by Daglish (2004) (marked by #D) and Collins et al. (2002) (marked by #C) and

estimated (E) by Eq. 8 or 9

Strain QRD14(1) QRD369 9 14 Comb F1 QRD369 QRD569

Genotype ss hs hh rs rr

LC50 value

R #C/#D: 0.0017 #D: 0.0042 #C: 0.00548 #D: 0.0520 #C: 1.0250a

E 0.0017 0.0040 0.00659 0.0518 1.0238a

a The observed and the estimated LC50 values for rr genotype obtained using Collins et al. (2002) data with the model (9). The predicted LC50

value obtained using the combined data sets at C from 0.1 to 1.0 mg/l with the model (8) was 0.9016

Table 7 The expressions of log-transformed resistance factors

f (x) for genotype x in terms of five parameters k, d1, d2, s1 and s2

1st

Gene

2nd Gene

s h r

s 0 d2s2 [=ln f (sh)] s2 [=ln f (sr)]

h d1s1 d2s2 ? d1s1 ? k(d1s1)

(d2s2)

d1s1 ? s2 ? k (d1s1)s2[=ln f (hr)]

r s1 d2s2 ? s1 ? k s1(d2s2)

[=ln f (rh)]

s2 ? s1 ? k s1 s2

458 J Pest Sci (2012) 85:451–468

123

Author's personal copy

48

only for the results obtained using the set in (3) and the set

in (5).

The results for the two-locus model are plotted in Fig. 3

and for the one-locus model in Fig. 4a. It can be seen from

the two figures that the frequencies of nine genotypes of

two loci and those of three genotypes of one locus statis-

tically approach equilibrium. These simulations confirm

the statistical presence of the discrete analogue of the

Hardy–Weinberg principle.

Themeans and standard deviations (std) over 26–50 weeks

(second generation) for the proportions of the nine genotypes

in the two-locus model are:

The sum of squared deviations from these means to the the-

oretical equilibrium proportions PE2 is 0.00024 which is very

small. For one locus, the simulated proportion means and

standard deviations over 26–50 weeks are as follows.

mean : ½0:3284; 0:4916; 0:1800�std : ½0:0016; 0:0012; 0:0012� ð12Þ

It can be seen again from (12) and (6) that the mean

values are very close to the equilibrium values as the sum

of squared deviations from this mean to PE1 is 0.00003

which is very small, and the std values are also very small,

indicating approximate equilibrium.

Comparing Fig. 4a and b, we see that there is a big

difference between the results of the two models. The

proportions of H are decreasing from the beginning to the

20th week then asymptotically and statistically approach an

equilibrium proportion, while the proportions of H2 are

increasing for a longer period then approach a different

equilibrium proportion. The proportions of S or R are

increasing and then tend to equilibrium but those of S2 or

R2 are decreasing first and approach different equilibriums.

Under 0.2 mg/l 9 8 days treatment

Figures 5 and 6 show that after 8 days fumigation only 22

female beetles (1.4%) in the two-locus model and 16

(1.1%) in the one-locus model remain alive, all of which

are strongly resistant (proportion of rr = 100%). After

fumigation, the proportion of genotype rr remains

unchanged at 100% since there is no possibility of other

alleles being reintroduced. The numbers of rr and R beetles

predicted by the two- and one-locus models at the end of

simulation are 353 and 425, respectively. Results from the

two models are thus very similar. Figure 11a, b in

Appendix 1 shows the non-aggregated results from the

original two-locus model.

Under 0.01 mg/l 9 14 days treatment

Figures 7 and 8 show the proportions and numbers of the

three genotypes in the one- and aggregated two-locus cases

under the low dose fumigation treatment (Fig. 12a, b in

Appendix 1 for the non-aggregated results from the origi-

nal two-locus model). For this treatment, the models give

(a) (b)

Fig. 2 Survival rates by substituting t = 24 h and a range of C values for genotypes a ss, sh, sr, hs and hh and b hh, hr, rs, rh and rr. Note

different x-axis scales. The survival curve for genotype hh is shown in (a, b) and for the purpose of comparison

ss hs rs sh hh rh sr hr rr

mean: ½0:0854 0:1455 0:0644 0:1268 0:2495 0:1196 0:0475 0:1035 0:0578�std: ½0:0036 0:0005 0:0026 0:0013 0:0015 0:0006 0:0031 0:0008 0:0032�

ð11Þ

J Pest Sci (2012) 85:451–468 459

123

Author's personal copy

49

different results. The proportion of R beetles in the one-

locus model is 100% after this low dose treatment while the

proportion of R2 beetles in the aggregated two-locus model

is about 63%, with the other 37% consisting of H2 beetles.

Also there are differences in total predicted numbers

between the two models.

Could the one-locus model be adapted to give similar

results to the two-locus model?

The one-locus model can indeed be adapted to give similar

results to the two-locus model for this low dose fumigation

in terms of predicted numbers. If we assume a greatly

raised survival rate of H beetle by letting SVH = 0.875

then very similar results are obtained by the two models:

883 and 936 beetles for H and H2, respectively, at the end

of fumigation (Fig. 9). The value of SVH = 0.875 under

the treatment 0.01 mg/l 9 14 days was chosen by

considering the fact that the total surviving number of H2

beetles depends on the initial frequencies and the various

genotypes that make up H2. In fact, after 14 days simulated

treatment, four genotypes, hr, rs, rh and rr, remained, of

which three hr, rs and rh are members of H2. The daily

survival rates, SV (see Table 1), and initial frequencies, P

(see Eq. 3), for these two genotypes are, respectively,

SV = [0.7558, 0.9470, 0.9620] and P = [0.1197, 0.0875,

0.0894]. Then the weighted average survival rate is

SVH ¼�0:7558ð0:1197Þ

þ0:9470ð0:0875Þ þ 0:9620ð0:0894Þ�= 0:1197þ 0:0875þ 0:0894½ � � 0:875

ð13Þ

Note that the rate 0.875 is far from the original 0.556 where

we set SVH to be the mean of the rates of the members of

H2 (see Eq. 2).

This method of resetting the value of SVH was able to

make the two models produce similar results during the

fumigation; both in terms of the number of beetles (com-

pare Fig. 9a and b) and the proportions (compare Figs. 10a

and 7b). This similarity is also evident in terms of total

numbers at the end of the simulation, but the numbers of

beetles in each genotype group were still very different

(Figs. 10b, 8b). In fact, there were none of genotype S2,

19,736 of H2, and 22,949 of R2 (42,685 in total) for the

aggregated two-locus model, but 1,341 of S, 12,424 of

H and 27,812 of R beetles (41,577 in total) for the one-

locus model after 140 days.

Discussion and conclusion

The simulation results showed that the two models produce

similar results for the short duration and high concentration

treatment. But they showed very different behaviours in the

Fig. 3 Statistically discrete analogue of the Hardy–Weinberg equi-

librium for the two-locus model

(a) (b)

Fig. 4 Proportions of genotypes over time in the absence of fumigation for the a one-locus model and b aggregated two-locus model

460 J Pest Sci (2012) 85:451–468

123

Author's personal copy

50

absence of fumigation and under the long duration with

low concentration treatment. We conclude that the one-

locus model is oversimplified and using such a model to

analyse and develop potential resistance management

strategies is likely to be less useful than using a more

realistic two-locus model. Thus, if a one-locus model is

used to develop possible control strategies, we would run a

risk of misidentifying optimal strategies, which could lead

to infestation even when the model predicts control of the

species.

As the treatment 0.2 mg/l 9 8 days was designed to kill

99.2% of strongly resistant beetles and to eradicate the

other genotypes of beetles (their predicted survival rates

after 8 days fumigation are all less than 1 9 10-11), we

would expect that the only group of beetles that could

survive at this high dose are the R2 = rr group in the two-

locus model and the R group in the one-locus model. Since

the initial proportions of R2 and R were the same, it was not

surprising that the two models showed very similar

behaviour in this case, as we observed.

In the absence of fumigation, both models tend towards

the Hardy–Weinberg equilibrium, but the one-locus model

exaggerates the proportion of strongly resistant beetles, on

average by about 3.6 times, compared to the two-locus

model. This matched the theoretical prediction based on

Eqs. 4a, 4b and 6, that the ratio between the equilibrium

frequency of strongly resistant beetles predicted by the

one-locus model, to that from the two-locus model, should

be PE1 Rð Þ=PE2 rrð Þ ¼ q21=q2v2 ¼ 0:1792=0:0496 ¼ 3:61;

which is much greater than one. Our individual-based

simulations of the population dynamic of R. dominica in

the absence of fumigation matched the statistically discrete

(a) (b)

Fig. 5 The proportions of beetles for each genotype under 0.2 mg/l 9 8 days fumigation treatment for the a one-locus model and b aggregated

two-locus model

(a) (b)

Fig. 6 The numbers of beetles for each genotype under 0.2 mg/l 9 8 days fumigation treatment for the a one-locus model and b aggregated

two-locus model. The numbers of S and H are zero after 1 day, so not visible in the plots

J Pest Sci (2012) 85:451–468 461

123

Author's personal copy

51

analogue of the Hardy–Weinberg equilibrium. Lilford et al.

(2009) confirmed the continuous analogue of the Hardy–

Weinberg equilibrium with their simulation model of the

population dynamics of R. dominica in the absence of

fumigation. Their plotted results did not show curves with

small perturbations, like ours, but horizontal lines in the

‘stable’ period. In fact, their models are more idealised

deterministic continuous models using differential equa-

tions while ours are stochastic discrete individual-based

models representing more aspects of biological reality.

The 0.01 mg/l 9 14 days treatment looks to represent a

dose lower than would be found in reality. However, we

included it for two reasons. It is important for theoretical

investigation of the differences between the models. Even

more importantly, it is likely to represent very relevant

aspects of real situations such as when the silo is not sealed

properly and leakage of PH3 happens, when poor circula-

tion of PH3 results in low doses in some parts of the silo at

some times, or when a malfunction in measurement

instruments results in incorrect maintenance of a full dose.

Even in a well-sealed silo, the PH3 doses are likely to be

much lower in the corners than those at other places. Hence

this low dose is still meaningful in practice. A model that

works correctly for such low doses is important for future

studies such as developing a spatially explicit 3-dimen-

sional model to study the evolution of resistance within a

spatially heterogeneous silo, which is much more realistic

than an idealised spatially homogenous silo. The effects of

partial low doses is likely to be especially important in

cases like this where effective resistance is ‘polygenic’,

depending on the presence of alleles at more than one locus

(Renton et al. 2011).

(a) (b)

Fig. 7 The proportions of beetles for each genotype under 0.01 mg/l 9 14 days fumigation treatment for the a one-locus model and

b aggregated two-locus model

(a) (b)

Fig. 8 The numbers of beetles for each genotype during and following a 0.01 mg/l 9 14 days fumigation treatment for the a one-locus model

and b aggregated two-locus model. Note that when numbers reach zero they are not shown because of the log scale

462 J Pest Sci (2012) 85:451–468

123

Author's personal copy

52

The results obtained using the one-locus model to sim-

ulate the low dose treatment would mislead us into a wrong

conclusion that, after fumigation at such a low concentra-

tion, strongly resistant beetles would make up the entire

population of 1,571 beetles at the end of fumigation and

26,533 beetles after 20 weeks. However, the more realistic

two-locus model predicts that the population of all beetles

is not reduced to such a low level as predicted by the one-

locus model in Fig. 8a. There is not only the same number

of R2 beetles as that of R beetles but also quite a substantial

number (939) of H2 beetles at the end of fumigation,

meaning that the proportion of R2 beetles is just 63%. After

20 weeks there are 19,736 H2 and 22,949 R2 beetles, or

42,685 in total, and thus the proportion of R2 beetles is only

about 53.8%. To sum up, the one-locus model

overestimated the proportion of strong-resistant individuals

within the population and underestimated the total popu-

lation numbers compared to the two-locus model.

If we set SVH = 0.875 under the treatment 0.01 mg/

l 9 14 days for the one-locus model then very similar

results to those from the aggregated two-locus model

during the fumigation period were obtained. This similarity

could lead one to attempt to use a simpler one-locus model.

However, after fumigation the proportion of genotypes is

still different, with a much larger pool of S beetles present

in the population at the end of the simulation. This follows

because the offspring proportions for the nine genotypes

cannot be simply aggregated to those for three genotypes;

the genotypes of generations from the two-locus model

after fumigation are only hh, hr, rh and rr all belonging to

(a) (b)

Fig. 9 The numbers of beetles during fumigation over 14 days under the 0.01 mg/l 9 14 days treatment predicted by a the one-locus model, re-

setting SVH = 0.875 and b the two-locus model, for each aggregated genotype

(a) (b)

Fig. 10 The proportions (a) and total numbers (b) of each genotype changing over time during the 0.01 mg/l 9 14 days fumigation and then a

subsequent period of 126 days of no fumigation, as predicted by the one-locus model after setting SVH = 0.875

J Pest Sci (2012) 85:451–468 463

123

Author's personal copy

53

H2 and R2 groups but those from the one-locus model are

possible to be S (see Table 4; Fig. 10). There is also the

obvious problem that the survival rate SVH = 0.875 is only

valid in this particular case for this particular combination

of genotypes, and would have to be recalculated in any

other case. The one-locus model is too simple to accurately

represent collected information such as the initial fre-

quencies and the probabilities of offspring genotypes from

two or more loci.

The fact that the two-locus model appears to be better

than the one-locus model raises the question of whether a

model with more than two loci would perform even better

or be even more realistic? In previously published resis-

tance modelling research ‘survivorship was not explicitly

included in the model because adequate data were not

available’ (Hagstrum and Flinn 1990), and thus a simple

single gene model was used, following the Occam’s

(or Ockham’s) razor principle, ‘Simpler explanations are,

other things being equal, generally better than more com-

plex ones’ (Gernert 2007). However, the research by Col-

lins et al. (2002) and Schlipalius et al. (2002) provides

strong evidence that more than one locus is involved in

phosphine resistance, at least in the populations they

investigated. They have thus followed Occam’s principle in

concluding that the best explanation for the observed data

is that resistance is conferred by two genes. As we have

now shown that the two-locus model gives different results

to a one-locus model, we conclude it is important that

models be based on this ‘best explanation’ of two loci. It is

possible that future studies may eventually indicate that

more than two loci are actually involved, in which case a

similar study to this one should be conducted to test

whether including more than two loci in a model makes a

difference to predictions and thus whether the additional

complexity is required. It is worth noting that a three-locus

model would include 27 possible genotypes (Shi and

Renton 2011) and models of mortality under different

concentrations and exposure durations would have to be

constructed for all these genotypes, involving substantial

additional experimental and modelling work. Therefore,

until experimental studies provide strong evidence that

more than two loci are involved, a two-locus model is the

best approach.

Another consideration that may be important is the

relative fitness of the different genotypes. If resistance to

phosphine carries a fitness penalty (lower fecundity or

reduced mating opportunities, for example), then resistant

genotypes will decrease in frequency in the absence of

fumigation and increase more slowly in the presence of

fumigation. If the fitness penalty is large enough then this

effect may be important and including fitness in the model

could make meaningful differences to model predictions.

Pimentel et al. (2007) found a correlation (r2 = 0.620)

between LC50 and r (a good measure of fitness) for a

number of phosphine-resistant population samples of R.

dominica and presented a linear model predicting r from

resistance factor. However, they did not provide any

information on the frequencies of resistance genes or

genotypes in their samples. It is therefore not possible to

directly assign a value for fitness to either major resistance

gene based on their work or to directly assign a fitness

value to each of the genotypes. Furthermore, the range of

resistance levels between their samples was much smaller

than the range of resistance levels between the genotypes in

our model and in the experimental studies on which it is

based, and therefore any attempt to indirectly assign fitness

values to our genotypes based on their model would

involve too much extrapolation. We also note that Schli-

palius et al. (2008) concluded that strongly phosphine-

resistant (two major genes) R. dominica suffer no fitness

disadvantage, after a population of resistant–susceptible

cross was reared in the absence of phosphine selection, and

the frequencies of resistant, susceptible and heterozygote

individuals were determined after 5, 15 and 20 generations.

The effects of fitness costs would be an important issue to

investigate in future, and if reasonable values for fitness

can be ascribed to the full range of genotypes, then fitness

can be readily accommodated in the model. However, at

this stage, conclusive information on fitness changes linked

to specific resistance genes is lacking, and the available

information appears to be somewhat contradictory.

Exploratory investigations showed that incorporating the

effect of fitness cost would result in lower populations

under no fumigation, but would not affect the patterns of

differences between the two models, which is the main

subject of this investigation. We therefore believe that an

assumption of no fitness cost is reasonable for this study.

Population- and individual-based approaches to model-

ling evolution of resistance have their various strengths and

weaknesses (Renton 2009). The former are quicker to run

and therefore investigate different hypotheses, because

they simply solve differential equations and do not require

a lot of computer memory capacity. However, our indi-

vidual-based simulation approach allows more aspects of

the individual variability and biological reality to be

included. The fact that our models run on a daily time step

also allows them to capture real conditions in more detail

and obtain more precise results than if they ran on a longer

weekly time step. In the real world, conditions are more

likely to change day by day rather than week by week.

Note that our individual-based models could be adapted for

other insect pests of stored grain quite easily, and the time

step can be adjusted to any appropriate value. Population-

based approaches are weak when the populations become

small and random difference between individuals cannot be

neglected. The advantages of individual-based approaches

464 J Pest Sci (2012) 85:451–468

123

Author's personal copy

54

are that they can be relatively easily adapted to incorporate

new biological attributes that we wish to investigate, such

as different genetics, different initial frequencies of geno-

types and/or different proportions of life stages, spatial

location, movement, interactions between individual

insects, etc.

Simulation modelling always relies on assumptions or

sub-models regarding processes at more basic levels of

organisation, and these assumptions or sub-models will

never be perfectly accurate representations of reality, par-

ticularly for a complex system that depends on a number of

underlying processes, such as the evolution of resistance to

pesticides. Nonetheless, it is valuable to develop a model,

as we have here, that represents and synthesizes the best

current understanding of the system. Such models can be

used to identify knowledge gaps and gain valuable insights

into the system dynamics and relationships through ‘virtual

experiments’ (Peck 2004, 2008). Further quantitative val-

idation of the model and its underlying assumptions and

sub-models would improve confidence in the quantitative

precision and accuracy of the model. However, for com-

plex systems, such as that being studied here, high levels of

quantitative precision may not be feasible and more qual-

itative relationships and patterns are likely to be of more

interest in any case (Peck 2004). In fact, the aim and value

of individual-based models are usually in gaining insights

into qualitative relationships and interactions, rather than

simple precise quantitative prediction (Grimm and Rails-

back 2004). While high levels of precision and accuracy

may seem to have intrinsic value, they may actually often

have little or no effect on more qualitative conclusions

(Renton 2011). The main purpose of this study was not to

show that the model gives highly accurate quantitative

predictions, but to test whether the two-locus model gives

different results to a simpler one-locus model. The con-

clusion that the number of loci represented does make a

difference is ‘robust’ in that it is unlikely to depend on any

particular model assumption. This is supported by the

failure of our attempt to make the models equivalent by

resetting the survival rate SVH. Furthermore, we believe

the current model represents the important processes

underlying resistance evolution with enough accuracy to

have confidence in the general qualitative patterns pre-

dicted by the model, and thus inform management of the

important issue of phosphine resistance, and also help

prioritize future experimental investigations that may fur-

ther improve our understanding and the model itself. We

thus plan to use the model to investigate the impact that

factors such as the initial proportions of different genotypes

and life stages; the dose and duration of fumigation and the

spatial and temporal homogeneity of dose within a silo will

have on the efficacy of phosphine fumigation and the

evolution of resistance.

Acknowledgements The authors would like to acknowledge the

support of the Australian Government’s Cooperative Research Cen-

tres Program. We also thank Rob Emery and Yonglin Ren and the

GRDC for their great help in provision of raw data and information

about beetle life cycles and silo fumigation.

Appendix 1

Figures for the original two-locus model

See Figs. 11 and 12.

(a) (b)

Fig. 11 The population of beetles for each genotype under 0.2 mg/l 9 8 days fumigation treatment for the original two-locus model. a The

weekly average proportions and b daily numbers

J Pest Sci (2012) 85:451–468 465

123

Author's personal copy

55

Table for resistance factors

See Table 8.

Appendix 2

Log-normal distribution as a model of time

within a life stage

Log-normal distribution is a continuous distribution in

which the logarithm of a variable has a normal distribution

(Limpert et al. 2001). It is appropriate for modelling the

time an individual spends within a given life stage because

it is bounded below at zero, and because it can be

parameterized to have a non-zero median and to simulta-

neously give a restricted range of likely values that does

not necessarily include values close to zero (unlike the

Weibull distribution for example).

The probability density function (pdf) and cumulative

distribution function (cdf) for the log-normal distribution

are, respectively,

pdf:

f ðt[ 0Þ ¼ 1

r

ffiffiffiffiffiffi2p

pÿ �texp ÿðln t ÿ lÞ2=2r2

� � ð14Þ

cdf:

FðtÞ ¼ 1

21þ erf

ln t ÿ l

r

ffiffiffi2

p� �� �

;

erfðxÞ ¼ 1ffiffiffip

pZx

0

expðÿu2Þ du

ð15Þ

where l and r are the mean and std of the corresponding

normal distribution, respectively, and erf(x) is the com-

plementary error function.

Given the mean and std values of a sample L from a ran-

domvariableTwith a log-normal distribution,ML and SL, the

parameters l and r can be estimated as follows. The

expectation E(T) and variation Var(T) are the estimates of

ML and (SL)2, respectively, andwe knowE(T) andVar(T) are

ML � EðTÞ ¼ expðlþ r2=2Þ; ð16Þ

S2L � VarðTÞ ¼ ðexpðr2Þ ÿ 1Þ½EðTÞ�2: ð17Þ

Substituting (16) into (17) we have

expðr2Þ ÿ 1 ¼ VarðTÞ=½EðTÞ�2 or

r2 ¼ ln½VarðTÞ=EðTÞ2 þ 1� ð18Þ

Solving Eqs. 16 and 18 yields

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln½VarðTÞ=EðTÞ2 þ 1�

qand

l ¼ ln½EðTÞ= expðr2=2Þ�ð19Þ

Thus, given the expected mean and standard deviation

of the times spent within different life stages by different

(a) (b)

Fig. 12 The population of beetles for each genotype under 0.01 mg/l 9 14 days fumigation treatment for the original two-locus model. a The

weekly average proportions and b daily numbers

Table 8 The resistance factors for the nine genotypes of the two-

locus model (LC50 at exposure time 48 h)

1st Gene 2nd Gene

s h r

s 1 (reference) f(sh) = 1.2537 f(sr) = 5.5307

h f(hs) = 2.4706 f(hh) = 3.2235 f(hr) = 18.4839

r f(rs) = 30.5882 f(rh) = 44.6005 f(rr) = 602.24

466 J Pest Sci (2012) 85:451–468

123

Author's personal copy

56

individuals, we have a simple method to calculate the

parameters of the log-normal distribution used to

stochastically generate the actual time spent within a

given life stage by a given individual.

For model verification, we used the Python built-in

function lognormvariate(l,r) to generate 10,000 random

numbers from log-normal distribution with mean ML and

std SL for the life duration of each stage of the beetle. The

results are listed in Table 9.

References

Andrewartha HG, Birch LC (1982) Selections from ‘‘the distribution

and abundance of animals’’. The University of Chicago

Arbogast RT (1991) Beetles: Coleoptera. In: Gorham JR (ed) Ecology

and management of food-industry pests. Association of Official

Analytical Chemists, Arlington, pp 131–176

Baldassari N, Martini A, Cavicchi S, Baronio P (2005) Effects of low

temperatures on adult survival and reproduction of Rhyzopertha

dominica. Bull Insectol 58(2):131–134

Banks HJ (1989) Behaviour of gases in grain storages. In: Fumigation

and controlled atmosphere storage of grain. Proceedings of an

international conference held at Singapore, pp 96–107

Beckett SJ, Longstaff BC, Evans DE (1994) A Comparison of the

demography of four major stored grain Coleopteran pests and its

implications for pest management. In: Proceedings of the 6th

international conference on stored-product protection, CAB,

Wallingford, Canberra, Australia, vol 1, pp 491–497

Birch LC (1945) The influence of temperature on the development of

the different stages of Calandra oryzae L. and Rhizopertha

dominica Fab. (Coleoptera). Aust J Exp BiolMed Sci 23(1):29–35

Birch LC (1948) The intrinsic rate of increase of an insect population.

J Anim Ecol 17:15–26

Birch LC (1953) Experimental background to the study of the

distribution and abundance of insects: I. The influence of

temperature, moisture and food on the innate capacity for

increase of three grain beetles. Ecology 34(5):698–711

Bunce NJ, Remillard RB (2003) Haber’s rule: the search for

quantitative relationships in toxicology. Human Ecol Risk

Assess 9(6):973–985

Collins PJ (1998) Resistance to grain protectants and fumigants in

insect pests of stored products in Australia. In: Banks HJ et al.

(eds) Proceedings of the first australian postharvest technical

conference, Canberra, Australia, pp 55–57

Collins PJ, Daglish GJ, Nayak MK, Ebert PR, Schlipalius DI, Chen

W, Pavic H, Lambkin TM, Kopittke RA, Bridgeman BW (2000)

Combating resistance to phosphine in Australia. In: Proceedings

of the international conference for controlled atmosphere and

fumigation in stored products, Fresno CA, pp 593–607

Collins PJ, Daglish GJ, Bengston M, Lambkin TM, Pavic H (2002)

Genetics of resistance to phosphine in Rhyzopertha dominica

(Coleoptera: Bostrichidae). J Econ Entomol 95(4):862–869

Collins PJ, Daglish GJ, Pavic H, Kopittke RA (2005) Response of

mixed-age cultures of phosphine resistant and susceptible strains

of lesser grain borer, Rhyzopertha dominica, to phosphine at a

range of concentrations and exposure periods. J Stored Prod Res

41:373–385

Daglish GJ (2004) Effect of exposure period on degree of dominance

of phosphine resistance in adults of Rhyzopertha dominica

(Coleoptera: Bostrychidae) and Sitophilus oryzae (Coleoptera:

Curculionidae). Pest Manag Sci 60(8):822–826

Driscoll R, Longstaff BC, Beckett S (2000) Prediction of insect

populations in grain storage. J Stored Prod Res 36:131–151

FAO (1975) Recommended methods for the detection and measure-

ment of resistance of agricultural pests to pesticides: tentative

method for adults of some major species of stored cereals with

methyl bromide and phosphine. FAO Method No 16, FAO Plant

Prot Bull 23:12–25

Gernert D (2007) Ockham’s razor and its improper use. J Sci Explor

21(1):135–140

Grimm V, Railsback SF (2004) Individual-based modeling and

ecology. Princeton University Press, Princeton

Hagstrum DW, Flinn PW (1990) Simulations comparing insect

species differences in response to wheat storage conditions and

management practices. J Econ Entomol 83(6):2469–2475

Hagstrum DW, Subramanyam B (2009) Stored-product insect source.

AACC International, St Paul

Herron GA (1990) Resistance to grain protectants and phosphine in

coleopterous pests of grain stored on farms in New South Wales.

Aust J Entomol 29:183–189

Lilford K (2009) Mathematical modelling of genetic resistance in

stored-grain pests. Honours Thesis, Queensland University of

Technology

Lilford K, Fulford G, Ridley A, Schlipalius DI (2009) Fumigation of

stored grain insects a two locus model of phosphine resistance.

In: 18th World IMACS/MODSIM Congress, Cairns, Australia

Limpert E, Stahel WA, Abbt M (2001) Log-normal distributions

across the sciences: keys and clues: on the charms of statistics,

and how mechanical models resembling gambling machines

offer a link to a handy way to characterize log-normal

distributions, which can provide deeper insight into variability

and probability—normal or log-normal: that is the question.

Bioscience 51(5):341–352

Longstaff BC (1999) An experimental and modelling study of the

demographic performance of Rhyzopertha dominica (F) I:

development rate. J Stored Prod Res 35:89–98

Matin ASM, Hooper GHS (1974) Susceptibility of Rhyzopertha

dominica to ionizing radiation. J Stored Prod Res 10:199–207

Peck SL (2004) Simulation as experiment: a philosophical reassessment

for biological modelling. Trends Ecol Evol 19(10):530–534

Peck SL (2008) The hermeneutics of ecological simulation. Biol

Philos 23(3):383–402

Pimentel MAG, Faroni LA, Totola MR, Guedes RNC (2007)

Phosphine resistance, respiration rate and fitness consequences

in stored-product insects. Pest Manag Sci 63:876–881

Rees D (2004) Insects of stored products. CSIRO, Collingwood

Renton M (2009) The weeds fight back: Individual-based simulation

of evolution of polygenic resistance to herbicides. In: Anderssen

RS, Braddock RD, Newham LTH (eds) 18th world IMACS

congress and MODSIM09 international congress on modelling

Table 9 A generated sample of random numbers from log-normal

distribution with mean ML & E(T) and (std SL)2& Var(T) for the

life duration of each stage of the beetle and the sample mean and std

with minimum and maximum values of the sample (sample

size = 10,000)

Stage Sample L E(T) Var(T)

Mean std min(L) max(L)

Egg 11.9135 1.5083 7.3603 19.7243 11.9 1.5

Larva 36.4324 4.6287 21.2575 55.5210 36.5 4.6

Pupa 9.6029 1.2005 5.9273 15.5685 9.6 1.2

Adult 116.9519 14.9946 70.6629 185.2574 117.0 15.0

J Pest Sci (2012) 85:451–468 467

123

Author's personal copy

57

and simulation. MSSANZ and IMACS, Cairns, Australia,

pp 574–580

Renton M (2011) How much detail and accuracy is required in plant

growth sub-models to address questions about optimal manage-

ment strategies in agricultural systems? AoB Plants. doi:

10.1093/aobpla/plr006

Renton M, Diggle A, Manalil S, Powles S (2011) Does cutting

herbicide rates threaten the sustainability of weed management

in cropping systems? J Theor Biol 283(1):14–27

Schlipalius DI, Cheng Q, Reilly PE, Collins PJ, Ebert PR (2002)

Genetic linkage analysis of the lesser grain borer Rhyzopertha

dominica identifies two loci that confer high-level resistance to

the fumigant phosphine. Genetics 161:773–782

Schlipalius DI, Collins PJ, Mau Y, Ebert PR (2006) New tools for

management of phosphine resistance. Outlooks Pest Manag

17:52–56

Schlipalius DI, Chen W, Collins PJ, Nguyen T, Reilly PEB, Ebert PR

(2008) Gene interactions constrain the course of evolution of

phosphine resistance in the lesser grain borer, Rhyzopertha

dominica. Heredity 100:506–516

Schwardt HH (1933) Life history of the lesser grain borer. J Kansas

Entomol Soc 6(2):61–66

Shi M, Renton M (2011) Numerical algorithms for estimation and

calculation of parameters in modelling pest population dynamics

and evolution of resistance in modelling pest population

dynamics and evolution of resistance. Math Biosci 233(2):77–89

Shi M, Renton M, Collins PJ (2011) Mortality estimation for

individual-based simulations of phosphine resistance in lesser

grain borer (Rhyzopertha dominica). In: Chan F, Marinova D,

Anderssen RS (eds) MODSIM2011, 19th international congress

on modelling and simulation. Modelling and Simulation Society

of Australia and New Zealand, December 2011, pp 352–358,

http://www.mssanz.org.au/modsim2011/A3/shi.pdf

Sinclair ER, Alder J (1985) Development of a computer simulation

model of stored product insect populations on grain farms. Agric

Syst 18:95–113

Stansfield WD (1991) Schaum’s outline of theory and problems of

genetics, 3rd edn. McGraw-Hill, New York

Tabashnik BE, Croft BA (1982) Managing pesticide resistance in

crop-arthropod complexes: interactions between biological and

operational factors. Environ Entomol 11:1137–1144

468 J Pest Sci (2012) 85:451–468

123

Author's personal copy

58

Individual-based modelling of the efficacy of fumigation tactics to control lesser

grain borer (Rhyzopertha dominica) in stored grain

Mingren Shi a,c,*, Patrick J. Collins c,e, James Ridsdill-Smith b,c,d, Michael Renton a,c,d

aM084, School of Plant Biology, FNAS, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, AustraliabM092, School of Animal Biology, FNAS, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, AustraliacCooperative Research Centre for National Plant Biosecurity, AustraliadCSIRO Ecosystem Sciences, Underwood Avenue, Floreat, WA 6014, AustraliaeAgri-Science Queensland, Department of Agriculture, Fisheries and Forestry, Ecosciences Precinct, GPO Box 267, Brisbane, QLD 4001, Australia

a r t i c l e i n f o

Article history:

Accepted 8 June 2012

Keywords:

Individual-based model

Two-locus simulation

Phosphine resistance

Lesser grain borer

Management tactics

a b s t r a c t

Increasing resistance to phosphine (PH3) in insect pests, including lesser grain borer (Rhyzopertha

dominica) has become a critical issue, and development of effective and sustainable strategies to manage

resistance is crucial. In practice, the same grain store may be fumigated multiple times, but usually for

the same exposure period and concentration. Simulating a single fumigation allows us to look more

closely at the effects of this standard treatment.

We used an individual-based, two-locus model to investigate three key questions about the use of

phosphine fumigant in relation to the development of PH3 resistance. First, which is more effective for

insect control; long exposure time with a low concentration or short exposure period with a high

concentration? Our results showed that extending exposure duration is a much more efficient control

tactic than increasing the phosphine concentration. Second, how long should the fumigation period be

extended to deal with higher frequencies of resistant insects in the grain? Our results indicated that if the

original frequency of resistant insects is increased n times, then the fumigation needs to be extended, at

most, n days to achieve the same level of insect control. The third question is how does the presence of

varying numbers of insects inside grain storages impact the effectiveness of phosphine fumigation? We

found that, for a given fumigation, as the initial population number was increased, the final survival of

resistant insects increased proportionally. To control initial populations of insects that were n times

larger, it was necessary to increase the fumigation time by about n days. Our results indicate that, in a 2-

gene mediated resistance where dilution of resistance gene frequencies through immigration of

susceptibles has greater effect, extending fumigation times to reduce survival of homozygous resistant

insects will have a significant impact on delaying the development of resistance.

! 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The safe storage and supply of cereal grains and foods depends

on control of potentially highly destructive insect pests, particularly

in tropical and subtropical regions. Owing to its international

market acceptance and a lack of acceptable, cost-effective alter-

natives, disinfestation with phosphine (PH3) fumigant is a funda-

mental tool used world-wide in the management of these insects.

However, the development of resistance to phosphine by the lesser

grain borer, Rhyzopertha dominica, a serious cosmopolitan pest of

stored cereal grains, seriously threatens effective insect pest

management (Collins, 2006; Emekci, 2010). The response of pest

managers to resistance has been to increase phosphine concen-

trations and exposure periods (Collins et al., 2005) and, in cases

where control cannot be achieved, apply residual insecticides. The

latter is a last resort, however, as it can limit market access. Other

measures adopted to combat resistant insects include strategic use

of sulfuryl fluoride, intensive storage hygiene and increased insect

population monitoring and resistance testing (Nayak et al., 2010). A

strategy relying primarily on a single fumigant backed by a limited

number of alternatives is highly risky, however, and will require

very careful management to sustain.

The development of resistance to pesticides in insects is affected

by a variety of interacting influences, including genetic, biological/

ecological and management (operational) influences (National

* Corresponding author. M084, School of Plant Biology, FNAS, The University of

Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia. Tel.: þ61 8

6488 1992; fax: þ61 8 6488 1108.

E-mail address: shi.mingren@gmail.com (M. Shi).

Contents lists available at SciVerse ScienceDirect

Journal of Stored Products Research

journal homepage: www.elsevier .com/locate/ jspr

0022-474X/$ e see front matter ! 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.jspr.2012.06.003

Journal of Stored Products Research 51 (2012) 23e32

Chapter 5

59

Research Council, 1986). Many factors can affect whether an insect

population will survive under phosphine fumigation treatments.

Some of the more important factors include gas concentration,

duration of fumigation and developmental stages of the insects

present.

Our research is aimed at contributing to the development of

viable, long-term strategies to support the management of phos-

phine resistance. Computer simulation models can provide a rela-

tively fast, safe and inexpensive way to project the consequences of

different assumptions about resistance and to weigh the merits of

various management options. We used stochastic individual-based

modelling that explicitly represents the fact that R. dominica pop-

ulations consist of individual beetles, each of a particular genotype

and a particular life stage. Individual-based approaches are rela-

tively easily adapted to incorporate the biological attributes that we

want to investigate, such as different initial frequencies of geno-

types and different proportions of life stages.

Resistance to phosphine is an inherited trait and two major

genes are responsible for the strong phosphine resistance in

R. dominica (Collins et al., 2002). These two genes act in synergy to

cause a significantly increased resistance to phosphine compared

with either one of the resistance genes on their own (Schlipalius

et al., 2008). Our previous research (Shi et al., in press) showed

the importance of basing resistance evolution models on realistic

genetics and that using simplified one-locus models to develop

pest control strategies runs the risk of not correctly identifying

tactics to minimise the incidence of pest infestation. Hence, our

simulations were carried out using the two-locus model.

In this paper we used our individual-based two-locus model to

address three questions about the management tactics of single

phosphine fumigation by investigating some biological and opera-

tional factors that influence the development of phosphine resis-

tance in R. dominica. First, which application tactic ismore effective:

long exposure time with low concentration, or short exposure time

with high concentration? This involved looking at the impact on

insect life stage (egg, larva, pupa, adult), each of which has an

inherently different tolerance to phosphine, on the efficacy of two

equi-toxic phosphine treatments: Fum1: “0.0685 mg/l " 16 days

(exposure for 16 days at a concentration of 0.0685mg/l)” and Fum2:

“0.2mg/l" 8 days (exposure for 8 days at a concentration of 0.2mg/

l)”. The second question is: within an insect population of a fixed

total size,what is the impact of different initial genotype frequencies

on the efficacy of phosphine fumigation? Following on from this,

how long should fumigation time be extended to provide an equally

effective level of control, if the initial frequency of strongly resistant

beetles is n times greater than the original one? This could represent

populations with different histories of exposure to phosphine

fumigation. Third, given afixed initial genotype frequency, howdoes

the level of infestation, i.e. the total number of insects within a grain

storage, impact on the effectiveness of phosphine treatment? This

may represent varying levels of grain hygiene maintenance or

varying amounts of movement of pests from infested places outside

to inside a storage facility.

2. Model and methods

An overview of the two-locus model is given here; full details of

the model have been provided previously (Shi and Renton, 2011;

Shi et al., in press, Shi et al. 2012). Themodel assumptions regarding

genotypes and resistance are described first, followed by explana-

tion of overall model dynamics and simulation processes. Then we

describe the methods we used to estimate the two kinds of finite

daily survival rates under the two PH3 treatments. Finally, we

describe in detail how we investigated the three central questions

posed in the Introduction.

2.1. Assumptions regarding genotypes and resistance

Resistance to phosphine is an inherited trait. Our simulation

model was constructed based on results from Collins et al. (2002)

and Schlipalius et al. (2002). Their research revealed that the

combination of alleles at two loci, rph1 and rph2, confers strong

resistance while rph1 by itself is responsible for theweak resistance

phenotype. It seems that both rph1 and rph2 individually express

a relatively low level of resistance but when they occur in the same

insect the resistance mechanisms synergise, producing a much

higher level of resistance. Based on this data, we assume there are

two possible alleles on each of these two loci, a susceptibility allele

and an incompletely recessive resistance allele. At both loci, the

susceptibility allele is assumed to be relatively common initially,

and the resistance allele to be relatively rare. At each locus there are

thus three possibilities, which we denote as s (homozygous

susceptible), h (heterozygous) and r (homozygous resistant). Our

two-locus simulation model thus includes nine possible genotypes,

which can be denoted as ss, sh, sr, hs, hh, hr, rs, rh and rr. For

example, ss denotes the genotype with both loci homozygous

susceptible, and rh denotes the genotype with the first locus

homozygous resistant, and the second locus heterozygous.

2.2. Overview of individual-based model dynamics and

assumptions

The simulated dynamics for each individual beetle at each daily

time step during the simulation are illustrated in Fig. 1, and the

default parameter values used in the model are listed in Table 1

with a brief description. The life stages include egg, larva, pupa

and adult. We separate the adult stage into two: adult 1 (immature

unable to lay eggs) and adult 2 (mature able to lay eggs) in simu-

lations but we merge the counts of adult 1 and adult 2 into a single

adult stage in the results. Life history parameters of each life stage

were estimated from published experimental data (Collins et al.,

2002, 2005; Daglish, 2004) based on the assumed temperature

and relative humidity. A number of processes within the simulation

are determined stochastically:

# The ‘time remaining within current life stage’ (TRICLS) for each

individual beetle is drawn randomly from a normal distribu-

tion, with mean and standard deviation depending on the life

stage (Table 1) in “Initialize” or “Enter the next stage” step

(Fig. 1).

# Whether the beetle survives through the day during a period of

fumigation is drawn from a Bernoulli distribution with survival

probability that depends on the genotype of the beetle, and the

concentration and exposure time of the fumigant, as explained

below.

# The sex of the egg is drawn from a Bernoulli distribution, with

an even probability of being male or female (male eggs

removed from simulation).

TRICLS <=

Current life

stage ends?

0?

Yes

next stageEnter the

TRICLSSet new

genotype

life stage TRICLS

[ ]

Is life time over?currently

underfumigation?

No

Yes

Dead

Able to

lay eggs?

Survival?

Yes

egg duration

genotype / sex

number of eggs

Determine:

No

YesNo No

Yes

No

next dayTo the

TRICLS 1

Initialise:

Fig. 1. The overall model dynamics occurring each day for each individual beetle

during a simulation.

M. Shi et al. / Journal of Stored Products Research 51 (2012) 23e3224

60

# The number of eggs produced by an individual in a day is

determined by drawing randomly from a Poisson distribution

with mean equal to the daily birth rate bD (Table 1).

# The genotype of each new egg is determined by drawing

randomly from a multinomial distribution based on the

maternal and paternal genotype and the offspring genotype

table (Shi and Renton, 2011).

Full details of model dynamics and parameter values are

included in Shi et al. (in press).

A 1:1 sex ratio was assumed, which allowed us to simulate the

female beetles only, with the assumption that total number and the

genotype frequency for the male beetles was the same as for the

females. The rate of development and survival of various life stages

of the lesser grain borer depend on several environmental factors,

among them temperature and moisture content of the grain (or

equilibrium relative humidity) were the main factors considered in

our simulations. For this model, we assumed grain type wheat

stored at a typical field temperature (T) of 25 $C (Cassells et al.,

2003) and relative humidity (r.h.) of 70%. The latter was chosen as

much of the published life history data are provided at 70% r.h. (Shi

et al. 2012). In addition, these environmental conditions are rec-

ommended by FAO for bioassay (FAO, 1975) and used by many

researchers in their experiments (e.g. Collins et al., 2000, 2002,

2005; Daglish, 2004; Herron, 1990; Pimentel et al., 2007;

Schlipalius et al., 2008).

It is assumed that no eggs are laid during periods of fumigation.

Mating occurs randomly; for example, resistance genetics does not

affect choice of mate. The grain food supply is non-limiting and its

quality does not affect the natural birth rate of the beetle. Fumigant

concentrations do not vary with time or locationwithin the storage

facility or silo. Variationwith time or space due to uptake or release

of gas from or into grain (sorptionedesorption), diffusion, or

leakage (Banks, 1989) is negligible. Temperature and relative

humidity within the store remains the same so that life stage

durations are constant.

2.3. Estimation of survival rates under fumigation

We used probit models, as we found they had smaller least

squares (numerical) errors for our data than logistic and Cauchy

models (Shi and Renton, unpublished results), to predict the total

mortality or survival (¼ 1 e mortality) rates of adult beetles under

different concentrations and exposure times. We used the four-

parameter probit model

Y ¼ aþ b1 logðtÞ þ b2 logðCÞ þ b3 logðtÞlogðCÞ (1)

for the beetles of genotypes hs, rs and rr, as it provided a better fit of

the available data than the 2- and 3-parameter models in terms of

smaller least squares (numerical) errors. Here, Y is the probit (the

inverse cumulative distribution function) value of mortality, C is

concentration (mg/l) and t is exposure time (hours). We used the

two-parameter probit model

Y ¼ aþ b logðCtÞ (2)

to predict the mortality rates for other genotypes, because t is

a constant (48 h) in the relevant data set. These models were fitted

to data on final mortality rates for different genotypes, concentra-

tions and durations (Collins et al., 2002, 2005; Daglish, 2004) and

the fitted parameters for the two models are listed in Table 2.

Directly substituting the fixed C value and the series of t values 1, 2,

., T into Eqs (1) and (2) results in the predicted cumulative survival

rates (CSR) after 1, 2, ., T days. The CSR for the final Tth day is the

total survival rate (TSR) for the full C " T treatment.

We wanted to design two phosphine treatments, Fum1 and

Fum2, to test whether long exposure time with low concentration,

Fum1, or short exposure time with high concentration, Fum2, is

more effective, i.e. results in lower survival rates, taking into

account that R. dominica tolerance to phosphine varies with life

stage. We wanted these two treatments to be equi-toxic, meaning

that for a population of only adult beetles they would result in the

same total mortality. We chose Fum2 to be the treatment “0.2 mg/

l " 8 days” (C ¼ 0.2 mg/l and t ¼ 8 days). Under this Fum2 treat-

ment, the total survival rate of the rr and rh beetles are 0.007888

and 7.2 " 10(10 respectively, and <1.04 " 10(21 for all other

genotypes. In other words, this treatment kills about 99.2%

(z1 ( 0.007888) of the rr beetles and effectively eradicates the

other genotypes. This ‘threshold’ treatment achieves a high kill rate

but may allow a small fraction of highly resistant beetles to survive.

Achieving this dose for the whole duration throughout the storage

facility can be seen as an ideal target fumigation strategy. In reality

doses achieved in storage facilities are likely to be more variable in

space and time (also see Discussion).

Table 2

The fitted parameters of two- and four-parameter probit models.

Genotype Two-parameter probit model Four-parameter probit model

a b a b1 b2 b3

ss 15.032386 9.229083

hs 11.284676 3.776399 6.964954 (1.010451

rs (10. 413046 15.575413 0.047656 4.701759

sh 10.854928 5.913329

hh 7.954711 5.913329

rh 1.682448 5.913329

sr 7.043248 5.913329

hr 3.944565 5.913329

rr (12.232356 10.386287 3.101974 1.190773

Table 1

Default parameters, variables and abbreviations for our two-locus model.

Parameter Description Value

mi Mean number of days at life stage i megg ¼ 11.9, mlarva ¼ 36.5,

mpupa ¼ 9.6, madult1 ¼ 15,

madult2 ¼ 102

si Standard deviation of number

of days at life stage i

Segg ¼ 1.5, Slarva ¼ 4.6,

Spupa ¼ 1.2, Sadult1 ¼ 0,

Sadult2 ¼ 15

bD Daily (finite) birth rate 0.2242

N0 Starting number of (female) beetles 100,000 (100 K)

TRICLS Time remaining in current life stage Variable

TSR Total survival rate at the end of a

fumigation treatment

Variable

CSR Cumulative survival rate each day

during a fumigation period

Variable

GADSR Geometrical average daily survival rate Variable

DDSR Different daily survival rate Variable

M. Shi et al. / Journal of Stored Products Research 51 (2012) 23e32 25

61

To set a treatment with a lower concentration C and longer

exposure time t to achieve the same 99.2% total kill rate of the adult

rr beetles we doubled the exposure time, i.e. setting t¼ 16 days, and

then the C value required tomake the treatment equi-toxic could be

estimated by rearranging Eq. (1):

logðCÞ ¼ ½Y ( a( b1 logðtÞ*=½b2 þ b3 logðtÞ* (3)

with t ¼ 16 " 24 (hours), Y ¼ 7.414 (corresponding to mortality

1 ( 0.007888), and the parameters listed in Table 2 for the rr

genotype. The result is C¼ 0.0685mg/l. Thus we had two equi-toxic

fumigation treatments:

Fumigation 1ðFum1Þ : “0:0685 mg=1" 16 days”;Fumigation 2ðFum2Þ : “0:2 mg=1" 8 days”:

(4)

As desired, these treatments predicted the same TSR for the

adult rr beetles. It can be seen from Table 3 that the CSR after 8 days

under Fum2 is indeed exactly equal to the CSR after 16 days under

Fum1, and in fact the nth day’s CSR under Fum2 is very close to the

(2n)th day’s CSR under Fum1 in general.

The probit models predicted total survival rate (TSR) from

a fumigation treatment lasting a number of days, but our

individual-based simulation needed predictions of the daily

mortalities within the fumigation period (the finite daily survival

rate). There are two ways to estimate the finite daily survival rate

under a PH3 treatment C " T at a fixed concentration C (mg/l) and

a range of times (1, 2,., T days). The first way to estimate the finite

daily survival rate is based on the assumption that the survival rate

is the same each day during a fumigation treatment, in which case

the finite daily survival rate of genotype x is [TSR(x)]1/T. We can call

this geometrical average daily survival rate (GADSR). The estimated

GADSRs for the nine genotypes under the two treatments Fum1

and Fum2 are listed in Tables 4 and 5. Note that if we use GADSR to

design Fum1, just simply set GADSRFum1 ¼ (GADSRFum2)1/2. In other

words, for the rr beetles the CSR for one day under Fum2 is exactly

equal to two-day’s CSR under Fum1 or the square of the GADSR

under Fum1.

The second way is to estimate a different daily survival rate

(DDSR) for each day: convert the ith day’s CSRi and the previous

(i ( 1)th day’s CSR i ( 1 into the ith day’s DDSRi(x) (for genotype x)

by setting DDSR1(x) ¼ CSR1(x) and then letting DDSRi(x) ¼ CSRi(x)/

CSRi ( 1(x), for i ¼ 2, ., T. (the current day’s DDSR is equal to the

ratio of the current day’s CSR to the previous day’s CSR). The DDSR

of the rr beetles under Fum1 and Fum2 thus derived are listed in

Table 6, and illustrated in Fig. 2.

If the simulation is concerned only with the results after fumi-

gation, then we can use the former approach with equal daily

survival rates (GADSR), as it is simpler. If the simulation is also

concerned with accurately representing what happens during the

fumigation process then we should use the latter approach (DDSR),

even though it is more complex and will increase the simulation

time. We used DDSR for the simulation of the impact of different

life stages on the effect of fumigation since we wanted to see the

details of what happens during the fumigation process in each life

stage. But we used GADSR for the other two simulations as wewere

concerned only with the survival numbers after fumigation.

Full details for the estimation of survival rates are included in

Shi and Renton (2011) and Shi et al. (2012).

2.4. Question 1: which is more important: concentration or

exposure time?

Flinn et al. (2001) reported that the days spent to reach 95%

mortality exposed to 180 ppm (0.25 mg/l) at 25 $C for life stages of

R. dominica were as follows:

Egg; Larva; Pupa; Adultð1:0; 0:4; 0:5; 0:3Þ (5)

For simplicity, we assume that the relative tolerance of stages

does not change with genotype. We also assumed that the relative

tolerance to PH3 of each life stage and therefore the TSR (total

survival rate) was equal to the ratio of the times to 95% mortality

provided in Eq. (5). The predicted time to 95% mortality of adults

exposed to 0.0685mg/l is approximately 12.5 days. If we regard this

as the TSR for adults then the DDSR of the rr beetles at other life

stages under Fum1 for each day are:

DDSRE ¼ ð1:0=0:3Þ1=12:5DDSRAz1:1055 DDSRA;

DDSRL ¼ ð0:4=0:3Þ1=12:5DDSRAz1:0243 DDSRA and

DDSRP ¼ ð0:5=0:3Þ1=12:5DDSRAz1:0435 DDSRA

(6)

where DDSRA is the daily survival rate of the rr beetles at adult

stage, and DDSRE DDSRL and DDSRP are the daily survival rate of the

rr beetles at the egg, larva and pupa stage, respectively. We can now

set the DDSR of the rr beetles at other life stages under Fum2 so that

the TSRs under Fum1 and Fum2 at each stage are almost the same.

That is, under Fum2,

DDSREz1:2222 DDSRA;DDSRLz1:0491 DDSRA andDDSRPz1:0889 DDSRA

(7)

We emphasize again that the beetles of other genotypes are all

killed under Fum1 and Fum2.

We ran themodel starting with equal proportions of the four life

stages PS ¼ (0.25, 0.25, 0.25, 0.25). We also started with a resistance

allele frequency of q¼ v¼ 0.62947, where q and v are the frequency

of the resistance alleles at the 1st and 2nd locus respectively, cor-

responding to a high rr genotype frequency of 0.157, representing

Table 3

Cumulative survival rates (CSR) under Fum1 up to the (2n)th day and Fum2 up to the

nth day for the adult rr beetles (ratio¼ [Fum2 at (n)th day]/[ Fum1 at (2n)th day] for rr

beetle at any stage).

n 1 2 3 4

Fum1/(2n)th day 0.99999999 0.99867 0.922132 0.6160571

Fum2/(n)th day 1. 0.99958 0.951103 0.6779515

Ratio 1.00000001 1.00091 1.031417 1.1004686

n 5 6 7 8

Fum1/(2n)th day 0.28196 0.09859 0.02924 0.007888

Fum2/(n)th day 0.32136 0.11116 0.03144 0.007888

Ratio 1.13973 1.12749 1.07548 1.000000

The bold values represents the total survival rates under Fum1 (16 days) and under

Fum2 (8 days) are the same 0.007888.

Table 4

The geometrical average daily survival rate (GADSR) for the nine genotypes for Fum2

treatment (0.2 mg/l " 8 days).

1st gene 2nd gene

s h r

s <1. " 10(11 <1. " 10(11 6.33 " 10(10

h <1. " 10(11 <1. " 10(11 0.00238

r 0.000304 0.007196 0. 54591

Table 5

The geometrical average daily survival rate (GADSR) for the nine genotypes for Fum1

treatment (0.0685 mg/l " 16 days).

1st gene 2nd gene

s h r

s < 1. " 10(11 < 1. " 10(11 0.00074

h 6.3 " 10(6 3.04. " 10(5 0.12849

r 0.02223 0.3801 0.7389

M. Shi et al. / Journal of Stored Products Research 51 (2012) 23e3226

62

a population on the threshold of exhibiting a serious resistance

problem (see Shi and Renton, 2011 for details on setting initial

genotype frequencies). We started to fumigate immediately and ran

the model for 140 days (20 weeks) in total, enough time for pop-

ulations to build up again after the fumigation.

2.5. Question 2: impact of initial genotype frequencies on

fumigation efficacy

To simulate the impact of different initial frequencies of geno-

types on the effectiveness of fumigation, we first set up nine sets of

initial frequencies of genotypes. We defined the resistance allele

frequencies for the nth set (n ¼ 1, 2,., 9) as qn ¼ vn ¼ (0.1 n)1/4. We

then calculated the equilibrium genotype frequencies for the nine

sets, resulting in initial frequencies for the rr beetles set of:

0:1; 0:2; 0:3;.; 0:8; 0:9

That is, the initial frequency of the nth set for rr beetles is n times

the original one (0.1). We set the initial proportions of the four life

stages to be equal PS ¼ (0.25, 0.25, 0.25, 0.25). We started to

fumigate immediately under Fum2 and ran the model for 140 days

(20 weeks) in total. We also ran the model to fumigate from the 1st

day to 40th day at the Fum1 concentration 0.0685 mg/l (and to

fumigate from the 1st day to 30th day at the Fum2 concentration

0.2 mg/l to learn how many extra days are required to reach the

same control level). Note that to separate the influence of the

different factors we ignored the impact of different life stages in

this simulation, that is, the GADSRs listed in Table 4 or Table 5 were

used for all life stages. The start population number of (female)

beetles for this simulation was also 100 K.

2.6. Question 3: impact of initial population size on fumigation

efficacy

This simulation was undertaken to test the impact of a range of

sizes of initial insect populations within the grain storage, repre-

senting various levels of storage hygiene (amounts of grain and

other residues around the storage facility that harbour insects and

provide sources of initial infestation). Initial genotype frequencies

were the same as for Question 1. The initial proportions of the four

life stages and the durations of fumigation and simulationwere the

same as for Question 2.We simulated the Fum1 treatment only, and

considered five different initial population numbers: 100 K, 200 K,

400 K, 800 K and 1,600 K.

3. Results

3.1. Question 1: which is more important: concentration or

exposure time?

The numbers of the rr beetles in all life stages decline at different

rates under the two fumigation treatments, and the proportions in

each life stage also vary in different ways (Fig. 3). These differences

then affect how the populations recover after fumigation ends

(Fig. 4). The number of rr beetles after fumigation under Fum2 are

about 1.4 times as many as those under Fum1 (Table 7), and this

difference carries through to a similar difference at the end of the

whole simulationperiod. Thenumbers andproportions under Fum1

for thefirst eight days and those under Fum2 for thefirst four days at

each stage are very similar (left parts of subplots in Fig. 3). In this

initial period, the numbers of rr beetle at all life stages decrease. The

adult beetles, the least tolerant life stage, decreasemost quickly, but

there is little difference between the life stages. After this initial

period, the differences between the four life stages and the two

treatments becomemore evident (right part of Fig. 3). The decline in

numbers under Fum1 is faster than under Fum2, with pupa and egg

numbers dropping to zero in Fum1 but not Fum2.

3.2. Question 2: impact of initial genotype frequencies on

fumigation efficacy

As the initial frequency of the rr beetles is increased, the

numbers of rr beetles at the end of fumigation and after simulation

both increase as well, as shown by the numbers in Table 8 and the

fact that the two curves in Fig. 5 (a) are both increasing. Further-

more, the two curves in Fig. 5(a) are nearly parallel, which indicates

that the number of beetles at the end of the simulation is almost

a constant multiple of the number of beetles following fumigation.

The nine curves in Fig. 5(b) are also parallel, indicating that the

effect of different initial genotype frequencies is a simple constant

multiplicative one. Each of the nine curves in Fig. 5(b) has two

parts; the first part, which is decreasing, is during the fumigation

and the second part, which is increasing, is after the fumigation.

The relationship between the initial frequency of the rr genotype

and the numbers of beetles at the end of fumigation and after

simulation is approximately linear (Table 8, Fig. 5(a)), and this is

actually true at any time during the simulation (Fig. 5(b)). It can be

seen from Table 9 that if the initial frequency is increased to n times

Table 6

Different daily survival rates for the nth day (n/DDSRn) of the adult rr beetles under Fum1 (16 days) and Fum2 (8 days).

Fum1 1/1. 2/0.99999999 3/0.99998 4/0.9986 5/0.984 6/0.938 7/0.860 8/0.777

9/0.705 10/0.649 11/0.607 12/0.576 13/0.553 14/0.536 15/0.524 16/0.515

Fum2 1/1. 2/0.9996 3/0.952 4/0.713 5/0.474 6/0.346 7/0.283 8/0.251

Fig. 2. The different daily survival rates (DDSR, the curve on the top) and its corre-

sponding cumulative survival rate (CSR from DDSR, having the same start point as the

DDSR curve; the 1st day’s DDSR is equal to the 1st day’s CSR), and the geometrically

average daily survival rate (GADSR, horizontal line) and the corresponding cumulative

survival rate (CSR from GADSR, also having the same start point as the GADSR line) of

the adult rr beetles under Fum2 (0.2 mg/l " 8 days). Note that the DDSR changes daily,

while the GADSR is constant, but both result in the same total survival/mortality (their

corresponding CSR curves end at the same point).

M. Shi et al. / Journal of Stored Products Research 51 (2012) 23e32 27

63

the original, then we need to extend fumigation duration approx-

imately n days under the Fum1 concentration to achieve a similar

control level, whereas under the Fum2 concentration we need to

extend fumigation duration approximately a half of n days.

3.3. Question 3: impact of initial population number on fumigation

efficacy

It can be seen by comparing Fig. 5 and Fig. 6 that the results of

this simulation are very similar to those obtained from the

previous simulation for Question 2; the two curves in Fig. 6(a) are

both increasing and nearly parallel, the five curves in Fig. 6(b) are

also parallel, and the ratio of the numbers at the end of fumigation

and at the completion of simulation are very similar; if the starting

population number is increased n times, the numbers at the end of

fumigation and at the completion of simulation also increase about

n times (Table 10). Note that the original frequency of rr beetles for

this question is 0.157 while that for Question 2 is 0.1; this is why

the numbers of rr beetles at the end of fumigation and at the

completion of simulation for the two questions differ, even though

the starting population number of both was 100 K. Also note that,

as our model is stochastic, for any particular model run the

increase will only be approximately n times, but when averaged

over a large number of model runs, the ratio will be exactly n

times.

4. Discussion

In this research we used our individual-based, two locus resis-

tance model to investigate three key questions about the impact of

a single application of phosphine on control of phosphine-resistant

R. dominica populations. As the fumigation parameters of exposure

time and concentration are usually under operational control, we

asked: at equi-toxic dosages, which is better for insect control;

a longer fumigation at lower concentration or a shorter fumigation

at a higher concentration? Our model showed that the former is

more effective, that is, will kill a higher proportion of insects than

the latter. The reason for this is that under a longer exposure period,

themore tolerant immature stages (eggs and pupae) have sufficient

time to develop to less tolerant stages (larvae and adults) (Fig. 3). As

there is no evidence available to indicate that exposure to phos-

phine may influence immature development time in R. dominica,

we have assumed in our model that this does not occur. Phosphine

has been shown to delay egg hatch in some insect pests of stored

products (Rajendran, 2000; Nayak et al., 2003) but not others (Price

and Bell, 1981; Pike, 1994).

It is already known that extending the fumigation period (while

lowering concentration) will increase toxicity of phosphine (Winks,

1985; Chaudhry, 2000; Collins et al., 2005; Mills and Athie, 1999;

Bond, 1984). High concentrations may not increase toxicity and, in

reality, they may cause insects to go into a protective ‘narcosis’. It

A B

C D

Fig. 3. Numbers (a, b) and proportions (c, d) of beetles with each life stage for the fumigation period under the two treatments Fum1 (a, c) and Fum2 (b, d). The vertical line x ¼ 8 for

Fum1 and the line x ¼ 4 for Fum2 separate the figures into two parts, emphasising the fact that the four curves (for four stages) on the left parts of (a) and (b) or of (c) and (d) have

very similar behaviour, but the right parts differ. (Note that when numbers reach zero they are not shown because of the log scale).

M. Shi et al. / Journal of Stored Products Research 51 (2012) 23e3228

64

appears that the toxic effects of phosphine accumulate slowly in R

insects and that the resistance mechanism can be overwhelmed

during long exposure periods. These effects are not accounted for in

this study, and provide additional support for a long duration

strategy, over and beyond the effects due to life stage transitions

that we found here.

The second key questionwe asked was: what impact does initial

resistance genotype frequency have on phosphine efficacy? In

addition, having shown that extending fumigation time is superior

to increasing concentration, we asked how long fumigation time

should be extended to provide greater control at higher resistance

gene frequencies. Our results show that the rate of survival of the

pest population will increase proportionally with initial resistance

genotype frequency, that is, by about n times if the initial frequency

of the rr genotype is increased n times. This is because we assumed

the same survival and development rates for all nine possible

genotypes, including rr. Thus, to achieve a similar level of control of

a population with an initial frequency of the rr genotype that is n

times higher than another population with a low initial frequency,

we need only increase the fumigation time by approximately n days

rather than having to multiply the original fumigation duration by

n times. This again shows that increased duration of fumigation is

an efficient way to increase the efficacy of fumigation.

Our third key question was: what impact does the level of

infestation or total number of insects within a grain storage have on

the effectiveness of a phosphine fumigation? Obviously, the

number of rr insects surviving a fumigation will increase about n

times if its initial population size is increased by n times. In addi-

tion, the initial number of each genotype is always the product of

two factors: the total initial population number and the initial

frequency of the genotype. If we fix one of these two factors and

increase the other factor by n times the original one, this always

A B

C D

Fig. 4. Numbers (a, b) and proportions (c, d) of beetles with each life stage for the whole simulation period (fumigation and recovery) under the two treatments Fum1 (a, c) (Note

that when numbers reach zero they are not shown because of the log scale.).

Table 7

Total survival numbers (TSN) of the rr beetles after fumigation and at the end of the

whole simulation (fumigation and recovery) under the Fum1 and Fum2 scenarios.

Treatment Total survival number (TSN)

After fumigation After simulation

Fum1 (0.0685 mg/l " 16 days) 160 2259

Fum2 (0.2 mg/l) " 8 days) 224 3077

Table 8

The numbers of the rr beetles at the end of Fum1 fumigation (EF) and at the end (or

completion) of the whole simulation (ES) for each of the initial genotype frequencies

of the rr beetles (f(rr)) tested. For each genotype frequency, the ratio of the EF and ES

results relative to the EF and ES results at the lowest initial frequency of 0.1 are also

given.

f(rr) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

EF 89 139 225 324 395 444 507 620 723

Ratio 1. 1.56 2.53 3.64 4.44 4.99 5.70 6.97 8.12

ES 1417 2173 3599 5328 6271 6834 8132 9577 11,593

Ratio 1. 1.53 2.54 3.76 4.43 4.82 5.74 6.76 8.18

M. Shi et al. / Journal of Stored Products Research 51 (2012) 23e32 29

65

results in an initial number of each genotype equal to n times the

original starting number of that genotype. To answer the second

key question discussed previously, we fixed the starting population

number at 100 K but varied the initial frequency of the rr genotype

beetles. To answer this third question, we fixed the initial frequency

but varied the initial population number. The results are simplified

by the fact that our two PH3 treatments killed all beetles of all

genotypes besides rr. Hence the simulations we used to answer key

questions 2 and 3 are quite alike, and gave very similar results

during the fumigation period (Figs. 5 and 6). Moreover, the daily

birth rates in the two simulations were the same, resulting in very

similar behaviours after fumigation. Thus we can draw a very

similar conclusion for the third question as for the second: to

achieve a similar level of control of a population that is n times

larger than another population, we need only to increase the

fumigation time by approximately n days rather than having to

multiply the original fumigation duration by n times. This once

again shows that increased duration of fumigation is an efficient

way to increase the efficacy of fumigation.

Our two- and four-parameter probit models (1) and (2) fit the

experimental data sets of Collins et al. (2002, 2005) and Daglish

(2004) very well and this allowed us to accurately predict

mortality of R. dominica under a variety of phosphine treatments

with different concentrations and exposure times (Shi and Renton,

2011). In previously published resistance modelling research

“survivorship was not explicitly included in the model because

adequate data were not available” (Hagstrum and Flinn, 1990), and

when different mortalities for different genotypes were included,

they were only roughly divided into a few levels (e.g. Tabashnik,

1989; Longstaff, 1988) and a simplified single gene model was

used. In other cases, mortalities were varied with temperature and

moisture in some detail but differences due to concentration,

exposure time, or genotype were not included (e.g. Flinn et al.,

1992). To our knowledge, no previous models have included

mortality predictions that vary with concentration, exposure time,

A

B

Fig. 5. The number of rr beetles at the end of the fumigation period and at the

completion of running the whole simulation (fumigation and recovery) for each of the

9 initial genotype frequencies (a), and the number of rr beetles over time during the

whole simulation (b).

Table 9

The number of extra days exposure needed under Fum1 and Fum2 to achieve

a similar control level to the first simulation when the rr genotype frequency is 0.1,

for various cases where the initial rr genotype frequency is higher that 0.1.

Initial frequency of rr 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Extra days under Fum1 3 4 6 6 7 7 8 8

Extra days under Fum2 1 2 3 3 3 4 4 4

A

B

Fig. 6. The numbers of rr genotype beetles at the end of the fumigation period and at

the end of the whole simulation (a), and over time during the whole simulation period

(b), for initial populations of 100 K, 200 K, 400 K, 800 K and 1600 K insects, given

a fixed initial genotype frequency and equal initial proportions of the four life stages.

M. Shi et al. / Journal of Stored Products Research 51 (2012) 23e3230

66

and genotype and based on extensive experimental data in the way

we have here, nor have they been based on data to the extent that

our predictions are. The experimental data sets we used were

collected over a long period of time, the results were confirmed in

field trials and are the basis for the current rates used to control

resistant insects in Australia. In addition, no previous studies of

resistance in stored grain pests have used detailed individual-based

models that allow this level of biological and genetic detail to be

represented (Renton, 2012), although similar approaches have been

employed to investigate the evolution of resistance to Bt transgenic

crops in insects in the field (Storer et al., 2003), and to herbicides in

weeds (Renton et al., 2011). These kind of improved models, based

on better data and including more biological detail, will help us

predict the evolution of phosphine resistance in R. dominica, and

weigh the merits of various management options in practice for

delaying or avoiding evolution of resistance as accurately as

possible.

As mentioned previously, our results show clearly that extending

the period of fumigation is an effective strategy for decreasing the

overall survival of R. dominica. These results also have important

implications formanagement of resistance evolution, as they indicate

how best to effectively control resistant genotypes, as well as

susceptible genotypes. Among the nine genotypes in the two-locus

model, there were five with relatively high levels of resistance: sr, hr,

rs, rh and rr. All but rr beetles were killed in our simulations, meaning

that the frequency of resistance alleles is always 100%. However, in

reality there is likely to be immigration of non-resistant insects from

outside the storage after fumigation that will dilute the frequency of

resistance alleles. In fact, previous simulations have shown that the

development of resistance may be suppressed by the immigration of

susceptible insects from unsprayed reservoirs of infestation and thus

the movement of insects within the farm or storage facility may be

very important, particularly when resistance is recessive (Sinclair and

Alder, 1985; Tabashnik and Croft, 1982). We argue that this suppres-

sion via immigration is also likely tomore effectivewhen resistance is

caused by a combination of two genes where both genes are required

to achieve significant levels of resistance. If such immigration occurs,

then the fewer rr beetles surviving at the end of fumigation, the less

developmentof resistance evolution inR. dominicawould be expected

tooccur. Thus, themanagement strategyof extendingexposureperiod

of phosphine fumigation is a very important one to control or delay

the resistance evolution in this pest. For example, a PH3 treatment of

0.53mg/l (350 ppm)" 7 days is often used by industry in Australia. It

may be better to instead use an equi-toxic treatment with a lower

concentration for a longer time, for example, “0.17 mg/l

(z112 ppm) " 14 days”.

We will conclude with some comments about the limitations

of our model and recommendations for further work. This study

was based on assumptions that the fumigation concentrations

within a storage facility or silo at any time are constant at their

target value and thus does not account for processes of diffusion

over time through the silo, sorptionedesorption into and out of

the grain, and leakage out of the silo. The PH3 concentration and

time duration applied would in practice be enough to kill all

insects if these assumptions were true and the evolution of

resistance was not already well advanced, and thus the evolution

of resistance in the pest would never occur. The representation

used in this study is thus a somewhat idealised spatially and

temporally homogenous storage facility that can be seen to

represent what managers should ideally be aiming to achieve.

This simplification was appropriate to allow us to address the

questions we wanted to focus on in this study. However, in reality

this homogeneity of dose is likely to be difficult to achieve

perfectly, and both temporal and spatial variability will be inev-

itable to some degree. Further studies representing spatial and

temporal heterogeneity of dose are needed to investigate what

effects such variability will have on the short-term and long-term

efficacy of different control options and the contribution of

resistance alleles to further generations. The effects of emigration

and immigration of insects into the storage will also be important

to consider in future studies.

Acknowledgement

The authors would like to acknowledge the support of the

Australian Government’s Cooperative Research Centres Program.

We also thank Rob Emery and Yonglin Ren for their great help in

provision of raw data and information about beetle life cycles and

silo fumigation, and the valuable comments and suggestions of an

anonymous reviewer.

References

Banks, H.J., 1989. Behaviour of gases in grain storages. In: Champ, B.R., Highley, E.,Banks, H.J. (Eds.), Fumigation and Controlled Atmosphere Storage of Grain,Proceedings of an International Conference Held at Singapore, 14e18 February1989, Singapore, pp. 96e107.

Bond, E.J., 1984. Manual of Fumigation for Insect Control. In: FAO Plant Productionand Protection Paper 54. Rome, FAO.

Cassells, J.A., Darby, J.A., Green, J.R., Reuss, R., 2003. Isotherms for Australian wheatand barley varieties. In: Black, C.K., Panozzo, J.F. (Eds.), Cereals 2003, Proceed-ings of the 53rd Australian Cereal Chemistry Conference, 7th to 10th Sept 2003.Glenelg, South Australia, pp. 134e137.

Chaudhry, M.Q., 2000. Phosphine resistance. Pesticide Outlook 11, 88e91.Collins, P.J., Daglish, G.J., Nayak, M.K., Ebert, P.R., Schlipalius, D.I., Chen, W., Pavic, H.,

Lambkin, T.M., Kopittke, R.A., Bridgeman, B.W., 2000. Combating resistance tophosphine in Australia. In: Donahaye, E.J., Navarro, S., Leesch, J.G. (Eds.),Proceedings of the International Conference for Controlled Atmosphere andFumigation in Stored Products, Fresno California, USA, pp. 593e607.

Collins, P.J., Daglish, G.J., Bengston, M., Lambkin, T.M., Pavic, H., 2002. Genetics ofresistance to phosphine in Rhyzopertha dominica (Coleoptera: Bostrichidae).Journal of Economic Entomology 95, 862e869.

Collins, P.J., Daglish, G.J., Pavic, H., Kopittke, R.A., 2005. Response of mixed-agecultures of phosphine-resistant and susceptible strains of lesser grain borer,Rhyzopertha dominica, to phosphine at a range of concentrations and exposureperiods. Journal of Stored Products Research 41, 373e385.

Collins, P.J., 2006. Resistance to chemical treatments in insect pests of stored grainand its management. In: Lorini, I., Bacaltchuk, B., Beckel, H., Deckers, D.,Sundfeld, E., Santos, J.P., Biagi, J.D., Celaro, J.C., Faroni, L.R., Bortolini, L.O.F.,Sartori, M.R., Elias, M.C., Guedes, R.N.C., Fonseca, R.G., Scussel, V.M. (Eds.),Proceedings of the 9th International Working Conference on Stored ProductProtection, 15e18 October 2006, Campinas, Brazil, pp. 277e282.

Daglish, G.J., 2004. Effect of exposure period on degree of dominance of phosphineresistance in adults of Rhyzopertha dominica (Coleoptera: Bostrychidae) andSitophilus oryzae (Coleoptera: Curculionidae). Pest Management Science 60,822e826.

Emekci, M., 2010. Quo vadis the fumigants? In: Carvalho, O.M., Fields, P.G.,Adler, C.S., Arthur, F.H., Athanassiou, C.G., Campbell, J.F., Fleurat-Lessard, F.,Flinn, P.W., Hodges, R.J., Isikber, A.A., Navarro, S., Noyes, R.T., Riudavets, J.,Sinha, K.K., Thorpe, G.R., Timlick, B.H., Trematerra, P., White, N.D.G. (Eds.),Proceedings of the 10th International Working Conference on Stored ProductProtection, 27 Junee2 July 2010, Estoril, Portugal, pp. 303e313.

FAO, 1975. Recommended methods for the detection and measurement of resis-tance of agricultural pests to pesticides: tentative method for adults of somemajor species of stored cereals with methyl bromide and phosphine. FAOMethod No 16. FAO Plant Protection Bulletin 23, 12e25.

Table 10

The number of the rr beetles at the end of the fumigation period and at the

completion of the whole simulation for the five initial population numbers, and the

corresponding ratios of numbers resulting from higher starting population numbers

to the numbers resulting from a starting population of 100 K insects, representing

the lowest starting number.

Start population number 100 K 200 K 400 K 800 K 1600 K

At the end of fumigation 126 235 459 1045 1992

Ratio 1. 1.87 3.64 8.29 15.81

At the end of simulation 1989 3804 7256 16,555 31,447

Ratio 1. 1.91 3.65 8.32 15.81

M. Shi et al. / Journal of Stored Products Research 51 (2012) 23e32 31

67

Flinn, P.W., Hagstrum, D.W., Muir, W.E., Sudayappa, K., 1992. Spatial model forsimulating changes in temperature and insect population dynamics in storedgrain. Environmental Entomology 21, 1351e1356.

Flinn, P., Phillips, T., Hagstrum, D., Arthur, F., Throne, J., 2001. Modeling the effects ofinsect stage and grain temperature on phosphine-induced mortality for Rhy-

zopertha dominica. In: Donahaye, E.J., Navarro, S., Leesch, J.G. (Eds.), Proceedingsof the International Conference for Controlled Atmosphere and Fumigation inStored Products, Fresno California, USA, pp. 531e539.

Hagstrum, D.W., Flinn, P.W., 1990. Simulations comparing insect species differencesin response to wheat storage conditions and management practices. Journal ofEconomic Entomology 83, 2469e2475.

Herron, G.A., 1990. Resistance to grain protectants and phosphine in coleopterouspests of grain stored on farms in New South Wales. Australian Journal ofEntomology 29, 183e189.

Longstaff, B.C., 1988. A modeling study of the effects of temperature manipulationupon the control of Sitophilus oryzae (Coleoptera: Curculionidae) by insecticide.Journal of Applied Ecology 25, 163e175.

Mills, K.A., Athie, I., 1999. The development of a same-day test for the detection ofresistance to phosphine in Sitophilus oryzae (L.) and Oryzaephilus surinamensis(L.) and findings on the genetics of the resistance related to a strategy toprevent its increase. In: Zuxun, Jin, Yongsheng, Liang, Xianchang, Tan,Lianghua, Guan (Eds.), Proceedings of the 7th International Working Confer-ence on Stored-Product Protection, Beijing, China, 14e19 October 1998,pp. 594e602.

National Research Council, Committee on Strategies for the Management of Pesti-cide Resistant Pest Populations, 1986. Pesticide Resistance: Strategies andTactics for Management. National Academy Press, Washington, D.C.

Nayak, M.K., Collins, P.J., Pavic, H., Kopittke, R.A., 2003. Inhibition of egg develop-ment by phosphine in the cosmopolitan pest of stored products Liposcelis

bostrychophila (Psocoptera: Liposcelididae). Pest Management Science 59,1191e1196.

Nayak, M.K., Holloway, J., Pavic, H., Head, M., Reid, R., Collins, P.J., 2010. Developingstrategies to manage highly phosphine resistant populations of flat grainbeetles in large bulk storages in Australia. In: Cavalho, M.O., Fields, P.G.,Adler, C.S., Arthur, F.H., Athanassiou, C.G., Campbell, J.F., Fleurat-Lessard, F.,Flinn, P.W., Hodges, R.J., Isikber, A.A., Navarro, S., Noyes, R.T., Riudavets, J.,Sinha, K.K., Thorpe, G.R., Timlik, B.H., Trematerra, P., White, N.D.G. (Eds.),Proceedings of the 10th International Working Conference on Stored ProductProtection, 27 Junee2 July 2010, Estoril, Portugal, pp. 396e401.

Pike, V., 1994. Laboratory assessment of the efficacy of phosphine and methylbromide fumigation against all life stages of Liposcelis entomophilus (Enderlein).Crop Protection 13, 141e145.

Pimentel, M.A.G., Faroni, L.A., Totola, M.R., Guedes, R.N.C., 2007. Phosphine resis-tance, respiration rate and fitness consequences in stored-product insects. PestManagement Science 63, 876e881.

Price, N.R., Bell, C.H., 1981. Structure and development of embryos of Ephestia

cautella (Walker) during anoxia and phosphine treatment. International Journalof Invertebrate Reproduction 3, 17e25.

Rajendran, S., 2000. Inhibition of hatching of Tribolium castaneum by phosphine.Journal of Stored Products Research 36, 101e106.

Renton, M., Diggle, A., Manalil, S., Powles, S., 2011. Does cutting herbicide ratesthreaten the sustainability of weed management in cropping systems? Journalof Theoretical Biology 283, 14e27.

Renton, M., 2012. Shifting focus from the population to the individual as a wayforward in understanding, predicting and managing the complexities ofevolution of resistance to pesticides. Pest Management Science (accepted07.04.12., doi pending).

Schlipalius, D.I., Cheng, Q., Reilly, P.E., Collins, P.J., Ebert, P.R., 2002. Genetic linkageanalysis of the lesser grain borer Rhyzopertha dominica identifies two loci thatconfer high-level resistance to the fumigant phosphine. Genetics 161, 773e782.

Schlipalius, D.I., Chen, W., Collins, P.J., Nguyen, T., Reilly, P.E., Ebert, P.R., 2008. Geneinteractions constrain the course of evolution of phosphine resistance in thelesser grain borer, Rhyzopertha dominica. Heredity 100, 506e516.

Shi, M., Renton, M., 2011. Numerical algorithms for estimation and calculation ofparameters in modelling pest population dynamics and evolution of resistancein modelling pest population dynamics and evolution of resistance. Mathe-matical Biosciences 233, 77e89.

Shi, M., Renton, M., Collins, P.J., 2012. Mortality estimation for individual-basedsimulations of phosphine resistance in lesser grain borer (Rhyzoperthadominica). In: Chan, F., Marinova, D., Anderssen, R.S. (Eds.), MODSIM2011, 19thInternational Congress on Modelling and Simulation. December 2011, Perth,Australia, pp. 352e358. http://www.mssanz.org.au/modsim2011/A3/shi.pdf.

Shi,M., Renton,M., Ridsdill-Smith, J., Collins, P.J., Constructing anew individual-basedmodel of phosphine resistance in lesser grain borer (Rhyzopertha dominica): dowe need to include two loci rather than one? Pest Science. in press.

Sinclair, E.R., Alder, J., 1985. Development of a computer simulation model of storedproduct insect populations on grain farms. Agricultural Systems 18, 95e113.

Storer, N.P., Peck, S.L., Gould, F., Van Duyn, J.W., Kennedy, G.G., 2003. Spatialprocesses in the evolution of resistance in Helicoverpa zea (Lepidoptera: Noc-tuidae) to Bt transgenic corn and cotton in a mixed agroecosystem: a biology-rich stochastic simulation model. Journal of Economic Entomology 96, 156e172.

Tabashnik, B.E., Croft, B.A., 1982. Managing pesticide resistance in crop-arthropodcomplexes: interactions between biological and operational factors. Environ-mental Entomology 11, 1137e1144.

Tabashnik, B.E., 1989. Managing resistance with multiple pesticide tactics: theory,evidence, and recommendations. Journal of Economic Entomology 82,1263e1269.

Winks, R.G., 1985. The toxicity of phosphine to adults of Tribolium castaneum(Herbst): phosphine-induced narcosis. Journal of Stored Products Research 21,25e29.

M. Shi et al. / Journal of Stored Products Research 51 (2012) 23e3232

68

Research Article

Received: 24 June 2012 Revised: 14 September 2012 Accepted article published: 1 November 2012 Published online in Wiley Online Library:

(wileyonlinelibrary.com) DOI 10.1002/ps.3457

Dosage consistency is the key factor inavoiding evolution of resistance to phosphineand population increase in stored-grain pests

Mingren Shi,a,b∗ Patrick J Collins,b,c T James Ridsdill-Smith,b,d

Robert N Emeryb,e and Michael Rentona,b,f

Abstract:

BACKGROUND: Control of pests in stored grain and the evolution of resistance to pesticides are serious problems worldwide.A stochastic individual-based two-locus model was used to investigate the impact of two important issues, the consistency ofpesticide dosage through the storage facility and the immigration rate of the adult pest, on overall population control andavoidance of evolution of resistance to the fumigant phosphine in an important pest of stored grain, the lesser grain borer.

RESULTS: A very consistent dosage maintained good control for all immigration rates, while an inconsistent dosage failed tomaintain control in all cases. At intermediate dosage consistency, immigration rate became a critical factor in whether controlwas maintained or resistance emerged.

CONCLUSION: Achieving a consistent fumigant dosage is a key factor in avoiding evolution of resistance to phosphine andmaintaining control of populations of stored-grain pests; when the dosage achieved is very inconsistent, there is likely to be aproblem regardless of immigration rate.c© 2012 Society of Chemical Industry

Keywords: dosage consistency; immigration rate; phosphine resistance; lesser grain borer; individual-based; two-locus

1 INTRODUCTIONThe lesser grain borer, Rhyzopertha dominica, is a very destructive

primary pest of stored grains. Fumigation with phosphine (PH3)

is a key component in the control of pest infestations.1 However,

heavy reliance on PH3 has resulted in the development of strong

resistance in several major pest species, including R. dominica.

There is a worldwide need for the development of sustainable

management strategies to avoid the evolution of resistance and to

control pest infestation. Computer simulation models can provide

a relatively fast, safe and inexpensive means to understand the

causative processes of effective management and to weigh the

merits of various management options.

Research has indicated the existence of two resistance

phenotypes in R. dominica, weak and strong,2 and revealed that

the presence of homozygous resistance alleles at two loci confers

strong resistance.3,4 These results motivated the construction of

an individual-based, two-locus model of population dynamics and

resistance evolution in stored-grain insect pests. This individual-

based modelling explicitly represents the fact that R. dominica

populations consist of individual beetles, each of a particular

genotype and a particular life stage. Individual-based models are

relatively easily adapted to incorporate the biological attributes of

interest, such as different initial frequencies of genotypes, different

proportions of life stages (egg, larva, pupa and adult), spatial

location, movement and interactions between individual insects.5

The details of the present model have been described in previous

publications.6–9 These studies have shown the importance of

including the full complexity of the polygenic resistance in the

model, rather than using a simplified one-locus model.6 The model

was also used to test the efficacy of different short-term fumigation

tactics in maintaining control.8 However, the model has not yet

been used to evaluate and identify viable, long-term strategies

to support the management of phosphine resistance and pest

infestation, which is the ultimate aim of the authors’ research.

It has been suggested that two factors are particularly important

in managing pest population number and the evolution of

resistance in stored-grain pests: (1) achieving a consistent fumigant

∗ Correspondence to: Mingren Shi, M084, School of Plant Biology, FNAS, The

University of Western Australia, 35 Stirling Highway, Crawley, WA 6009,

Australia. E-mail: shi.mingren@gmail.com

a School of Plant Biology, FNAS, The University of Western Australia, Crawley,

WA, Australia

b Cooperative Research Centre for National Plant Biosecurity, Canberra, Australia

c Agri-Science Queensland, Department of Agriculture, Fisheries and Forestry,

Brisbane, Qld, Australia

d School of Animal Biology, FNAS, The University of Western Australia, Crawley,

WA, Australia

e Entomology Branch, Department of Agriculture and Food, Western Australia,

Bentley, WA, Australia

f CSIRO Ecosystem Sciences, Floreat, WA, Australia

Pest Manag Sci (2013) www.soci.org c© 2012 Society of Chemical Industry

Chapter 6

69

www.soci.org M Shi et al.

dosage (or concentration) within the storage facility; (2) controlling

immigration of insects from outside the facility.10 In this study, the

individual-based two-locus model is extended to include spatial

variability in dosage and immigration into the storage of adult

insects. This enables two significant management questions to be

addressed:

Q1 How does the consistency of dosage achieved within the

storage facility affect the evolution of resistance to phosphine

and insect population numbers?

Q2 What is the impact of the immigration rate of adult beetles

on the evolution of resistance to phosphine, and population

numbers?

A third question is also investigated:

Q3 Do the answers to the first two questions regarding

management options depend on the frequency of the strongly

resistant genotype of the pest in the general population (that

is, in initial and immigrating populations)?

2 METHODS2.1 Overview of the model

The overall model dynamics for an individual is illustrated in

Fig. 1. The present models run on a daily time step allowing

them to capture real conditions in more detail and obtain more

precise results than if they ran on a longer weekly time step. For

simulation purposes, the adult life stage is separated into two: a

pre-oviposition period when the female beetle is unable to lay

eggs and a subsequent, mature egg-laying period. However, the

counts of adult 1 and adult 2 are merged into a single adult stage

in the results. The simulation steps are described below:

1. Initialise the genotype, life stage and dosage experienced

(described below).

2. Check whether the current life stage ends. If so, enter the next

life stage and go to step (3), otherwise go to step (4).

3. Check whether (whole) lifetime is over. If so, the beetle is dead

and removed from the population, otherwise go to step (4).

4. Check whether currently under fumigation. If yes, go to step

(5); if no, go to step (6).

5. Check whether the beetle has survived. If so, go to the next day

and then repeat from step (2), otherwise the beetle is dead and

removed.

6. Check whether the (female) beetle is able to lay eggs (i.e.

whether it is in the adult 2 stage). If so, determine the number

of eggs and each egg’s sex (remove the males), life duration,

genotype and dosage experienced and then go to the next day.

Otherwise go directly to the next day. Then repeat from step 2.

A number of processes within the simulation are determined

stochastically; some of them are described below, and full details

of others have been provided previously.6,8

2.2 Parameters

Unless otherwise noted, the model parameters take the same

values as those in the authors’ previous work.6–9 For example:

• Natural growth rate = 0.4109/week,11 which was estimated

for grain stored at a temperature of 25 ◦C and a relative

humidity (RH) of 70%, conditions that are recommended by

FAO for bioassay12 and used by many researchers in their

experiments,3,13–18

• Mean (and standard deviation) number of days at the four life

stages (from published experimental data19–25) are:

Stage Egg Larva Pupa Adult Total

Mean 11.9 36.5 9.6 117 175

(std) (1.5) (4.6) (1.2) (15.0) (25 weeks)

(1)

where the mean number of adult stages includes 15 days (std = 0)

for the pre-oviposition period.26,27

• Natural death rate (average) = 1/25 = 0.04/week as the

average lifetime is 25 weeks [see text table (1) above].

Then the natural birth rate (per week) = natural growth

rate + natural death rate = 0.4509.28 Finally, the (natural)

daily finite birth rate (eggs per day per female parent) is

bD = exp(0.4509)/7 = 1.5697/7 = 0.2242.

• Equal initial proportion of four life stages: each = 1/4 = 0.25.

As before,6,8 a 1:1 sex ratio was assumed, making it possible

to set the start number and count the population of female

beetles, with the assumption that the total number and the allelic

frequency for the male beetles were the same as for the females. In

this study, the starting female beetle number is set to be 100 000

(100 K).

2.3 Two loci and nine genotypes

In the present two-locus model, for simplicity it is assumed that

there are two possible alleles (resistance or susceptibility) at each

of the two loci, meaning nine genotypes in total (Table 1).

Figure 1. The simulated dynamics for individual beetles at each daily time step during the simulation.

wileyonlinelibrary.com/journal/ps c© 2012 Society of Chemical Industry Pest Manag Sci (2013)

70

Dosage consistency is the key to avoiding resistance evolution www.soci.org

Table 1. The identifiers of nine genotypes (ss, sh, . . . , rr) in the two-locus model: s – homogeneous (‘homo’) susceptible (‘suscept’); r – homogeneousresistant (‘resist’); h – heterozygous (‘hetero’)

Second gene

First gene s homo suscept h heterozygous r homo resist

s homo suscept ss Both homo suscept sh First homo suscept, secondheterozygous

sr First homo suscept, second homoresist

h heterozygous hs First hetero, second homo suscept hh Both heterozygous hr First hetero, second homo resist

r homo resist rs First homo resist, second homosuscept

rh First homo resist, second heterozygous rr Both homo resist

2.4 Addressing Q1 regarding the impact of dosageconsistency

Rather than explicitly representing the spatial distribution of

the beetles and fumigants, it is assumed that, if the fumigant

concentration is variable, then the concentration experienced by

individual beetles is variable too. For simplicity it is assumed that

the dosage experienced by an individual beetle is constant over

time. Thus, the present model does not account for the possibility

of the concentration changing with time at a given point in the silo,

or of beetles moving during the course of phosphine fumigation

between locations with different concentrations.

In this study, the starting target fumigation dosage is selected as

C0 = 0.15 mg L−1 (108 ppm) for 14 days, which is close to the target

dosage in a real treatment used in Cooperative Bulk Handling

(CBH), Western Australia.1,29 The total survival rates (TSRs) for

this treatment according to previously developed probit mortality

models are:7

Genotype rh rr others

TSR 3.341 × 10−12 3.114 × 10−5 <5.14 × 10−29

(2)

However, the 0.15 mg L−1 is only the nominal target dosage.

Each beetle actually experiences a different dosage to every other

beetle owing to spatial heterogeneity within the silo, and the actual

dosage experienced by an individual beetle is always less than the

nominal target dosage to a lesser or greater degree. To model this

variability, the dose to which each beetle is exposed is generated

individually according to a power law distribution defined by

a parameter k. This parameter k depends on the maximum or

nominal target dosage (dmax) and the median dosage (dm) in the

following way:

k = log (0.5) /log (dm/dmax) (3)

A uniformly distributed random number p is generated for each

individual, and then the dosage d experienced by the individual is

yielded from

d = dmaxp1/k (4)

This ensures that the expected median dosage over many

individuals is indeed dm. It can be seen from the histograms in

Fig. 2 that, if the median dosage is closer to the maximum dosage,

then there is less variability and the vast majority of beetles receive

a high dosage. This corresponds to a situation where there is better

fumigant distribution within the silo, for example. On the other

hand, when the median dosage is further from the maximum

dosage, then there is more variability and the beetles receive a

wide range of dosages, including some cases of very low dosages.

This corresponds to a situation where there is poor circulation

within the silo.

The above three median dosages (dm = 0.14, 0.11 and 0.08 mg

L−1 see Fig. 2) were used to test how different amounts of

variability in dosage affect the evolution of resistance and

population increase. These different levels of variability in fumigant

concentration represent real factors in a spatially heterogeneous

storage facility. These factors include leakage of PH3 from the

silo, PH3 dispersion through the silo and degree of physical and

chemical reactions such as uptake or release of gas from or into

grain (sorption–desorption) and diffusion.30

2.5 Addressing Q2 regarding the impact of immigration rate

Immigration was represented by simply adding a number of adult

beetles into the population each day of the simulation. Four

different immigration rates were considered: 0 (no immigration),

20, 100 and 500 adult beetles day−1. The factors represented

by these different rates include hygiene conditions, the degree

of proper seal of the facility and the movement of pests from

places outside, where hygiene conditions are poor, to inside a

storage facility. These are wide but biologically reasonable ranges,

based on the limited data available,31,32 and are sufficient fully to

represent the above factors.

The initial proportions for the immigrating adult beetles are

0.4 and 0.6 for adult 1 and adult 2 stages respectively, as the

life duration for adult 1 is much shorter. The frequencies of the

nine genotypes for the immigrating adult beetles are the same as

the initial frequencies for the beetles inside the facility described

below.

2.6 Addressing Q3 regarding the frequency of the rr beetlesin the general population

A molecular analysis of phosphine resistance gene frequency in

R. dominica undertaken in July 2011 in eastern Australia found

a frequency of population samples with the rr genotype (that

is, homozygous for both resistance genes) of 0.01.33 In general

there appears to be variability between populations, and also

between regions, in terms of the frequency of the strongly

resistant genotypes. Therefore, two frequencies of the strongly

resistant genotype f (rr) = 0.01 and f (rr) = 0.1 were considered, to

see whether the effects of the above two management options

depended on the frequency of the rr beetles in the general

population. This frequency was assumed to be the ‘background’

frequency of the rr genotype in the general population in the area

of the storage being simulated, and thus the frequency is used

for both the initial population in the storage and the external

population providing all immigrating insects. In both cases,

f (rr) = 0.01 and f (rr) = 0.1, the frequencies of other genotypes

were calculated by assuming that the general population

is at Hardy–Weinberg equilibrium,34 giving the following

Pest Manag Sci (2013) c© 2012 Society of Chemical Industry wileyonlinelibrary.com/journal/ps

71

www.soci.org M Shi et al.

Figure 2. Histograms for distribution of dosages among 100 K individual beetles with dmax = 0.15 mg L−1 and (a) dm = 0.14 mg L−1 (k = 10.0466),(b) dm = 0.11 mg L−1 (k = 2.2348) and (c) dm = 0.08 mg L−1 (k = 1.1027).

Table 2. The short-hand identifiers for the 12 combinations ofmedian dosages and immigration rates considered in this study,including three cases without immigration and nine cases withimmigration (maximum dosage = 0.15 mg L−1)

With immigration

(number of immigrations per day)Median

dosage

(mg L−1)

No

immigration 20 100 500

0.14 (D14) D14N0 D14N20 D14N100 D14N500

0.11 (D11) D11N0 D11N20 D11N100 D11N500

0.08 (D08) D08N0 D08N20 D08N100 D08N500

results:

ss hs rs sh hh rh

0.21860 0.20219 0.04675 0.20219 0.18702 0.04325

sr hr rr

0.04675 0.04325 0.01(5)

ss hs rs sh hh rh

0.03669 0.09428 0.06057 0.09428 0.24229 0.15566

sr hr rr

0.06057 0.15566 0.1(6)

2.7 Simulations

All combinations of median dosages (three levels) and

immigration numbers (four levels) were considered. This meant

12 combinations in total, which are listed in Table 2. All 12

combinations were simulated for both the initial frequencies of

genotypes shown in equations (5) and (6), so there were 24

simulations in total.

The simulation period was 732 days (≈ 2 years), and six

fumigations (F1, F2, . . . , F6) were implemented, each lasting

14 days, followed by a natural fumigant-free growth period lasting

108 days. Thus, the simulation period consisted of six phases (P1,

P2, . . . , P6), each lasting 14 + 108 = 122 days (Table 3). This

represents the preferred practice in places such as Australia of

using no more than three conventional fumigations per year on

undisturbed grain.10

3 RESULTSAll of the 24 simulations were run 6 times to check for

stochastic variation, and the results were very similar each time.

Table 3. The duration of each phase: 14 day fumigation followed by108 day natural growth period

Phase

Fumigation period

(14 days long)

Natural growth period

(108 days long)

P1 1–14 15–122

P2 123–136 137–244

P3 245–258 259–366

P4 367–380 381–488

P5 489–502 503–610

P6 611–624 625–732

Total population numbers (TPN) always decreased during each

fumigation period and increased during each natural growth

period, and so a local minimum value of daily population numbers

was reached at the end of each fumigation, and a local maximum

value of daily population numbers was achieved at the end of each

phase (or natural growth period) (Fig. 3). Note that the TPN at the

end of a fumigation can be zero; if there is no immigration, then

TPN will remain at zero, but for the treatments with immigration

the population will increase again with immigration and also with

reproduction as these immigrants start to reproduce.

3.1 Effect of dosage consistency and concentration whenthe general frequency f (rr) = 0.01

The results for the 12 cases when the general frequency of rr

beetles was equal to one in a hundred are shown in Table 4 and

Fig. 3, and the important patterns in these results are summarised

in Table 5. It can be seen from Table 5 and Fig. 3 that:

• In the four cases with a very consistent high dosage (D14),

population numbers were zero or close to zero.

• In the case of medium consistent dosage (D11):

• when immigration was zero or relatively low (D11N0 and

D11N20), population numbers decreased;

• when immigration was medium (D11N100), population

numbers were stable, varying over a small interval;

• when immigration was high (D11N500), population numbers

increased.

• In the four cases with a very inconsistent dosage (D08),

population numbers increased, each from about 1000. Note,

however, that the top values ranged from 19 000 (D08N0)

to 9600 (D08N20) and then to 8600 (N08N100), and then

increased to 23 700 (D08N500); this is due to a dilution effect

from immigration.

wileyonlinelibrary.com/journal/ps c© 2012 Society of Chemical Industry Pest Manag Sci (2013)

72

Dosage consistency is the key to avoiding resistance evolution www.soci.org

Figure 3. The daily population numbers (log scale) and the rr proportions in the four cases with the same median dosage and f (rr) = 0.01 over 732 daysfor (a), (b) dm = 0.14 mg L−1 , (c), (d) dm = 0.11 mg L−1 and (e), (f) dm = 0.08 mg L−1. Note that for D14N0 in (a) the daily population numbers at the end ofF1 (the 14th day) and afterwards are zero, as are the population numbers at the end of each fumigation in the cases of D14N20 and D14N100. These zerovalues are not shown because of the log scale. Also note the different scale of the y-axis in (a), (c) and (e).

There is an important transition point at D11N100 (100

immigrants per day) where total insect population number (TPN)

is stable. TPN decreased where immigration rates were less than

100 and increased when they were higher than 100 per day.

On the other hand, Table 5 and Fig. 3 show that the

corresponding proportions of the rr beetles (Prrs) at the end

of each of six fumigations were:

• In the four cases with a very consistent high dosage (D14), the rr

proportions were zero or stable. Note that, when the population

reaches zero, the rr proportion is recorded as zero.

• In cases with a medium consistent dosage (D11):

• when there was no immigration (D11N0), the rr proportions

increased to 100%;

• when there was immigration (D11N20, D11N100 and

D11N500), the rr proportions were stable at similar levels.

• In the four cases with a very inconsistent dosage (D08), the

rr proportions all increased from similar low values, but the

increase was smaller for higher immigration numbers.

Note that each fumigation duration is only 14 days, which is

only 2% of the whole simulation period (732 days), and so it may

look as if the plot in Fig. 3b does not match with the rr proportions

at the end of each fumigation listed in Tables 4 and 5. Hence,

for clarification, the second and fourth fumigation periods are

plotted for the D14 scenarios in detail in Fig. 4. Here it is seen

that rr proportions increase at the start of the fumigation as the

less resistant genotypes die, but then fall to zero as the high,

consistent dosage kills the rr insects as well, so that by the end of

the fumigation TPN and Prrs have both reached zero for D14N0,

D14N20 and D14N100.

To gain a full understanding of the dilution effect of immigration,

more information was needed about the population numbers of

Pest Manag Sci (2013) c© 2012 Society of Chemical Industry wileyonlinelibrary.com/journal/ps

73

www.soci.org M Shi et al.

Table 4. Total population number (TPN) and proportion of rr beetles (Prr) initially and at the end of each fumigation (EndF1, EndF2, . . . , EndF6) inall 12 cases when f (rr) = 0.01

D14N0 D14N20 D14N100 D14N500

Time (day) TPN Prr TPN Prr TPN Prr TPN Prr

Initially 100 000 0.01 100 020 0.01 100 100 0.01 100 500 0.01

EndF1(14th) 0 0.000 0 0.000 0 0.000 7 0.714

EndF2(136th) 0 0.000 0 0.000 0 0.000 10 0.500

EndF3(258th) 0 0.000 0 0.000 0 0.000 10 0.700

EndF4(380th) 0 0.000 0 0.000 0 0.000 9 0.556

EndF5(502nd) 0 0.000 0 0.000 0 0.000 13 0.692

EndF6(624th) 0 0.000 0 0.000 0 0.000 16 0.562

D11N0 D11N20 D11N100 D11N500

EndF1(14th) 76 0.671 83 0.566 98 0.541 116 0.612

EndF2(136th) 51 0.843 53 0.623 111 0.649 386 0.552

EndF3(258th) 29 1.000 26 0.731 108 0.676 480 0.623

EndF4(380th) 25 1.000 21 0.667 128 0.703 498 0.641

EndF5(502nd) 19 1.000 21 0.714 116 0.655 492 0.618

EndF6(624th) 14 1.000 18 0.556 110 0.618 464 0.619

D08N0 D08N20 D08N100 D08N500

EndF1(14th) 975 0.222 998 0.235 929 0.229 1066 0.216

EndF2(136th) 1196 0.444 1305 0.363 1778 0.292 4716 0.238

EndF3(258th) 1742 0.703 1682 0.511 2609 0.366 8433 0.288

EndF4(380th) 3445 0.857 2647 0.643 3662 0.432 12 227 0.342

EndF5(502nd) 7958 0.939 4692 0.750 5450 0.523 16 869 0.420

EndF6(624th) 19 104 0.976 9640 0.847 8619 0.622 23 673 0.494

Figure 4. The proportions of the rr genotype (a) during the second fumigation period from day 123 to day 136 and (b) during the fourth fumigationperiod from day 367 to day 380.

other genotypes, and these are provided in Tables 6 and 7. The

nine genotypes can be divided into three groups: SH (ss, sh, hs and

hh), R1 (sr, hr, rs and rh) and rr – resistant homozygotes. Almost

all SH beetles died after each fumigation. Mating SH beetles with

rr beetles produces 14/15 SH and R1 offspring and 1/15 rr beetles;

mating R1 × rr produces 3/4 R1 and 1/4 rr progeny; only mating

rr × rr will reproduce 100% rr genotypes. A detailed explanation is

provided in Shi and Renton.9

3.2 Effect of dosage consistency and concentration whenthe general frequency f (rr) = 0.1

In general, the results for f (rr) = 0.1, summarised in Table 8, are

similar to those for f (rr) = 0.01 (Table 5). However, the cases where

population numbers changed from decreasing to stable or from

stable to increasing or from decreasing to increasing occurred

earlier in the table when the fully resistant genotype was present at

a higher frequency in the initial and immigrating populations. For

example, the first case where numbers are increasing is D11N100,

when f (rr) = 0.01, but this occurs at D11N500, when f (rr) = 0.1. The

overall population numbers when f (rr) = 0.1 were much bigger

than in the corresponding cases when f (rr) = 0.01. This pattern is

also repeated for rr proportions (Prrs) in all cases of D14. Finally,

the proportions of Prr when f (rr) = 0.1 were very similar to all

corresponding cases of D11 and D08, when f (rr) = 0.01, but the

variations in frequencies were less when f (rr) = 0.1 than when

f (rr) = 0.01.

wileyonlinelibrary.com/journal/ps c© 2012 Society of Chemical Industry Pest Manag Sci (2013)

74

Dosage consistency is the key to avoiding resistance evolution www.soci.org

Table 5. Patterns of total (local minimal) population numbers (TPNs)and corresponding rr proportions (Prrs) at the end of fumigationslisted in Table 4.

Number of immigrations (per day)

Median dosage N0 N20 N100 N500

TPN

D14 #0 #0 #0 ∼[7, 16]

D11 76↓1483↓18 ∼[100, 130] 116↑

500

D08 ∼1000↑19 000

1000↑9640

1000↑8620

1100↑23 700

Prr

D14 #0 #0 #0 ∼[0.50, 0.71]

D11 0.67↑1.0 ∼[0.56, 0.73] ∼[0.54, 0.70] ∼[0.55, 0.64]

D08 0.22↑0.98

0.24↑0.85

0.23↑0.62

0.22↑0.49

• Notation key:• N1↑

N2: TPNs or Prrs increasing over time with each fumigation from N1to N2

• N1↓N2: TPNs or Prrs decreasing over time with each fumigation from N1to N2

• ∼[N1, N2]: TPNs or Prrs remain relatively stable within a small interval[N1, N2]

• #0: TPNs or Prrs are all zeros at the end of each fumigation

Table 6. The pattern of population numbers in the three groupsat the end of each of five phases (EndP1–EndP5) in the cases ofD08N0 and D08N20 (low median dosage with either no immigrationor immigration of 20 adult beetles day−1)

Case SH R1 rr In total

D08N0 2521↓36839↓5126 2178↑

514611 538↑

86 581

D08N20 12 914↓10 360 9742↑21 495

1925↑34 489

25 199↑65 726

Table 7. The pattern of population numbers in the three groups atthe end of each of five fumigations (EndF2–EndF6) in the cases ofD08N0 and D08N20. Note that the population number at the end of aphase in Table 6 is the starting number of the next fumigation

Case SH R1 rr In total

D08N0 80↓0585↓455 531↑

18 6491196↑

19 104

D08N20 81↓35 750↑1438

474↑8167

1305↑9640

4 DISCUSSIONThe analyses of the results demonstrate clearly that consistency

in phosphine dosage is the key factor in avoiding evolution of

resistance to phosphine and suppressing population increase in

R. dominica. When a high and consistent dose is achieved, there

is no increase in population numbers or increase in frequency of

resistance, regardless of immigration rate. When the dose achieved

is very inconsistent, however, population numbers will increase

and frequency of resistance will increase, regardless of immigration

rate.

This simulation has highlighted the potential trade-offs that

occur when pest management practitioners must control insect

pest population numbers and limit resistance development at

the same time. Their aim is to prevent insect infestation so that

the commodity is maintained at the required standard of quality.

The present simulations show that applying a consistent dose

Table 8. Patterns of total (local minimal) population numbers (TPNs)and corresponding rr proportions (Prrs) at the end of fumigationswhen f (rr) = 0.1

Number of immigrations (per day)

Median dosage N0 N20 N100 N500

TPN

D14 #0 #0 #[10, 17] ∼[53, 74]

D11 610↓110580↓320 610↑

1160770↑

5400

D08 3900↑223 000

3900↑240 000

3900↑274 000

4340↑659 000

Prr

D14 #0 #0 ∼[0.50, 0.70] ∼[0.46, 0.55]

D11 0.84↑1.0 ∼[0.85, 0.92] ∼[0.84, 0.89] ∼[0.82, 0.89]

D08 0.54↑0.995

0.56↑0.99

0.55↑0.96

0.54↑0.94

of phosphine high enough to overcome resistance mechanisms,

in a storage that prevents immigration, will achieve the desired

result. However, in practice, consistently high dosages can often

be difficult to achieve and maintain in large grain bulks, resulting in

incomplete kill of resident insect populations. When immigration

is excluded, selection of resistance develops faster in storages with

moderate dosage inconsistency (D11N0) than in those with high

consistency (D14N0), simply because with high consistency all

insects are killed, including the strongly resistant ones; however,

in these cases the very low numbers of insects present in the grain

would probably not be detected. In storages with moderate dosage

inconsistency, overall insect numbers increase when immigration

is high, but this increase is relatively slow; in these storages,

resistance frequency only increases when immigration is excluded.

It is only in storages that have low dosage consistency (D08N0)

and in practice would be poorly sealed that insect numbers grow

to very high levels. It is also likely that fumigations would occur

more often in these storages, as insects would be more frequently

detected. In addition, the present analysis reveals that, in storages

with low dosage consistency, resistance frequencies continue to

increase with every fumigation undertaken; immigration has a

dilution effect, but this is not enough to counteract the increase

in resistance frequencies, even at the highest immigration rates.

Therefore, the most practical strategy for the pest management

practitioner to adopt is to ensure that appropriately high dosages

are applied evenly to all parts of the storage. High dosage

consistency is achieved by ensuring that storages are well sealed

and active mechanisms are used to distribute phosphine evenly

throughout the storage.

4.1 The impacts of dosage consistency and immigrationrates on population increase

The results revealed that whether overall population numbers

increased, decreased or remained relatively stable was determined

by an interaction between dosage consistency (median phosphine

concentration) and immigration rates, with the former being the

most important factor. This is because the number of insects

(TPN) surviving a fumigation period was largely determined by the

interaction between the dosage consistency (median phosphine

concentration) and the initial number of beetles at the start of

the treatment, with the latter being strongly affected by the

immigration rate over the preceding fumigant-free period. The

lower the median dose, the more insects survived the fumigation,

and the higher the immigration rates, the higher the initial number

of insects, and the higher the number surviving. When the median

Pest Manag Sci (2013) c© 2012 Society of Chemical Industry wileyonlinelibrary.com/journal/ps

75

www.soci.org M Shi et al.

Table 9. TPN and Prr in all 12 cases at the end of each phase (denoted by EndP1, EndP2, . . . , EndP6) [f (rr) = 0.01]

D14N0 D14N20 D14N100 D14N500

Time (day) TPN Prr TPN Prr TPN Prr TPN Prr

Initially 100 000 0.01 100 020 0.01 100 100 0.01 100 500 0.01

EndP1(122nd) 0 0.000 13 420 0.011 73 146 0.012 371 644 0.010

EndP2(244th) 0 0.000 13 459 0.011 73 097 0.011 371 443 0.010

EndP3(366th) 0 0.000 13 455 0.010 73 101 0.012 371 446 0.010

EndP4(488th) 0 0.000 13 408 0.009 73 237 0.011 371 488 0.010

EndP5(610th) 0 0.000 13 482 0.011 73 129 0.012 371 456 0.010

EndP6(732nd) 0 0.000 13 442 0.009 73 113 0.012 371 676 0.010

D11N0 D11N20 D11N100 D11N500

EndP1(122nd) 868 0.753 14 407 0.025 74 277 0.015 373 053 0.011

EndP2(244th) 430 0.970 13 950 0.019 74 298 0.015 375 760 0.012

EndP3(366th) 311 1.000 13 757 0.016 74 263 0.016 376 750 0.013

EndP4(488th) 245 1.000 13 632 0.013 74 585 0.017 376 745 0.013

EndP5(610th) 185 1.000 13 639 0.013 74 523 0.015 376 891 0.013

EndP6(732nd) 179 1.000 13 670 0.013 74 382 0.016 376 593 0.012

D08N0 D08N20 D08N100 D08N500

EndP1(122nd) 11 538 0.189 25 199 0.076 83 896 0.025 384 248 0.013

EndP2(244th) 12 772 0.403 27 668 0.134 92 654 0.043 424 795 0.024

EndP3(366th) 18 741 0.687 31 551 0.222 100 912 0.067 462 925 0.038

EndP4(488th) 37 353 0.851 42 234 0.352 113 112 0.103 502 931 0.058

EndP5(610th) 86 581 0.941 65 726 0.525 133 100 0.168 553 838 0.088

EndP6(732nd) 208 569 0.977 119 172 0.702 167 392 0.261 628 198 0.133

dose was very high (D14), dose was the dominant factor, as it was

high enough to kill all or nearly all of the insects. When the median

dose was lower (D11), the dominance of dosage became weaker

and immigration rates became more important. If immigration

rates were zero or relatively low (D11N0, D11N20 and D11N100),

then, even though population numbers could reach high levels

by the end of the fumigant-free period (Table 9), the mortality

from fumigation was still high enough to balance this growth

and immigration, so that overall numbers did not increase. With

high immigration (D11N500), the very high population numbers

reached by the end of the fumigant-free period (Table 9) tipped

the balance, so that the kill rates achieved by the moderately

consistent fumigation were not high enough to stop the overall

population numbers from increasing. When the median dosage

was lowest, indicating high inconsistency (D08), the overall

population increased in all cases because kill rates were never

high enough to balance the high numbers reached by the start of

the fumigation owing to growth and immigration (Table 9).

Achieving consistently high dosage throughout the storage

facility should thus be a primary management aim. Many factors,

such as leakage from the silo, sorption–desorption, convection

currents and diffusion, affect phosphine dosage consistency in a

spatially heterogeneous storage facility. The results of the present

simulations confirm that increasing the toxic level, i.e. increasing

the phosphine concentration (here, the median dose), will result

in practical benefits. Increasing the target dose (the maximum

dose in terms of this study) would be likely to have similar

benefits. Furthermore, in a previous study, simulations suggested

that extending the exposure time is an even more effective way

of increasing the toxic effect than increasing the concentration.8

However, there are definite limits to the extent to which maximum

dosage can be increased and fumigation duration can be extended,

so achieving high consistency is still essential.13 These results show

that fumigation in a well-sealed silo at a reasonable target dosage

will allow phosphine to be held at the required concentration

for long enough to ensure destruction of resistant homozygotes

(the rr individuals)14,35 and minimise the opportunity for insects to

escape the toxicant.

When the dosage of phosphine achieved is moderately

inconsistent, immigration rates become critical. If immigration is

high, insect population numbers increase markedly, even though

the frequency does not increase. Thus, the present results support

industry pest management practices that reduce the possible

movement of insects into storages. These include inspection of

grain for insect infestation before inloading, maintaining a high

standard of silo structural sealing throughout the storage period

and employing assiduous sanitation around and in storages

to remove harbourages for insects. Maintaining high standards

of sealing also contributes significantly to ensuring dosage

consistency, so quality sealing addresses both these important

factors.

4.2 The impact of dosage consistency and immigration rateon resistance frequency within the storage

The present results indicate that the evolution of resistance to

phosphine in the lesser grain borer is not a major problem if

consistent high dosages can be achieved over an adequate period

of time. When a very consistent dosage is always achieved, then

pesticide resistance evolution is not a problem; resistance does

not evolve because the achieved mortality of the rr genotype

beetles is high enough to keep their population frequency from

increasing. When a moderately consistent dosage is achieved, the

wileyonlinelibrary.com/journal/ps c© 2012 Society of Chemical Industry Pest Manag Sci (2013)

76

Dosage consistency is the key to avoiding resistance evolution www.soci.org

frequency of the rr beetles reaches high levels, particularly when

there is no immigration, but does not increase over time. Kill rates

achieved when there is no immigration or low immigration are

still high enough to keep population numbers under control, even

with the high level of resistance. At high immigration rates and

a moderately consistent dosage, population numbers increase

even though the frequency of resistance does not increase. In

this moderately consistent dosage case, the presence of the

resistant genotype in the overall population is a problem, but

the selection for this genotype (the evolution of resistance) is not

in itself a problem. However, when a low consistency of dosage

is achieved, the frequency of the rr beetles does increase over

time. This exacerbates the problem of low kill rates and increasing

population numbers, contributing to the observed population

explosions.

The results of the present simulations also confirmed the

potentially important effect of immigration in diluting resistance

gene frequency. This is evident from comparison of the different

immigration levels for those highly variable fumigations with a low

median dose and low consistency. The overall effect of immigration

on resistance frequency is indicated clearly in Fig. 3f, where

frequency increases more slowly for cases where immigration is

higher. The effects of this difference in frequency increase rate

on population numbers are then shown in Fig. 3e, where initially

population numbers increase more rapidly in the cases where

immigration is higher, but after some time the opposite pattern

in resistance frequency starts to dominate and the population

numbers increase faster in the cases where immigration is lower.

By the end of the simulated period, it can be seen that numbers in

the no immigration case (D08N0) have overtaken those in the lower

immigration cases (D08N20 and D08N100) and are approaching

those of the high immigration case (D08N500).

This dilution effect can be understood in more detail by focusing

on two particular cases, one with low immigration (D08N20 – 20

immigrant beetles day−1) and one with no immigration (D08N0).

After the first natural growth period (EndP1), the numbers of rr

beetles in the two situations were nearly the same, but the numbers

of SH (ss, sh, hs and hh), R1 (sr, hr, rs and rh) and the total number

in the latter were much bigger than those in the former (Table 6).

When these beetles laid eggs, most were SH and R1 genotypes, and

the proportion of rr beetles was reduced (Table 7). These numbers

were the starting numbers in the second fumigation, in which

almost all SH and the greater part of R1 beetles died. Immigration

resulted in an increase in the numbers of SH and R1 beetles from

EndP2 to EndP5 in D08N20, while the numbers of SH and R1 insects

decreased in D08N0. In addition, the number of rr beetles increased

at a faster rate in D08N0 than in D08N20. That is why, after the

last (sixth) fumigation, the total number of survivors (TPN) and the

proportion of rr individuals in D08N0 were both greater than in

D08N20 (Tables 6 and 7). Similarly, after the last fumigation, owing

to the relative decrease in rr frequency from higher immigration

in D08N100, the total population number was higher in D08N20

than in D08N100; although the difference between them was

not yet great, the trends shown in Table 4 clearly indicate that

numbers are growing much faster for D08N20 than for D08N100,

and so the difference would be expected to continue to grow.

These simulations demonstrate that the flow of susceptible genes

through immigration into populations under selection will reduce

the proportion of strongly resistant rr individuals.10 However, as

demonstrated by D08N500, the numbers of insects required to

reduce rr genotype frequency is very high and would likely result

in damage to the commodity and be inconsistent with market

standards.10

4.3 The impact of frequency of strongly resistant beetles ininitial and immigrating populations

When the dose achieved is very inconsistent (D08), population

numbers increase more rapidly and resistance evolves more

quickly if the frequency of the rr beetles in the initial and

immigrating populations is high (Tables 5 and 8). This is because

the proportions of rr genotypes in populations at the start of each

fumigation are bigger when f (rr) = 0.1 than when f (rr) = 0.01,

and so numbers surviving fumigations were more than 10 times

greater (Tables 5 and 8, and Figs 3 and 5). As the numbers surviving

fumigation are so much higher, growth in the fumigant-free period

starts from a higher base and reaches much higher levels by the

next fumigation, even when immigration is the same. When the

‘background’ frequency of strongly resistant beetles is higher, the

diluting effect of immigration is also much lower (contrast Figs 3

and 5). For example, the difference between the four immigration

rates is much smaller in Fig. 3f than in Fig. 5f.

These results emphasise the potential benefit of reducing the

frequency of rr genotypes in the general population. If rr genotypes

already exist within a storage, then the practical solution is either

to avoid using phosphine or to apply it consistently at a dosage

high enough to control these insects.14,35 (Another solution would

be to switch to a different fumigant with a totally different mode of

action if an equally acceptable or efficient alternative is available,

but in many situations this may not be the case.)

4.4 Strengths, limitations and future work

To the present authors’ knowledge, previous models of population

dynamics or evolution of resistance to insecticides in stored-grain

pests (36–42 for example) have not included the level of biological

and genetic detail and realism accounted for in this study, which

is enabled by the individual-based approach used here.5 This is

also the first model to consider evolution of fumigant resistance.

As mentioned in the descriptions of this model, previous studies

have not represented the way in which resistance is conferred by

alleles at two different loci, resulting in nine different genotypes

with different levels of resistance and thus different patterns

of mortality from various dosages, and have not parameterised

these mortality by genotype by dosage by duration relationships

from detailed experimental data in the way this model does.

Furthermore, previous studies have not been able to represent the

way in which dosages vary between individual beetles to present

the spatial heterogeneity within the silo in the way this study

does, and thus have not been able to predict the effects of such

heterogeneity in dose on population dynamics and the evolution

of resistance over multiple fumigations over a long time period.

The authors believe that these kinds of improved individual-based

model, based on better data and including more biological detail,

will help to gain insights into the processes of the evolution of

phosphine resistance in R. dominica. Importantly, they will also

allow better evaluation of the merits of various management

tactics and strategies for suppressing population numbers and

delaying or avoiding the evolution of resistance.

In conclusion, some comments about the limitations of the

present study and recommendations for further research. Here,

the authors’ previous individual-based two-locus model has been

extended to include spatial variability in dosage and immigration

of adult insects. This was based on assumptions that the fumigation

Pest Manag Sci (2013) c© 2012 Society of Chemical Industry wileyonlinelibrary.com/journal/ps

77

www.soci.org M Shi et al.

Figure 5. The daily total population number (TPN) (in log10 scale) and rr proportion (Prr) in the four cases with the same median dosage and f (rr) = 0.1(in total 728 days): (a), (b) for median dosage dm = 0.14 mg L−1 ; (c), (d) for dm = 0.11 mg L−1 ; (e), (f) for dm = 0.08 mg L−1. Note that the daily populationnumbers at the end of F1 (14th day) and afterwards in the case of D14N0 are all zero, and the population numbers at the end of each fumigation in thecase of D14N20 are all zero.

dosage experienced by an individual beetle within a storage facility

or silo at any time is constant and thus does not account for the

reality of changing concentrations within grain storages over time.

Further studies representing temporal heterogeneity of dosage

are needed to investigate what impacts such variability will have

on the short- and long-term efficacy of different control options

and the contribution of resistance alleles to further generations.

There are a number of other biological and operational factors

that should also be considered. For example, emigration of insects

from the storage and insects moving within the storage to escape

the toxicant may be important. In this study, emigrating adults are

assumed to be lost from the simulation, but, if external populations

were modelled explicitly, the life stage and resistance genetics of

immigrating and emigrating insects would need to be considered.

The distribution of insects in grain bulks is not random and varies

with species; this may affect dosage experienced and therefore

influence mortality, selection rates and population growth. The

assumption that actual dosage is always less than or equal to the

target dose may be incorrect, and real data on actual dosages

achieved across the spatial and temporal extent of the fumigation

could be incorporated into the model. Temperature, and thus

growth and reproduction rates, are assumed to be constant in this

study, but in reality they are likely to vary in both time and space;

this could have implications for resistance evolution, which should

be considered in future studies. The impact of initial population

size was not covered here, but previous study showed that this

does not greatly affect resistance dynamics unless populations are

very small.8 Only a fixed schedule of three fumigations per year

was considered, but the frequency of fumigation is also likely to

affect populations and resistance evolution, as is switching to an

adaptive or flexible strategy where fumigation is applied when

populations reach critical threshold levels. Differences in fitness

between resistant and susceptible insects have been identified,17

but the effect of this phenomenon on long-term control and

wileyonlinelibrary.com/journal/ps c© 2012 Society of Chemical Industry Pest Manag Sci (2013)

78

Dosage consistency is the key to avoiding resistance evolution www.soci.org

resistance frequency is not yet clear; fitness costs could be

incorporated into the model to address this issue. In addition,

if the aim is to preserve the useful life of phosphine, then the

impact on resistance selection and IPM of the use of alternative

pesticides and non-chemical treatments such as cooling should

be investigated. The costs and benefits from different strategies

will be an important factor in deciding in the field which approach

to take. The authors recommend that, owing to its flexibility

and ability to represent important aspects of biological reality

and inter-individual variation, an individual-based approach such

as the one employed in this study is the best way to address

these different issues in the future.5 Finally, the present model is

parameterised for one particular species, the lesser grain borer,

and its particular resistance patterns, but it is likely that many of

the results could be generalised to other stored-grain pest species,

especially if their resistance patterns are similar; the present model

could be relatively easily modified to test the validity of such

generalisation and to develop species-specific recommendations

when such generalisation fails.

ACKNOWLEDGEMENTSThe authors would like to acknowledge the support of

the Australian Government’s Cooperative Research Centres

Programme. They also thank Yonglin Ren and the Grains Research

and Development Corporation for their great help in provision

of raw data and information about beetle life cycles and silo

fumigation.

REFERENCES1 Emery RN and Nayak MK, Cereals – pests of stored grains, in

Pests of Field Crops and Pastures: Identification and Control, ed.by Bailey P, Ch. 2. [Online]. CSIRO Publishing (2007). Available:http://www.publish.csiro.au/pid/3165.htm.

2 Collins PJ, Resistance to grain protectants and fumigants in insect pestsof stored products in Australia, in Proc First Australian PostharvestTechnical Conference, ed. by Banks HJ et al. Canberra, Australia, pp.55–57 (1998).

3 Collins PJ, Daglish GJ, Bengston M, Lambkin TM and Pavic H, Geneticsof resistance to phosphine in Rhyzopertha dominica (Coleoptera:Bostrichidae). J Econ Entomol 95(4):862–869 (2002).

4 Schlipalius DI, Cheng Q, Reilly PE, Collins PJ and Ebert PR, Geneticlinkage analysis of the lesser grain borer Rhyzopertha dominicaidentifies two loci that confer high-level resistance to the fumigantphosphine. Genetics 161:773–782 (2002).

5 Renton M, Shifting focus from the population to the individual asa way forward in understanding, predicting and managing thecomplexities of evolution of resistance to pesticides. Pest Manag SciDOI: 10.1002/ps.3341 (2012).

6 Shi M, Renton M, Ridsdill-Smith J and Collins PJ, Constructing a newindividual-based model of phosphine resistance in lesser grainborer (Rhyzopertha dominica): do we need to include two loci ratherthan one? Pest Sci 85(4):451–468 (2012).

7 Shi M, Renton M and Collins PJ, Mortality estimation for individual-based simulations of phosphine resistance in lesser grainborer (Rhyzopertha dominica), in MODSIM2011, 19th InternationalCongress on Modelling and Simulation, ed. by Chan F, MarinovaD and Anderssen RS, pp. 352–358. [Online]. Modelling andSimulation Society of Australia and New Zealand (2011). Available:http://www.mssanz.org.au/modsim2011/A3/shi.pdf.

8 Shi M, Collins PJ, Ridsdill-Smith J and Renton M, Individual-basedmodelling of the efficacy of fumigation tactics to control lessergrain borer (Rhyzopertha dominica) in stored grain. J Stored Prod Res51:23–32 (2012).

9 Shi M and Renton M, Numerical algorithms for estimation andcalculation of parameters in modelling pest population dynamicsand evolution of resistance in modelling pest population dynamicsand evolution of resistance. Math Biosci 233(2):77–89 (2011).

10 Opit G, Collins PJ and Daglish GJ, Resistance management, in StoredProduct Protection, ed. by Hagstrum DW, Thomas W, Phillips TW andCuperus G. Kansas State Research and Extension Publication S156,Ch. 13 (2012).

11 Driscoll R, Longstaff BC and Beckett S, Prediction of insect populationsin grain storage. J Stored Prod Res 36:131–151 (2000).

12 Recommended methods for the detection and measurement ofresistance of agricultural pests to pesticides: tentative methodfor adults of some major species of stored cereals with methylbromide and phosphine. FAO Method No. 16. FAO Plant Prot Bull23:12–25 (1975).

13 Collins PJ, Daglish GJ, Nayak MK, Ebert PR, Schlipalius DI, Chen W et al.,Combating resistance to phosphine in Australia. Proc Int Conf onControlled Atmosphere and Fumigation in Stored Products, Fresno,CA, pp. 593–607 (2000).

14 Collins PJ, Daglish GJ, Pavic H and Kopittke RA, Response of mixed-agecultures of phosphine resistant and susceptible strains of lessergrain borer, Rhyzopertha dominica, to phosphine at a range ofconcentrations and exposure periods. J Stored Prod Res 41:373–385(2005).

15 Daglish GJ, Effect of exposure period on degree of dominance ofphosphine resistance in adults of Rhyzopertha dominica (Coleoptera:Bostrychidae) and Sitophilus oryzae (Coleoptera: Curculionidae). PestManag Sci 60(8):822–826 (2004).

16 Herron GA, Resistance to grain protectants and phosphine incoleopterous pests of grain stored on farms in New South Wales.Aust J Entomol 29:183–189 (1990).

17 Pimentel MAG, Faroni LA, Totola MR and Guedes RNC, Phosphineresistance, respiration rate and fitness consequences in stored-product insects. Pest Manag Sci 63:876–881 (2007).

18 Schlipalius DI, Chen W, Collins PJ, Nguyen T, Reilly PEB and Ebert PR,Gene interactions constrain the course of evolution of phosphineresistance in the lesser grain borer, Rhyzopertha dominica. Heredity100:506–516 (2008).

19 Baldassari N, Martini A, Cavicchi S and Baronio P, Effects of lowtemperatures on adult survival and reproduction of Rhyzoperthadominica. Bull Insectol 58(2):131–134 (2005).

20 Rees D, Insects of Stored Products, CSIRO Publishing, Collingwood, Vic,Australia (2004).

21 Andrewartha HG and Birch LC, Selections from ‘The Distribution andAbundance of Animals’. University of Chicago Press, Chicago, IL(1982).

22 Birch LC, The influence of temperature on the development of thedifferent stages of Calandra oryzae L. and Rhizopertha dominica Fab.(Coleoptera). Aust J Exp Biol Med Sci 23(1):29–35 (1945).

23 Birch LC, Experimental background to the study of the distributionand abundance of insects. I. The influence of temperature, moistureand food on the innate capacity for increase of three grain beetles.Ecology 34(5):698–711 (1953).

24 Beckett SJ, Longstaff BC and Evans DE, A comparison of the demo-graphy of four major stored grain Coleopteran pests and itsimplications for pest management. Proc 6th Int Conf on Stored-Product Protection. Vol. 1, CAB International, Wallingford, Oxon, UK,pp. 491–497 (1994).

25 Longstaff BC, An experimental and modelling study of thedemographic performance of Rhyzopertha dominica (F.). I.Development rate. J Stored Prod Res 35:89–98 (1999).

26 Matin ASM and Hooper GHS, Susceptibility of Rhyzopertha dominicato ionizing radiation. J Stored Prod Res 10:199–207 (1974).

27 Schwardt HH, Life history of the lesser grain borer. J Kans Entomol Soc6(2):61–66 (1933).

28 Birch LC, The intrinsic rate of increase of an insect population. J AnimEcol 17:15–26 (1948).

29 Kostas E, Fumigation, pest control and bio-security practices. SoutheastAsia District 2nd Conference and Expo, Bali, Indonesia. [Online].International Association of Operative Millers (2011). Available:http://www.iaom.info/southeastasia/08cbh.pdf [18 June 2012].

30 Banks HJ, Behaviour of gases in grain storages, in Proc Int Conf onFumigation and Controlled Atmosphere Storage of Grain, Singapore,pp. 96–107 (1989).

31 Edde PA, Phillips TW, Nansen C and Payton ME, Flight activityof the lesser grain borer Rhyzopertha dominica F. (Coleoptera:Bostrichidae), in relation to weather. Environ Entomol 35:616–624(2006).

Pest Manag Sci (2013) c© 2012 Society of Chemical Industry wileyonlinelibrary.com/journal/ps

79

www.soci.org M Shi et al.

32 Toews MD, Campbell JF, Arthur FH and Ramaswamy SB, Outdoor flightactivity and immigration of Rhyzopertha dominica into seed wheatwarehouses. Entomol Exp Appl 121(1):73–85 (2006).

33 Kaur R, Molecular genetics and ecology of phosphine resistance in lessergrain borer, Rhyzopertha dominica (F.) (Coleoptera: Bostrichidae).PhD Dissertation, University of Queensland, Brisbane, Qld, Australia(2012).

34 Stansfield WD, Schaum’s Outline of Theory and Problems of Genetics, 3rdedition. McGraw-Hill, New York, NY (1991).

35 Daglish GJ, Collins PJ, Pavic H and Kopittke KA, Effects of timeand concentration on mortality of phosphine-resistant Sitophilusoryzae (L.) fumigated with phosphine. Pest Manag Sci 58:1015–1021(2002).

36 Sinclair ER and Alder J, Development of a computer simulation modelof stored product insect populations on grain farms. Agric Syst18:95–113 (1985).

37 Comins HN, Tactics for resistance management using multiplepesticides. Agric Ecosyst Environ 16:129–148 (1986).

38 Longstaff BC, Temperature manipulation and the management ofinsecticide resistance in stored grain pests: a simulation studyfor the rice weevil, Sitophilus oryzae. Ecol Model 43:303–313(1988).

39 Flinn PW and Hagstrum DW, Simulations comparing the effectivenessof various stored-grain management practices used to controlRhyzopertha dominica (Coleoptera: Bostrichidae). Environ Entomol19:725–729 (1990).

40 Flinn PW, Hagstrum DW and Muir WE, Effects of time of aeration, binsize, and latitude on insect populations in stored wheat: a simulationstudy. J Econ Entomol 90:646–651 (1997).

41 Clift A, Herron G and Terras MA, DEMANIR, a simulation modelof insecticide resistance development and management. MathComput Simulation 43:243–250 (1997).

42 Flinn PW, Hagstrum DW, Reed C and Phillips TW, Simulation model ofRhyzopertha dominica population dynamics in concrete grain bins.J Stored Prod Res 40:39–45 (2004).

wileyonlinelibrary.com/journal/ps c© 2012 Society of Chemical Industry Pest Manag Sci (2013)

80

Chapter 7

Summary and discussion

The studies underlying this thesis have developed new approaches to modelling the

evolution of resistance and population dynamics of stored grain insect pests. They have

used these to gain novel insights into the best ways to manage resistant populations, and

to slow the rate at which populations evolve to become more resistant. They have also

developed new algorithms for estimation and calculation of model parameters.

Numerical algorithms developed in this study for estimation and calculation of model

parameters, especially those for estimation of mortality, worked very well and the

results of numerical experiments demonstrated that these algorithms are valid and

efficient. Thus these algorithms established a solid basis for model construction and in

future can be used not only for individual-based models, such as the one presented in

this thesis, but also for other types of models, such as population-based models.

The results of mortality estimations show that in all cases it was better to use log-

transformed concentration and time as explanatory variables in models of mortality due

to phosphine fumigation, rather than use the untransformed variables. Moreover, for all

data sets, a probit model provided a better or equally as good fit to the data as

alternative Cauchy or logistic models. Therefore, the probit models based on log-

transformed explanatory variables provide the best predictions of mortality of R.

dominica. To my knowledge, no previous models have included mortality predictions

that vary with concentration, exposure time, and genotype, based on extensive

experimental data in the way this research has.

With these more accurate mortality estimations and other key parameters, we have

compared the differences between the predictions of the one- and two-locus models. We

have also investigated how fumigation tactics based on extending the duration of

fumigation or increasing the concentration of fumigation influence the control of

resistant and non-resistant lesser grain borer, and how these are the impacted by

different initial gene frequencies. In addition, we evaluated the impact of phosphine

81

dose consistency and immigration rate on the effect of fumigation and the consequences

for the development of phosphine resistance. The results of these investigations will

allow stored grain managers to continue to use the relatively safe and effective

phosphine fumigant for the control of infestations of this serious pest, and thus help

safeguard world-wide grain supplies.

Simulating the population dynamics of R. dominica in the absence of phosphine

fumigation, and under high and low dose phosphine treatments, shows the importance

of basing resistance evolution models on realistic genetics; predictions of our two-locus

individual-based model vary significantly from those of an equivalent model based on a

simplifying assumption that resistance is conferred by a single gene. In the case of no

fumigation, initiating the same frequencies of genotypes, the two models tended to

different stable frequencies, although both reached Hardy-Weinberg equilibrium. The

one-locus model exaggerated the equilibrium proportion of strongly-resistant beetles,

compared to the aggregated predictions of the two-locus model. Under a low dose

treatment the one-locus model overestimated the proportion of strongly-resistant

individuals within the population and underestimated the total population numbers

compared to the two-locus model. Using over-simplified one-locus models to develop

pest control strategies runs the risk of not correctly identifying tactics to minimise the

incidence of pest infestation. Thus, we used two-locus individual-models for the

subsequent simulations.

Testing the efficacy of different short-term fumigation tactics suggested that extending

exposure duration is a much more efficient control tactic than increasing the phosphine

concentration. This result is consistent with the reality well known by experienced

managers and farmers. Our quantitative results indicated that if the original frequency of

resistant insects is increased n times, then the fumigation needs to be extended, at most,

n days to achieve the same level of insect control. Also, to control initial populations of

insects that were n times larger, it was necessary to increase the fumigation time by

about n days. Our results indicate that, for a 2-gene mediated resistance where dilution

of resistance gene frequencies through immigration of susceptibles has greater effect

than for 1-gene mediated resistance, extending fumigation times to reduce survival of

homozygous resistant insects will have a significant impact on delaying the

development of resistance.

82

Evaluating and identifying viable, long-term strategies indicated that achieving a

consistent fumigant dosage is a key factor in avoiding evolution of resistance to

phosphine and maintaining control of populations of stored grain pests. A very

consistent dosage maintained good control for all immigration rates, while an

inconsistent dosage failed to maintain control regardless of immigration rate. At

intermediate dosage consistency, immigration rate became a critical factor in whether

control was maintained or resistance emerged.

Previous studies have not represented the way that resistance is conferred by alleles at

two different loci, resulting in nine different genotypes with different levels of

resistance and thus different patterns of mortality from various dosages, and have not

parameterised these mortalities by genotype, dosage and duration relationships from

detailed experimental data in the way we have with this study.

Previous models of population dynamics or evolution of resistance to insecticides in

stored grain pests (e.g. Clift et al., 1997; Comins, 1986; Flinn and Hagstrum, 1990;

Flinn et al., 2004; Flinn et al., 1997; Longstaff, 1988; Sinclair and Alder, 1985) have not

included the level of biological and genetic detail and realism accounted for in this

research, although similarly detailed approaches have been employed to investigate the

evolution of resistance to Bt transgenic crops in insects in the field (Storer et al., 2003),

and to herbicides in weeds (Renton et al 2011). In both these cases, the high level of

biological, spatial and genetic detail was enabled by an individual-based approach

similar to that used here (Renton, 2012).

Furthermore, previous studies have not been able to represent the way that dosages vary

between individual beetles to present the spatial heterogeneity within the silo in the way

we have here in this study, and thus have not been able to predict the effects of such

heterogeneity in dose on population dynamics and the evolution of resistance over

multiple fumigations over a long time period.

I believe that these kinds of improved individual-based models, based on better data and

including more biological detail, will help us develop insights into the processes of the

evolution of phosphine resistance in R. dominica. Importantly, they will also allow us to

better evaluate the merits of various management tactics and strategies for suppressing

population numbers and delaying or avoiding the evolution of resistance.

83

The current model is parameterised for one particular species, the lesser grain borer and

its particular resistance patterns, but it is likely that many of the results could be

generalised to other stored-grain pest species, especially if their resistance patterns were

similar; the current model could be relatively easily modified to test the validity of such

generalisation and to develop species-specific recommendations when such

generalisation fails.

We conclude with some comments about the limitations of this study and

recommendations for further research. Temperature and relative humidity, and thus

growth and reproduction rates, are assumed to be constant in this study, but in reality

are likely to vary in both time and space; this could have implications for resistance

development which should be considered in future studies. There are a number of other

biological and operational factors that should also be considered. For example,

immigration of resistant insects, or susceptible insects, to the storage will influence the

proportion of the population carrying resistance alleles and insects moving within the

storage to escape the toxicant will influence the spatial distribution of resistance alleles.

Emigration may also be important if different genotypes emigrate at different rates.

Differences in fitness between resistant and susceptible insects have been identified

(Pimentel et al., 2007), but the effect of this phenomenon on long term control and

resistance frequency is not yet clear; fitness costs could be incorporated into the model

to address this issue. The spatially heterogeneous model was based on assumptions that

the fumigation dosage experienced by an individual beetle within a storage facility or

silo at any time is constant and thus does not account for the reality of changing

concentrations within grain storages over time. Further studies representing temporal

heterogeneity of dosage are needed to investigate what impacts such variability will

have on the short and long term efficacy of different control options and the

contribution of resistance alleles to further generations. Modelling predictions are only

as good on which they are based; one of the strengths of the modelling in this thesis was

the quality of the recently available data on resistance genetics and mortality levels of

the different genotypes. As new data becomes available, such as data on actual dosages

achieved across the spatial and temporal extent of the fumigation or the fitness costs of

the different resistance alleles, this could be incorporated into the model relatively

easily, thus further improving our understanding and predictions. In addition, if we are

aiming to preserve the useful life of phosphine then the model should be extended to

84

account for integrated pest management options such as alternative pesticides and non-

chemical treatments such as cooling, so that their impact on the development of

resistance can be investigated as well.

References

(1) Clift A., Herron G. and Terras M.A. (1997), DEMANIR, a simulation model of

insecticide resistance development and management. Mathematics and computers in

simulation 43, 243–250.

(2) Comins H.N. (1986), Tactics for resistance management using multiple pesticides.

Agriculture, Ecosystems & Environment, 16, 129–148.

(3) Flinn P.W. and Hagstrum D.W. (1990), Simulations comparing the effectiveness of

various stored-grain management practices used to control Rhyzopertha dominica

(Coleoptera: Bostrichidae). Environmental Entomology 19, 725–729.

(4) Flinn P.W., Hagstrum D.W., Reed C. and Phillips T.W. (2004), Simulation model of

Rhyzopertha dominica population dynamics in concrete grain bins. Journal of Stored

Products Research 40, 39–45.

(5) Flinn P.W., Hagstrum D.W. and Muir W.E. (1997), Effects of time of aeration, bin

size, and latitude on insect populations in stored wheat: a simulation study. Journal of

economic entomology 90, 646–651.

(6) Longstaff B.C. (1988), Temperature manipulation and the management of

insecticide resistance in stored grain pests: a simulation study for the rice weevil,

Sitophilus oryzae. Ecological Modelling 43, 303–313.

(7) Pimentel M.A.G., Faroni L.A., Totola M.R. and Guedes R.N.C. (2007), Phosphine

resistance, respiration rate and fitness consequences in stored-product insects. Pest

Management Science 63, 876-881.

(8) Renton M. (2011), How much detail and accuracy is required in plant growth sub-

models to address questions about optimal management strategies in agricultural

systems? AoB Plants. doi: 10.1093/aobpla/plr006.

(9) Renton M. (2012), Shifting focus from the population to the individual as a way

forward in understanding, predicting and managing the complexities of evolution of

resistance to pesticides. Pest Management Science (accepted 7/4/2012, DOI

10.1002/ps.3341).

(10) Sinclair E.R. and Alder J. (1985), Development of a computer simulation model of

stored product insect populations on grain farms. Agricultural Systems 18, 95-113.

85

(11) Storer, N.P., Peck S.L., Gould, F., Van Duyn, J.W., Kennedy, G.G. (2003), Spatial

processes in the evolution of resistance in Helicoverpa zea (Lepidoptera: Noctuidae) to

Bt transgenic corn and cotton in a mixed agroecosystem: a biology-rich stochastic

simulation model. Journal of Economic Entomology 96, 156-172.

86

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � !" � � � � # � � � � � � � � � � � � � � $ � � � � � � � � � � � � % � � � � � � � � � � � � � � & � � � � � � � � � � ' ' ! � � ( � � � � !� � � ' ) * * � � � ! � � � ! � � � * + � � � � � * � � � * , � & ! ' � ' - . / 0 1 2 3 4 5 . 67 1 8 9 : 9 ; 6< = > ? @ A B ? C D E F G F H I J K L G M N O K H N F P F F N Q L G R E F N F S F M L T U F P R L QI V I R J H P J W M F U J P J X F U F P R I R G J R F X H F I R L Y L P R G L M T F I R H P Q F I R J R H L P J P N R E FN F S F M L T U F P R L Q T E L I T E H P F Z [ \ ] ^ G F I H I R J P Y F H P M F I I F G X G J H P W L G F GZ _ ` a b c d e f g ` h i c j k l k m h ^ n o L U T V R F G I H U V M J R H L P U L N F M I Y J P T G L S H N F JG F M J R H S F M p Q J I R q I J Q F J P N H P F r T F P I H S F K J p R L K F H X E R E F U F G H R I L QS J G H L V I U J P J X F U F P R L T R H L P I n \ L K F S F G q R E F V I F Q V M P F I I L Q I H U V M J R H L PU L N F M I G F M H F I L P R E F J Y Y V G J R F F I R H U J R H L P L Q H U T L G R J P R U L N F MT J G J U F R F G I q I V Y E J I U L G R J M H R p n o L P Y F P R G J R H L P J P N R H U F L Q F r T L I V G FJ G F W L R E H U T L G R J P R H P N F R F G U H P H P X U L G R J M H R p H P G F I T L P I F R L J R L r H YJ X F P R n s F Y F P R G F I F J G Y E H P N H Y J R F N R E F F r H I R F P Y F L Q R K L G F I H I R J P Y FT E F P L R p T F I H P _ t i c j k l k m h H P u V I R G J M H J q K F J v J P N I R G L P X q J P NG F S F J M F N R E J R R E F T G F I F P Y F L Q G F I H I R J P Y F J M M F M F I J R R K L M L Y H Y L P Q F G II R G L P X G F I H I R J P Y F q R E V I U L R H S J R H P X R E F Y L P I R G V Y R H L P L Q J R K L O M L Y V IU L N F M L Q G F I H I R J P Y F n w r T F G H U F P R J M N J R J I F R I L P T V G H Q H F N T F I R I R G J H P I qF J Y E Y L G G F I T L P N H P X R L J I H P X M F X F P L R p T F L Q L V G R K L O M L Y V I U L N F M qK F G F J M I L J S J H M J W M F n \ F P Y F H R W F Y J U F T L I I H W M F R L F r T M H Y H R M p H P Y M V N FU L G R J M H R H F I L Q R E F N H Q Q F G F P R X F P L R p T F I H P R E F U L N F M n x P R E H I T J T F G K FN F I Y G H W F N E L K K F V I F N R K L X F P F G J M H y F N M H P F J G U L N F M I Z z { | ^ q d f c } k gJ P N ~ c � k � g k m U L N F M I q R L Q H R R E F J S J H M J W M F F r T F G H U F P R J M N J R J I F R I n � FV I F N J N H G F Y R J M X F W G J H Y J T T G L J Y E � e l e f h ~ k b e i k l � e f � e j h g f k �g e m ` l k � � e q G J R E F G R E J P R E F R G J N H R H L P J M U J r H U V U M H v F M H E L L N F I R H U J R H L P qR L F I R H U J R F R E F U L N F M T J G J U F R F G I n D E F G F I V M R I I E L K R E J R W L R E T G L W H RJ P N M L X H I R H Y U L N F M I Q H R R E F N J R J I F R I K F M M W V R R E F Q L G U F G H I U V Y E W F R R F GH P R F G U I L Q I U J M M M F J I R I � V J G F I Z P V U F G H Y J M ^ F G G L G I n | F J P K E H M F q R E FX F P F G J M H y F N H P S F G I F U J R G H r R F Y E P H � V F J Y E H F S F N I H U H M J G J Y Y V G J Y pG F I V M R I R L R E L I F Q G L U R E F U J r H U V U M H v F M H E L L N F I R H U J R H L P q W V R H I M F I IR H U F Y L P I V U H P X J P N Y L U T V R J R H L P J M M p N F U J P N H P X n� � � � � � � � � U L G R J M H R p F I R H U J R H L P q T G L W H R U L N F M I q M L X H I R H Y U L N F M qX F P F G J M H y F N H P S F G I F U J R G H r J T T G L J Y E q T F I R Y L P R G L M I H U V M J R H L P n� ! � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �   ¡ ¢ £ ¤ � ¥ ¦   § ¨ © ¨ ª ¥ « � , � � �� � � � � � � � , � ' � � � � � ' � � � � � � � � � � � � ! ¬ � � � � � � � � � � � �' � � ' � � � � ­ � � ® ¯ � ° � � � � � ' � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � ' � � � � � � � � � � � � ! � � � � , � � � � , �� � � � � � � � � � � ® � � � � � � � � � � � � � � � , � � � ' � � � � � � � � � �� � � � � � � � � � , � � � � + � � ' � � ' � � � � � � � � � � � � � � ± ¦   § ¨ © ¨ ª ¥ !² ³ ´ µ ¶ ¶ · ¸ ¶ ¹ µ ¹ µ º � » ¶ ¼ º ½ · ¶ ¹ ¾ ¿ À Á º · ¹ º ½ »  à · ¹ ½ Ä Å ¶ Ä Æ � ½ Ä ¸ Å º ¾ Æ Á Â Æ Ç È È É ÆÄ » Ê � � � À ¿ ½ � Ä ¹ ¶ ¿ » Ä Å Ë Å Ä » ¹ Ì ¶ ¿ · º Í Ã ½ ¶ ¹ ¾ Æ Â Ã · ¹ ½ Ä Å ¶ Ä Î Ï µ ¿ » º Ð Ñ Ç Ò Ç Ó Ô Ô Õ Ò É É Ö ×À Ä Ø Ð Ñ Ç Ò Ç Ó Ô Ô Õ Ò Ò È Ô × º Õ Ù Ä ¶ Å Ð · µ ¶ ³ Ù ¶ » Ú ½ º » Û Ú Ù Ä ¶ Å ³ Í ¿ Ù Ü ³² ³ � º » ¹ ¿ » ¶ · ¸ ¶ ¹ µ ¹ µ º � » ¶ ¼ º ½ · ¶ ¹ ¾ ¿ À Á º · ¹ º ½ »  à · ¹ ½ Ä Å ¶ Ä Æ � ½ Ä ¸ Å º ¾ Æ Á  ÆÇ È È É Æ � ´ � � � � Í ¿ · ¾ · ¹ º Ù ´ Í ¶ º » Í º · Ä » Ê � � � À ¿ ½ � Ä ¹ ¶ ¿ » Ä Å Ë Å Ä » ¹ Ì ¶ ¿ · º Í Ã ½ ¶ ¹ ¾ Æ à · ¹ ½ Ä Å ¶ Ä Î º Õ Ù Ä ¶ Å Ð Ù ½ º » ¹ ¿ » Û Í ¾ Å Å º » º ³ à ¸ Ä ³ º Ê Ã ³ Ä Ã Ü ³� � � ' � � � � � � � � � � � � � � � � � � � ' � � , � � � � � � � � , � � � � � � � � � � � Ý ' � � � , � � � � � � � � � � � � � � � � � � � , � � � � � � � � � � � � � ' � � � � ! Þ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ' � � � � � � � � � �' � � � � � � !

� � ' � � , � � � � � ' � � � � � � � � � � � � � � � � � � � � � � ß � � , � , � � � � '� � � � � Ý ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � à � � � � � � � � � � � � , � � � � � á â ã ä � � � � � � � � ' � � � � � � � � � � � � � � � �� � � � ! � � � � , � � � � � � � � � � � ' � � � � � � � � � � ®� � � � � � � � � � ± ¦   § ¨ © ¨ ª ¥ � � # � � � � � � , � � � � � � � � � � � � � � �� � � � � � � ' � � � � � � ' � � � � � � � � � � � � � � � � " � ° � � � � � � � �å � � � � � � â ( ä ! % � � � � � � � � � � � � ° � � � � � � � � � � � � ° � � � �� � � � � ' � � � ¢ ¤ ¥ æ ! â � � � � � ä � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � ! % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � , � � � � � � � ' � � � � � � � � � � � ' � ! % � � � Ý ' � � � � � � � � �� � � � � � ¢ ¤ ¥ æ ± â ç � � ä � � � � ' � � � � � � � � � � � � � � � � � � � � � � �� Ý ' � � � � � � � � � � � � � � � � � � � � � � � ' � � � � � � � � ' � � � � � � � � � � è � � � � � � � � ' � � � � � � � � � � � � � � � � � � � ' � � � � � � � � �� � � � � � � � � ! % � � � � � � � � � � � � � Ý ' � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � # � � � � � ! % � � � � Ý ' � � � � � � � � � � � � � � � � � � � ­ � ! � ! â é ä ¯ � � � � � � � � � � � � ' � ' � � � � � � � ' � � � � � � � � � � � � � � � � � � � � � , � � � � � � Ý � � � � � � � � � � � �� � � � ! � � � � � � � � � � � � � ' � � � � � � � � � � � Ý ' � � � � � � � � � � � � �� � � � � � � � � � � � � � , � � � � � � � � � ­ � � � � � ' � � � � � � � � � � � � � � � � ' � � � � � � � � � � � ' � ¯ !� � � ' � � � � � � � � � � � � � � � � � � � � � � � � Ý ' � � � � � � � � � �� � ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ' � � � � � � � Ý � � � � � � ! � � ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ý � � �� � � � � � � � � � � � � � � � � � � � � � � ê ­ � � * � ¯ � � � Ý ' � � � � � � � � ¤ ! % � � � � � � � � � � � � � � � � � � � � � � � � � � � � , � , � � � � ­ � ë � � � � � � � � ¯ � � � � � � � � � � � � � � � � � � � � � Ý ' � � � � � � � � � � � � �� Ý ' � � � � � � � � � � � � � � � � � � � � � � � � � , � � � ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � !� � � � � ' ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ¡ £   ì ¨ ¤ � � æ   í ¨ î ¤ ¨ ª � � � � � � � � � � � � � � � � � � � � � � � ¢ ¤ ¥ æ� Ý ' � � � � � � � ­ � � � � � � � � � ¯ â ç � � ä � � � � � � � � � � � ï å ð � ç �ï å ð � � é � � � � � � � � � � � � � � � ¬ � ­ ï å ð � ç ñ ï å ð � � é ¯ � � � � �� � � � � ' � � � � � � � � � � � � � � � � ' � î î ­ � � � � � � � � � � � � � � � � � � � � � � � ' � � � � � ¯ � £ £ ­ � � � � � � � � � � � � � � � � � � � � � � � � � � ¯ � � � �­ � � � � � � � � � � � � � � � � � � � � � � ¯ � � ' � � � � , � � � â � ( ä ! " � � �� � � ' � � � � � � � � � à � � � � � � � � � � � � � � � � � � � , � � � � � � � � � � � � � ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � !� � ! � $ % � ò ð # � ð � # % $ å � # ó� � � � � � � � � � � � í ¢ © ¢ £ ¥ æ ¨ � ¢ ¦ æ ¨ © ¢ ¥ £ §   ¦ ¢ æ ­ ô ó � ¯ � � � � � � � � � õ ö ¥ ÷ ì ø ù ø ÷ ì ú ù ú ÷ û ÷ ì ü ù ü � ­ � ¯� � � Ý � � � � � � � � � � � � � � � � � � � � � � � æ ¢ ¥ î ¤ î ý þ ¥ £ ¢ î£ ¢ í £ ¢ î î ¨   © � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � ' � � � , � � � � � , � æ ¨ © ÿ � þ © ª ¤ ¨   © � � � � � � � � � � � � � � � � �, � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ' � � � � � � � �, � � � â � ä ! ô ó � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � �Appendix

87

� � � � � � � � � æ   í ¨ î ¤ ¨ ª £ ¢ í £ ¢ î î ¨   © � � ¡ £   ì ¨ ¤ £ ¢ í £ ¢ î î ¨   © ! " � � � �' � � � � � � � � � � � � � � � � � � � � � � � � � � � Ý ' � � � � � � � � � � � � !" � � � � � � � � � ' � � � � � � � � � � � � � � � � � � � , � ' ' � � � � ë§ ¥ ù ¨ § þ § æ ¨ ÿ ¢ æ ¨ �     ¦ ¢ î ¤ ¨ § ¥ ¤ ¨   © � � � � � � � � � � � � � � � , � � � � � � � � � � � � � � � ' ' � � � � ë í ¢ © ¢ £ ¥ æ ¨ � ¢ ¦ ¨ © � ¢ £ î ¢ § ¥ ¤ £ ¨ ù¤ ¢ ª � © ¨ ý þ ¢ � � � � � � � � � � � � � � � � � ' � � � � � � â � � ä ! % � � � � � � � �� � , � � � � � , � � � � � � � � � � � � � ) � � � � � ' � � � � � � � � � � � � �° � � � � � � � � � ' � � , � � � � � � � � � � � � � � � � � � � � ' � � � � � � � � � � , � � � � � � � � � � � � � � � � � � � Ý � � � � � , � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ­ � � � � � � ¯ � � �� � ° � � � � � � � � � � � � � � � ° � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � ú � � � � � � � â � ä !� ± � £   ì ¨ ¤ §   ¦ ¢ æ î% � � ' � � � � � ­ � � � � � � ß ' � � � � � � � � � � � � � á ¯ � � � ° � � � � � � � ­ � ¯­ õ � ­ � ¯ � � ¯ � � � � � � , � � � � � � � � � � , � � � � � � � � � � � � � � � � � � �­ � ð ¬ ¯ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � â � � & ä )� �� ���� � � �� !"# $ % &&'( ­ � ¯� � � � � � � ß ' � � � � � ­ � ¯ á + � � � ° � � � � � � õ , � � � � �' � � � � , � � � ' � � � � � � � � � � � � � � � � � � ' � � � � � � ¥ � � � � � � � � � � � � �' � � � � � � � � � � � � � � � � � � � � � � � � � � � ' � � � � � � � � � � � � � � !) � � � � � � � � � ' � � � � � � ' � � � � � � � � � � â � ä � ' � � � � � ' � � �õ ö ­ ­ � ¯ � � � ¯ ¥ ÷ ì ø � � � ­ ¤ ¯ ÷ ì ú � � � ­ ê ¯ ­ ( ¯� � � � � � � � � � � � � � � � � � � � � � ¤ � � ê � � � � ' � � � � , � � �� Ý ' � � � � � � � � � � � � � � � � � � � � � � � � � õ � � � � ' � � � � � � � � � � � � � !� � � � � � � � � � � � � , � � � � � � � � � ' � � � � � � � � � � � � � � � � �� � � � ' � � � � � � ê ¤ ­ � ! � ! � � � � � ê � � � � Ý � � � � � � � � � � � �¤ ¯ � � � � � � � � � � ' � � � � � � � ' � � � � � � , � � � � � ê � � ¤ � � � �' � � � � � � ì ø � � ì ú � � � � � � � � � � � � � � � � � � � ' � � � � � � � ì­ � � � � ' � � � � � � ' � � � � � � � � � � ¯ )õ ö ¥ ÷ ì � � � ­ ê ¤ ¯ * + ," � � � � � � � � � � � � � � � � � � � � ­ � � � � ¯ � � � � � � � � � � � � � � � ­ � � ¢ ¯ � � � � � � � ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � , � � � � ì !# � � Ý � � � � � � ì ® � � � ­ ¤ ¯ � � � ­ ê ¯ � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � , � � � � � ¤ � � ê � � � � � � � � � � � � � � � �' � � � � � � ' � � � � � � � � � �õ ö ¥ ÷ ì ø � � � ­ ¤ ¯ ÷ ì ú � � � ­ ê ¯ � ì ® � � � ­ ¤ ¯ � � � ­ ê ¯ * - ,. ± �   í ¨ î ¤ ¨ ª §   ¦ ¢ æ î% � � � � � � � ' � � � � � � ° � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � æ   í ¨ ¤ � � � ° ) ­ % � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � õ � � $ à ! ­ � ¯ ¯� ö ­ � ¯ � � � â � * ­ � � � ¯ ä � � � ö / ø ­ � ¯ � � * ­ � � � / 0 ¯ ­ � ¯� � � � � � � � � � � � � � � � � � � $ à ! ­ � ¯ ! � � � � � � � � � � � � � � �' � � � � � � � � � � � � � � � � � � � � � � � ! % � � � � � � ' � � � � � � � � � � � �' � � � � � � � � � � � � � � � � � � � � � � � ' � � � � � � � � $ à ­ ç ¯ � � ­ � ¯ � �� � ' � � � � , � � � �� ö ¥ ÷ ì ­ ê ¤ ¯ ­ & ¯

� ö ¥ ÷ ì ø ­ ¤ ¯ ÷ ì ú ­ ê ¯ � ì ® ­ ê ¤ ¯ ­ ã ¯ê ± 1 ¢ © ¢ £ ¥ æ ¨ � ¢ ¦ ¨ © � ¢ £ î ¢ § ¥ ¤ £ ¨ ù ¥ ¡ ¡ £   ¥ ª �# � � � � � � � � � � � � � � � � � � � � � � � � � � , � � � � � � � � � � � �� � � � � � � � � � , � � � , � � � � � � � � � � � � � � � � � � � � � � �� à � � � � � 2 ¨ ¤ � £ ¢ î ¡ ¢ ª ¤ ¤   ¤ � ¢ ¡ ¥ £ ¥ § ¢ ¤ ¢ £ î � � � � � � � � � � � ! ¬ � �� Ý � ' � � � � � � � � � � � � � ­ ( ¯ � � � � 3 � à � � � � � � � � � ( , � � � � � ­ ¥ �ì ø � ì ú ¯ � � � � � ' � � � � � � � � � � � � � � � 4 õ 5 è ¤ 5 � ê 5 6 7 8 9 � � � � � � � )õ 5 ö � : ¥ ÷ â � � � ­ ¤ 5 ¯ ä : ì ø ÷ â � � � ­ ê 5 ¯ ä : ì ú ­ ¨ ö � � � � ; � 3 ¯ < ­ é ¯% � � � � � � Ý � � � � � � � � � , � � à � � � � � � � = ö > � � � � � = � ­ ¥ �ì ø � ì ú ¯ ? � � � @ A BC C C DE F ­ � � ¯% � � � � � � � � � � � � à � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � â � ç ä ! � � � � � � � � � � � � � � � GH GHGH I J KL M N OJL M N OPQR RRRQR RR STUVUWSS ­ � � ¯� � � � � X YZ � [\ ] ^ _[\ ] ^ _ `a bb cdedf � � � � ¨ g h ' � � � � � � � � , � � � �� � � � � � � � � � � � � � � � � � � � � ¢ ¥ î ¤ i ý þ ¥ £ ¢ î � � � � � � � ! � � � � � � � �� � � � � � à � � � � � � � � � � � � � � � � � � � � , � � � � � � � � � � � � �� à � � � � � � � = ö > � ! � ! � ? � = ö � ? > � ' � � , � � � � � � � � ? � � � � , � � � � � � � ! # � � � � � � � � � � � � � � j � � � � � � � � � � � � � � � � , � � � � � � � � Ý � « � � � � � j > � � � � � � � � � � � â � ä ! � � � � � � � � � � � � � � � � � � � � à � � � � � � � Ý � � � � � j � � k ø ! � � � � � � � � � � � � �� � ° � � � � � � � � ? � � � � � � � � � � � � � � � j � ­ � ? � ¯ k ø � ? ! Þ � � � � � � �� � � � à � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � j � � � � � � � � � � � � ' � � � � � � � � � � � � � � � � � � � � j � � � � � � � � � � � � � � � �� � � � l � � � � � � ' � � � � � � � � � � � � � � , � � � � � � � � ' � � � � � �­ i m ê ¯ � � � � � � � � j � � � � � � , � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ­ � ? � ¯ k ø � ? â � ä !n ± 3 ¢ ¥ ¤ ¢ © ¢ ¦ £ ¥ 2 ¦ ¥ ¤ ¥ î ¢ ¤ î� � � � � � ¢ ¤ ¥ æ ± � � � � � � â ç ä � � � � , � � � � � � � � � � � � � � � � � � � �� ' � � ' � � � � � � � � � � � � � � � � ­ ê ) � � * � ¯ � � Ý ' � � � � � � � � ­ ¤ ¯ ç ã� � � � � � � ' � � � � � ­ � � � � ï å ð � ç ë � � � � � ' � � � � � � � � � � � � � � ' �î î ¯ � � � � � � � � � � � � � ­ � � � � ï å ð � � é ë £ £ ¯ ' � � � � � � ' � � �� � � � � � � � � � � � � ¬ ø ' � � � � � � ­ ­ � � é ñ � ç ¯ � ­ � ç ñ � � é ¯ ë � � ¯ ! % � �� � � � � � � � � � � � � � � � � � � � % � � � � ! � � � � � � � � � � �� � ' � � � ­ ° � � � ¯ � � � � � � � � � � � � � � � � � � � � � � � , � � � � � � � � ' � � � � � � � � � � ­ � ! � ! � � � � � � o p q q q q � � � � � � � � o p o o o r ¯ !ò � � � � � � � � � � � � � � � ' � � � � � � ' � � � � � � � � � � � � , � � � � � � � � � � � � !% � � � s t t u t , � � � ­ � � � � � � � � � � � � � � � , � é é ! é v � � � � � � � � ¯ � � � � � � � )ê ) � ! � � � ! � � � � ! � � � ! ( � � ! ç � � ! � � � ! & � � � ! �� s t t u t ) � ç ! � � � � � ! & ç � ã ! � � é � & ! � ç ç � � ! � � � � ! � � ã � ç ! � ( ( � ( ! & ç ­ � � ¯

ww x x xy y yz { | }{ | }{ | }{ | }{ | }{ | }~~~ ����

88

% � � � , � � � � � � � � � , � � � � � � � � � Ý ' � � � � � � � � � � � � � � ¢ ¤ ¥ æ ±â � ä � � � � � � � , � � � � � � � � � � ï å ð � � é � � � � � � � Ý ' � � �� � � � � � � � Ý � � � � � � � � � � � � � � � � � � ! � � � � ! � � � * � � � � � � � � � Ý ' � � � � ' � � � � � !% � � � � � � ' � � � � � � ' � � � � � � � � � � ­ ç ¯ � � � � � � � � � � � � � � ­ & ¯� � � � � � � � � � � � � � � � � � � � î î � � � � � � � � � � � � � � �� Ý ' � � � � � � � � � , � � � � � ! % � � � � � � ' � � � � � � ' � � � � � � � � � �­ � ¯ � � � � � � � � � � � � � � ­ ã ¯ � � � � � � � � � � � � � � � � � � � � � � �£ £ � � � � � � ' � â ç � � ä � � � � � � � � � � � � � � � � � � � � � � � Ý ' � � � �� � � � � � � � � � � � � � � � � � � � � � , � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ¤ ­ � � ¯ , � � � � � � � � � � � � � � � � � � � � � � ) ¤ � � ç ñ ­ � � � � � � � � � � � ç ! � � � � � ! & ç � ã ! � � é � & ! � ç ç � � ! � � �� ! � � ã � ç ! � ( ( � ( ! & ç ¯ ­ % � � � � � � â � � ä ¯ !� Â Ì � � � ³ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Æ � � Æ � � � � � � � � � � � � �� � � � � � � � � � � � ? � Ó Ô � � � � Ó �� � � � � Î � � Ü� ¿ · º Î Ù Ú   Å Ü È ³ È È Ò È ³ È È Ò ¡ È ³ È È Ö È ³ È È ¢ È ³ È È Ó² ¿ ½ ¹ Ä Å ¶ ¹ ¾ È ³ È Ö È Ò È ³ ¢ Ö È ³ £ È Ó £ È ³ É £ ¢ ¢ � ¤ ¥ ¥ ¥ ¥¦ § ¨ © ª � Î « « Ü� ¿ · º Î Ù Ú   Å Ü È ³ È È Ö ¡ È ³ È È Ó È ³ È È ¡ È ³ È È £ ¡ È ³ È Ò È ³ È Ö² ¿ ½ ¹ Ä Å ¶ ¹ ¾ ¥ ¤ ¥ ¥ ¥ ¥ È ³ ¢ Ó Ó ¡ È ³ ¢ É Ó È È ³ Ô È Ó £ È ³ Ô ¡ É Ò È ³ É Ô Ç Ô� � � ¬ ­ ® Î ¯ ¯ Ü� ¿ · º Î Ù Ú   Å Ü È ³ Ò È ³ Ö ¡ È ³ ¡ Ò ³ È Ò ³ Ö ¡² ¿ ½ ¹ Ä Å ¶ ¹ ¾ ¥ ¤ ¥ ¥ ¥ ¥ È ³ È Ö È È È ³ Ö Ö ¡ Ó È ³ ¡ Ö È ¢ È ³ ¡ £ È ¡� � � ! å $ � ) ó % � # � ð � ò � � ó ) � � ò �� Â Ì � � � � ³ � � � � � � � � � � ° � � � � � � � � � � � � � � � � � � � � ± � � Ò Ó Î � � Ü � � �� � � ² ³ Ò Î ´ ´ Ü � � � � � � � � � � � � � � � � � � � � ± � � ¡ Ç É Î � � Ü � � � � � � � � � � �� µ � � � � � � � � � � � ² � � ° � � � � ² � � � ¶ � � � � � � � � � � · � � � � � � � � � � � � � �� ² � � � � � � � � � � � � � � ² � � � � � � � � � � � � · � � � � � �% � � � � � � � ' � � � � � � � � � � � à � � � � � � � � � � � � � � �� � � � , � � � � ' � � � � � � � � � � � � � � � � � ­ � � $ à ! ­ � � ¯ ¯ � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � % � � � � ! # � � � � � � � � � � � � � , � ­ � � � � � � � ¯ � � � � � � � � � � � � � � � � ' � � � � � � � � ¬ � � � � ( !� � � � � � � � � � � � % � � � � � � ¬ � � � � � � � � � � � � � � � � � � � � � , � � , � � � � � � � ' � � � � � � î î � � � � � � � � � � ' �� � � � � � � � � � ' � � � � � � � � � � � � � � � � � � � � � � Ý ' � � � � � � � � � � � � � � � � � � � � � � ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � , � � , � � � !

³ ¶ Ú Ã ½ º Ò ³ � ¸ · º ½ ¼ º Ê Ù ¿ ½ ¹ Ä Å ¶ ¹ ¶ º · Ä » Ê Ï ½ º Ê ¶ Í ¹ º Ê Ù ¿ ½ ¹ Ä Å ¶ ¹ ¶ º · ¸ ¾ Ï ½ ¿ ¸ ¶ ¹ Ä » ÊÅ ¿ Ú ¶ · ¹ ¶ Í Ù ¿ Ê º Å · À ¿ ½ ¹ µ º · ¹ ½ Ä ¶ » ± � � Ò Ó Î � � ܳ ¶ Ú Ã ½ º Ö ³ � ¸ · º ½ ¼ º Ê Ù ¿ ½ ¹ Ä Å ¶ ¹ ¶ º · Ä » Ê Ï ½ º Ê ¶ Í ¹ º Ê Ù ¿ ½ ¹ Ä Å ¶ ¹ ¶ º · ¸ ¾ Ï ½ ¿ ¸ ¶ ¹ Ä » ÊÅ ¿ Ú ¶ · ¹ ¶ Í Ù ¿ Ê º Å · À ¿ ½ ¹ µ º · ¹ ½ Ä ¶ » � ¿ Ù ¸ ³ Ò Î « « Ü% � � � � � � � � � � � � � , � � � � � � £ £ � � � � � � � � � � � � � �' � � � � � � � � , � � � � � � � � � ' � � � � � � � � � � � � � � � � � � � � � �� � � � , � � � � � � � � � � � � � � � � � � � � � � � � � � � !% � � � � � à � � � � � � � � ­ % � � � � ¯ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ' � � � � � � � � � � è � � � � ç � � � � � � � � � � � � � � � � � � � � � � � � � � � � � £ £ � � � � � � � � � ç � � � � � �� � � î î � � � � � � !% � � � � � ' � � � � � � � � � � � ï å ð � ç � � � � � � � � � � � � � � � �­ ¬ � � ! ç ¯ ! Þ � � � � � � � � � � � � � � � � � ' � � � � � � � � � � à � � � � � � � � � � � � � � � � � � � � � � � � � , � � � � � � � Ý ' ' � � � � � � � � � �� � � � � � � � � � � � � ­ � � � � � ' � � � � � � � � ' � � � � � , � � � ¯ � � � � � � � � � � � � � � ­ � � ¯ ) � ! � � � ç � � � ' � � � � � � � � ! ( ã � � ­ � Ý � � � �� � ° � � � � � � � ¯ � � � � � î î � � � � � � ­ � � � � � � � � � � � � � � � � � � � � � ' �� � � � � � ¯ !

¹ § º » ¼ ½ ¾ º ¿ À Á Á » º  ½ à ½ ¨ » Á » à Ĵ ¹ ½ Ä ¶ » Ë ½ ¿ ¸ ¶ ¹ � ¿ Ú ¶ · ¹ ¶ ͱ � � Ò Ó Å � Ò Ó ³ È É Ç ¢ Ñ Ô ³ Ó Ö Ó Ô � Å ¿ Ú Î Æ Ç Ü � È � Õ Ç ³ Ó Ó Ç Ò Ñ £ È ³ ¡ Ô É £ Î Æ Ç Ü� ¿ Ù ¸ ³ Ò Å � £ ³ Ç Ò È Ò Ñ Ó ³ £ £ Ó È � Å ¿ Ú Î Æ Ç Ü � È � Õ ¢ ³ È Ó Ö ¡ Ñ Ô ³ ¡ ¡ ¡ Ô Î Æ Ç Ü± � � ¡ Ç É Å � Õ Ò Ò ³ Ô Ó É Ö Ñ Ò È ³ È ¢ Ç ¢ Å ¿ Ú Î Ç ÜÕ ¢ ³ Ó ¡ Ç ¢ Å ¿ Ú Î Æ ÜÑ ¢ ³ Ç ¢ ¡ £ Å ¿ Ú Î Ç Ü Å ¿ Ú Î Æ Ü È � Õ Ó ³ É £ Ç É Ñ È ³ È Ò É ¢ Î Ç ÜÕ Ó ³ Ç Ô £ £ Î Æ Ü Ñ È ³ Ò £ Ó Ò Î Æ Ç ÜÉ » ½ Ä Á Ä Ê Ë ½ à » Ä » à à § ñ � � Ò Ó È ³ È È È ¡ Ó È ³ È Ö É Ö Ó� ¿ Ù ¸ ³ Ò È ³ È ¢ ¢ Ç Ô È ³ Ò Ó Ô £ Ô± � � ¡ Ç É È ³ È È ¢ Ö ¡ È ³ È ¢ ¢ ¡ £

89

³ ¶ Ú Ã ½ º ¢ ³ � ¸ · º ½ ¼ º Ê Ù ¿ ½ ¹ Ä Å ¶ ¹ ¶ º · Ä » Ê Ï ½ º Ê ¶ Í ¹ º Ê Ù ¿ ½ ¹ Ä Å ¶ ¹ ¶ º · ¸ ¾ Ï ½ ¿ ¸ ¶ ¹ Ä » ÊÅ ¿ Ú ¶ · ¹ ¶ Í Ù ¿ Ê º Å · À ¿ ½ ¹ µ º · ¹ ½ Ä ¶ » ± � � ¡ Ç É Î ¯ ¯ Ü Ä ¹ Ó Ô µ ½ · º Ø Ï ¿ · à ½ º ¹ ¶ Ù º³ ¶ Ú Ã ½ º Ó ³ Ë ½ ¿ ¸ ¶ ¹ Å ¶ » º · ¿ ¸ ¹ Ä ¶ » º Ê Ã · ¶ » Ú Ï ½ ¿ ¸ ¶ ¹ Ù ¿ Ê º Å ¸ ¶ ¹ µ Ú º » º ½ Ä Å ¶ È º Ê ¶ » ¼ º ½ · ºÙ Ä ¹ ½ ¶ Ø Ä Ï Ï ½ ¿ Ä Í µ Î Ì » ½ � » ¼ ² Ä ¹ ½ ¶ Ø Ü Ä » Ê Ù Ä Ø ¶ Ù Ã Ù Å ¶ Í º Å ¶ µ ¿ ¿ Ê º · ¹ ¶ Ù Ä ¹ ¶ ¿ »% � � � � ' � � � � � ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � à � � � ­ � � � � � � � � ¯ � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � , � � �� � � � Ý � � � � � � à � � � � � � � � Ý � � � � � � ° � � � � � � � � � � � � � � � � � � � , � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ' ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � , � ' ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ' � � � � � � � � �� � � � � � � � ! # � Î � � Á � � � Ì ² � � �

% � � � � � � � � � � � � � � ° � � � � ° � � � � � � � � � � � � ' ' � � � � � � �# � � � � � � ô � , � � � � � � � Ï � � � ' � � � � , � å � � � � � � � � � � � � � � � � � ! " � � � � � � ° � ! � ! � � � � � � � � � � � � � � � � � ' � � � � �� � � � � � � � � ' � � , � � � � � � � � � !å � ³ � � � � � � ´� Ò � Ì º » Õ � · ½ Ä º Å Â Æ Ì ½ º ¼ ¶ Å Å º � � � Î Ö È È ¢ Ü Ì º » º ½ Ä Å ¶ È º Ê � » ¼ º ½ · º · Ð � µ º ¿ ½ ¾ Ä » ÊÂ Ï Ï Å ¶ Í Ä ¹ ¶ ¿ » · Æ Ö Ð Ñ º Ê ³ � º ¸ Ò ¿ ½ Í Æ � Ò Ð ´ Ï ½ ¶ » Ú º ½� Ö � Ì Å ¶ · · � � Î Ò É Ó È Ü � µ º ½ º Å Ä ¹ ¶ ¿ » ¸ º ¹ ¸ º º » º Ø Ï ¿ · à ½ º ¹ ¶ Ù º Æ Í ¿ » Í º » ¹ ½ Ä ¹ ¶ ¿ » Ä » ʹ ¿ Ø ¶ Í ¶ ¹ ¾ ¶ » º Ø Ï º ½ ¶ Ù º » ¹ · ¿ » ¶ » · º Í ¹ ¶ Í ¶ Ê º · ³  » » Ä Å · ¿ À ¹ µ º � » ¹ ¿ Ù ¿ Å ¿ Ú ¶ Í Ä Å´ ¿ Í ¶ º ¹ ¾ ¿ À  ٠º ½ ¶ Í Ä ¢ ¢ Î Ó Ü Ð Ï Ï ³ £ Ö Ò Õ £ Ç Ç� ¢ � � ¿ Å Å ¶ » · Ë Ó Î Ò É É Ô Ü � º · ¶ · ¹ Ä » Í º ¹ ¿ Ú ½ Ä ¶ » Ï ½ ¿ ¹ º Í ¹ Ä » ¹ · Ä » Ê À à ٠¶ Ú Ä » ¹ · ¶ » ¶ » · º Í ¹Ï º · ¹ · ¿ À · ¹ ¿ ½ º Ê Ï ½ ¿ Ê Ã Í ¹ · ¶ »  à · ¹ ½ Ä Å ¶ Ä ³ � » Ð Ô ¯ Õ Ö × Ø ¯ � Ç Ù Ú � Ç ¯ Û Ü Ø Û ÝÔ Õ � Ç « Û ¯ Þ ß � Ç à ß Ö « Ý Ø Ö Û Ü Æ Õ Ý á ß ¯ ß Ý Ö ß Î º Ê ¸ ¾ Ì Ä » Í · â Ó Æ ß Ç Û Ü Ü � Ä » ¸ º ½ ½ Ä Æ à · ¹ ½ Ä Å ¶ Ä Æ Ï Ï ¡ ¡ ã ¡ £� Ó � � ¿ Å Å ¶ » · Ë Ó Æ � Ä Ú Å ¶ · µ Ì Ó Æ Ì º » Ú · ¹ ¿ » ² Æ � Ä Ù ¸ Í ¶ » � ² Æ Ë Ä ¼ ¶ Í â Î Ö È È Ö ÜÌ º » º ¹ ¶ Í · ¿ À ½ º · ¶ · ¹ Ä » Í º ¹ ¿ Ï µ ¿ · Ï µ ¶ » º ¶ » ä « å æ Õ ç ß ¯ Ç « Û è Õ é Ø Ý Ø Ö ÛÎ � ¿ Å º ¿ Ï ¹ º ½ Ä Ð Ì ¿ · ¹ ½ ¶ Í µ ¶ Ê Ä º Ü ³ Ó ¿ à ½ » Ä Å ¿ À � Í ¿ » ¿ Ù ¶ Í � » ¹ ¿ Ù ¿ Å ¿ Ú ¾ É ¡ Î Ó Ü ÐÔ Ç Ö ã Ô Ç É� ¡ � � ¿ Å Å ¶ » · Ë Ó Æ � Ä Ú Å ¶ · µ Æ Ì Ó Æ Ë Ä ¼ ¶ Í Æ â Æ Î ¿ Ï ¶ ¹ ¹ Í º Æ � Â Î Ö È È ¡ Ü � º · Ï ¿ » · º ¿ ÀÙ ¶ Ø º Ê Õ Ä Ú º Í Ã Å ¹ à ½ º · ¿ À Ï µ ¿ · Ï µ ¶ » º ½ º · ¶ · ¹ Ä » ¹ Ä » Ê · à · Í º Ï ¹ ¶ ¸ Å º · ¹ ½ Ä ¶ » · ¿ ÀÅ º · · º ½ Ú ½ Ä ¶ » ¸ ¿ ½ º ½ Æ ä « å æ Õ ç ß ¯ Ç « Û è Õ é Ø Ý Ø Ö Û Æ ¹ ¿ Ï µ ¿ · Ï µ ¶ » º Ä ¹ Ä ½ Ä » Ú º ¿ ÀÍ ¿ » Í º » ¹ ½ Ä ¹ ¶ ¿ » · Ä » Ê º Ø Ï ¿ · à ½ º Ï º ½ ¶ ¿ Ê · ³ Ó ¿ à ½ » Ä Å ¿ À ´ ¹ ¿ ½ º Ê Ë ½ ¿ Ê Ã Í ¹ ·� º · º Ä ½ Í µ Ó Ò Ð ¢ £ ¢ ã ¢ Ô ¡� Ç � � ¿ ¸ · ¿ » Â Ó Æ Ì Ä ½ » º ¹ ¹ Â Ì Î Ö È È Ô Ü � » ¹ ½ ¿ Ê Ã Í ¹ ¶ ¿ » ¹ ¿ Ì º » º ½ Ä Å ¶ È º Ê � ¶ » º Ä ½² ¿ Ê º Å · Î ¢ ½ Ê º Ê ³ Ü Æ Ì ¿ Í Ä � Ä ¹ ¿ » Æ ³ � Ð � µ Ä Ï Ù Ä » Ä » Ê â Ä Å Å   � � �� £ � ³ ¶ » » º ¾ � Ó Î Ò É £ Ò Ü Ë ½ ¿ ¸ ¶ ¹  » Ä Å ¾ · ¶ · Î ¢ ê Ñ º Ê ³ Ü � Ä Ù ¸ ½ ¶ Ê Ú º � » ¶ ¼ º ½ · ¶ ¹ ¾ Ë ½ º · ·� Ô � â Ä Ú · ¹ ½ Ã Ù Æ � ³ Á ³ Æ ³ Å ¶ » » Æ Ë ³ Á ³ Î Ò É É È Ü ´ ¶ Ù Ã Å Ä ¹ ¶ ¿ » · Í ¿ Ù Ï Ä ½ ¶ » Ú ¶ » · º Í ¹· Ï º Í ¶ º · Ê ¶ À À º ½ º » Í º · ¶ » ½ º · Ï ¿ » · º ¹ ¿ ¸ µ º Ä ¹ · ¹ ¿ ½ Ä Ú º Í ¿ » Ê ¶ ¹ ¶ ¿ » · Ä » ÊÙ Ä » Ä Ú º Ù º » ¹ Ï ½ Ä Í ¹ ¶ Í º · ³ Ó ¿ à ½ » Ä Å ¿ À � Í ¿ » ¿ Ù ¶ Í � » ¹ ¿ Ù ¿ Å ¿ Ú ¾ Ô ¢ Î Ç Ü Ö Ó Ç É ÕÖ Ó £ ¡ ³� É � Ë ¶ Ù º » ¹ º Å ² Â Ì Æ ³ Ä ½ ¿ » ¶ � Â Æ � ¿ ¹ ¿ Å Ä ² � Æ Ì Ã º Ê º · � � � Î Ö È È £ Ü ³ Ë µ ¿ · Ï µ ¶ » º½ º · ¶ · ¹ Ä » Í º Æ ½ º · Ï ¶ ½ Ä ¹ ¶ ¿ » ½ Ä ¹ º Ä » Ê À ¶ ¹ » º · · Í ¿ » · º ë à º » Í º · ¶ » · ¹ ¿ ½ º Ê Õ Ï ½ ¿ Ê Ã Í ¹¶ » · º Í ¹ · ³ Ë º · ¹ ² Ä » Ä Ú ´ Í ¶ Ç ¢ Ð Ô £ Ç Õ Ô Ô Ò� Ò È � ´ Í µ Å ¶ Ï Ä Å ¶ à · � � Æ � ¿ Å Å ¶ » · Ë Ó Æ ² Ä Ã Ò Æ � ¸ º ½ ¹ Ë � Î Ö È È Ç Ü � º ¸ ¹ ¿ ¿ Å · À ¿ ½Ù Ä » Ä Ú º Ù º » ¹ ¿ À Ï µ ¿ · Ï µ ¶ » º ½ º · ¶ · ¹ Ä » Í º ³ � à ¹ Å ¿ ¿ Í · ¿ » Ë º · ¹ ² Ä » Ä Ú º Ù º » ¹ Ò £ С Ö ã ¡ Ç� Ò Ò � ´ Í µ Å ¶ Ï Ä Å ¶ à · � � Æ � µ º » Á Æ � ¿ Å Å ¶ » · Ë Ó Æ � Ú Ã ¾ º » � Æ � º ¶ Å Å ¾ Ë � Ì Æ � ¸ º ½ ¹ Ë �Î Ö È È Ô Ü Ì º » º ¶ » ¹ º ½ Ä Í ¹ ¶ ¿ » · Í ¿ » · ¹ ½ Ä ¶ » ¹ µ º Í ¿ à ½ · º ¿ À º ¼ ¿ Å Ã ¹ ¶ ¿ » ¿ À Ï µ ¿ · Ï µ ¶ » º½ º · ¶ · ¹ Ä » Í º ¶ » ¹ µ º Å º · · º ½ Ú ½ Ä ¶ » ¸ ¿ ½ º ½ Æ ä « å æ Õ ç ß ¯ Ç « Û è Õ é Ø Ý Ø Ö Û ³ â º ½ º Ê ¶ ¹ ¾ Ò È È Ð¡ È Ç ã ¡ Ò Ç� Ò Ö � ´ µ ¶ ² Æ � º » ¹ ¿ » ² Î Ö È Ò Ò Ü � à ٠º ½ ¶ Í Ä Å Ä Å Ú ¿ ½ ¶ ¹ µ Ù · À ¿ ½ º · ¹ ¶ Ù Ä ¹ ¶ ¿ » Ä » ÊÍ Ä Å Í Ã Å Ä ¹ ¶ ¿ » ¿ À Ï Ä ½ Ä Ù º ¹ º ½ · ¶ » Ù ¿ Ê º Å Å ¶ » Ú Ï º · ¹ Ï ¿ Ï Ã Å Ä ¹ ¶ ¿ » Ê ¾ » Ä Ù ¶ Í · Ä » ʺ ¼ ¿ Å Ã ¹ ¶ ¿ » ¿ À ½ º · ¶ · ¹ Ä » Í º ¶ » Ù ¿ Ê º Å Å ¶ » Ú Ï º · ¹ Ï ¿ Ï Ã Å Ä ¹ ¶ ¿ » Ê ¾ » Ä Ù ¶ Í · Ä » ʺ ¼ ¿ Å Ã ¹ ¶ ¿ » ¿ À ½ º · ¶ · ¹ Ä » Í º ³ ² Ä ¹ µ º Ù Ä ¹ ¶ Í Ä Å Ì ¶ ¿ · Í ¶ º » Í º · Æ Ö ¢ ¢ Î Ö Ü Ð £ £ Õ Ô É� Ò ¢ � ´ µ ¶ ² Æ � º » ¹ ¿ » ² Æ � ¶ Ê · Ê ¶ Å Å Õ ´ Ù ¶ ¹ µ Ó Æ � ¿ Å Å ¶ » · Ë Ó Î Ö È È Ö Ü � ¿ » · ¹ ½ Ã Í ¹ ¶ » Ú Ä» º ¸ ¶ » Ê ¶ ¼ ¶ Ê Ã Ä Å Õ ¸ Ä · º Ê Ù ¿ Ê º Å ¿ À Ï µ ¿ · Ï µ ¶ » º ½ º · ¶ · ¹ Ä » Í º ¶ » Å º · · º ½ Ú ½ Ä ¶ » ¸ ¿ ½ º ½Î ä « å æ Õ ç ß ¯ Ç « Û è Õ é Ø Ý Ø Ö Û Ü Ð Ê ¿ ¸ º » º º Ê ¹ ¿ ¶ » Í Å Ã Ê º ¹ ¸ ¿ Å ¿ Í ¶ ½ Ä ¹ µ º ½ ¹ µ Ä » ¿ » º ìË º · ¹ ´ Í ¶ º » Í º Æ ¶ » Ï ½ º · · Æ � � � Ò È ³ Ò È È £   · Ò È ¢ Ó È Õ È Ò Ö Õ È Ó Ö Ò Õ Ç� Ò Ó � Á ¿ Å ¸ º ½ Ú Ó � Î Ö È È ¡ Ü � Ä ¹ Ä Ä » Ä Å ¾ · ¶ · à · ¶ » Ú ¹ µ º Ù º ¹ µ ¿ Ê ¿ À Å º Ä · ¹ · ë Ã Ä ½ º · к Ø ¹ ½ Ä Í ¹ ¶ » Ú ¹ µ º Ù ¿ · ¹ ¶ » À ¿ ½ Ù Ä ¹ ¶ ¿ » À ½ ¿ Ù º Ø Ï º ½ ¶ Ù º » ¹ · ³ ´ Ï ½ ¶ » Ú º ½

90