tm

Accepted 26 May 2006

strategy were to stock at low levels and to use fire to control tree abundance and thereby

range management although intuitively appealing are not optimal.

Grass-tree interactions

Livestock production

Savannas are defined as tropical ecosystems where grasses concerning the use of natural resources. Hence a thorough

E C O L O G I C A L E C O N O M I C S 6 2 ( 2 0 0 7 ) 1 0 2 1 1 4

ava i l ab l e a t www.sc i enced i rec t . com

mand trees co-dominate (Huntley andWalker, 1982). They coverlarge proportions of the tropical continents, that is 65% ofAfrica, 60%ofAustralia and40%of SouthAmerica (Huntley andWalker, 1982). The large area covered by savannas means thattheir sustainable management is of regional and globalconcern. One of the main land-use activities practised in

evaluation of sustainable development requires the consider-ation of economic, ecological and ethical factors in anintegrated framework. Economic sustainability typicallymeans that resources should be managed in such a way thatthe utility does not decline over time (Perman et al., 2003).Ecological sustainability may mean preserving ecological 2006 Elsevier B.V. All rights reserved.

1. Introduction the Brundtland Report (WECD, 1987), sustainable developmentaims to guarantee inter- and intragenerational fairnessFiresavannas is livestockproduction, yet conseconstitute sustainable livestock productioelusive (Vetter, 2005).

Defining sustainable land-use systemconsideration of what one means by susta

Corresponding author.E-mail address: higgins@wzw.tum.de (S.I. H

0921-8009/$ - see front matter 2006 Elseviedoi:10.1016/j.ecolecon.2006.05.019maintain the system in a grass dominated state.We conclude that opportunistic strategies ofAvailable online 31 July 2006

Keywords:Bush encroachmentDisequilibrium theoryOpportunistic strategiesA B S T R A C T

Classic rangeland theory advocates stocking rangelands at relatively lowand constant levels.This theory has been labelled inappropriate for savanna rangelands, because savannas arestrongly influenced by stochastic processes. Opportunistic strategies that force animalnumbers to track available forage have been proposed as an alternative managementparadigm. However, no studies have examined whether these opportunistic strategies aresustainable or optimal.We developed a simulationmodel of a savanna rangeland to identifyoptimal, sustainable strategies for the management of extensive rangelands. We optimisedthe utility of agents who are motivated by economic, production or ecological factors underboth deterministic and stochastic conditions. In all cases we found that it was optimal tomanage the system conservatively and not opportunistically. Moreover, it was optimal tomanage more conservatively under stochastic conditions. Key elements of the conservativeA R T I C L E I N F O

Article history:Received 14 November 2005Received in revised form24 April 2006aLehrstuhl fr Vegetationskologie, Technische Universitt Mnchen, 85350 Freising-Weihenstephan, GermanybLehrstuhl fr Wirtschaftslehre des Landbaues, Technische Universitt Mnchen, 85350 Freising-Weihenstephan, GermanyANALYSIS

Sustainable management of exrangelands

Steven I. Higginsa,, Jochen Kantelhardtb, Si

www.e l sev i e r. consusonwhatmightn systems remains

s requires carefulinability. Following

iggins).

r B.V. All rights reservedensively managed savanna

on Scheitera, Jan Boernerb

/ l oca te /eco l econresilience over time or ensuring that the flow of someecological service does not decline over time (Daily, 1997)For instance, in a grazing system maintaining ecological resiliencemight involve preserving the soil layer or preventing thesystem from moving into a tree dominated state; while

..-

tainability also varies among actors, because different actors

important difference between our model and previousmodelsis that we model, for both grasses and trees, two biomasscompartments, roots and shoots. This allows us to simulatethe fact that herbivores and fire cannot consume roots, andallows a separation of below ground and above ground com-petition (Fig. 1). The model of Perrings and Walker (1997) doesnot separate above ground from below ground processes,whereas Janssen et al. (2004) only make this distinction forgrasses. In the paragraphs that follow we describe growthfunctions for the grass and tree biomass components of themodel (all parameters and their default values are listed inTable 1).

The grass shoot biomass in the next time step (GSt+1) isdetermined by a growth rate parameter (gGS) and grass rootbiomass (GRt). Growth is negatively influenced by competitionfrom grass shoot biomass and woody shoot biomass. The

103M I C S 6 2 ( 2 0 0 7 ) 1 0 2 1 1 4have different risk aversions, discount rates and constraints.For instance, a farmer might maximise utility by maximisingprofit while discounting future earnings strongly; whereas asection of society might maximise utility by maximising thestability of production.

The focus of this study is the sustainable management ofextensive rangeland systems in savanna regions. That is, weconsider systems in which supplementary feeding of livestockis not economically viable and where grass and livestockproduction are primarily limited by rainfall. Rainfall in sa-vanna regions varies with season; most rain falls in thesummer months, very little rain falls in the winter months.Rainfall also varies stochastically between years. As a conse-quence, livestock farmers in savanna regions must developstrategies for dealing with variability and uncertainty inresource supply. Theory based on the harvesting of a stockthat has density dependent and deterministic growth hasstrongly influenced the management of savanna rangelands(Mentis, 1984). However, more recent studies (Ellis and Swift,1988; Behnke and Scoones, 1993) have argued that thestochastic nature of rainfall, which is the primary driver ofresource supply in savannas, make such theories inappropri-ate for savannas.

This tensionbetween classic and new rangeland theories(Cowling, 2000) has often been cast as a conflict betweenequilibrium and non-equilibrium paradigms, or between sto-chastic and deterministic paradigms. However, such polarisa-tion is counter productive, because in any dynamical systemboth non-equilibrium or transient dynamics and equilibriumdynamics are important. Similarly, many dynamical systemsare influencedby amixture of both stochastic and deterministicprocesses. In savannas, the equilibriumnon-equilibrium di-chotomy is particularly inappropriate because even when asavanna is not at equilibrium, its trajectory in state space isdetermined by the system's equilibriumpoints. The stochasticdeterministic dichotomy is also inappropriate in savannasbecause while the stochastic effects are apparent (e.g. rainfallon grass production), deterministic linkages are equally unde-niable (e.g. grass consumption on animal production).

The broad aim of this study is to propose a model ofsavanna rangelands that combines a mix of the deterministicand stochastic, equilibrium and non-equilibrium processesthat form the keystones of the classic and new rangelandparadigms. We use this model to explore guidelines for thesustainable management of livestock production systems insavannas. Our intent is to explore the extent towhich differentaspects of sustainability are compatible with one another.Specifically, we aim to define strategies that optimise thesustainability from the perspective of agents that are motivat-ed by economic, ecological and production factors under de-terministic and stochastic environmental conditions.

2. Ecological sub-modelsustaining ecological services may involve maintaining largetrees that act as keystone species. The interpretation of sus-

E C O L O G I C A L E C O N OWe create a discrete time model, inspired by the models ofPerrings and Walker (1997) and Janssen et al. (2004). The mostparameter WG describes the intensity of the competitiveeffect of woody shoots on grass shoot growth, while determines the strength and linearity of both the intra- andinter-specific competitive effects (when >1 competitiveeffects are only felt at higher levels of competitor density).Grass shoot biomass can be consumed by both herbivores andfire. Herbivore consumption is the product of the rate of shootconsumption per livestock unit per time step () and thenumber of livestock units (Zt). Note that we assume that ouranimals are grazers and that they therefore do not consumesignificant amounts of woody shoot biomass. Shoot biomassis also reduced by decomposition (dGS). We assume that theproportion of grass shoot biomass consumed by fire increaseswith fire intensity (Ft), but that above a certain fire intensitythe consumption of shoot biomass asymptotes. We describethis asymptotic relationship using a function that takes theform

f x;a;b xb

ab xb :

For x0 the function f(x; a, b) is bounded by 0 and 1 and theparameter a defines the value of x where f(x; a, b)=0.5. Theparameter b describes how rapidly f(x; a, b) changes as xapproaches a. We use the function f(x; a, b) to describe several

Fig. 1 Conceptual outline of the grasstreemodel. Themodelhas four main compartments allowing the separation ofabove ground from below ground processes. The modelsimulates growth and decomposition of these compartmentsas well as competition between the grass and tree

compartments. Grazing and fire can remove grass shoots,while fire can remove tree and grass shoots.

and

M I CTable 1 Descriptions of the parameters used in the model

Symbol Name

Ecological parametersgGS Growth rate grass shootgGR Growth rate grass rootgWS Growth rate tree shootgWR Growth rate tree roots Coefficient rainfall on growth rategZ Growth rate animalsdGS Decomposition rate grass shootsdGR Decomposition rate grass roots

104 E C O L O G I C A L E C O N Orelationships in the model. The equation for shoot biomass attime t+1 is then,

GSt1 GSt gGSGRt1GShtxWGWSht lZtf Ft;abbbbGStdGSGSt:

1

Here a is the threshold fire intensity where grass shootconsumption increases, b describes how rapidly grass shootconsumption changes as a is approached, and is themaximum grass shoot biomass removed by a fire.

The grass root biomass in the next time step, GRt+1, isinfluenced by a growth rate parameter (gGS), grass shootbiomass, and negatively by grass root biomass and tree root

dWS Decomposition rate tree shootsdWR Decomposition rate tree rootsWG Competition coefficient tree on grass rootsGW Competition coefficient grass on tree rootsWG Competition coefficient tree on grass shoots Weighting of tree shoot biomass for wWG Grazing ratef Energy transfer of ignited grassa Moisture weighting for fire ignition Maximum consumption of grass shoots by firea Location parameter grass consumption by fireb Shape parameter grass consumption by fire Maximum consumption of tree shoots by firea Location parameter tree consumption by fireb Shape parameter tree consumption by fire

Management parametersa Location parameter selling animalsb Shape parameter selling animalsmin Minimum proportion of animals solda Location parameter purchasing animalsb Shape parameter purchasing animalsmax Maximum level of fire suppressionn Natural fire ignition thresholda Location parameter fire suppressionb Shape parameter fire suppression Technology levela Location parameter technologyb Shape parameter technology

Economic parametersp Price of animals Price band factorq Cost of technologyc Cost of fire suppression Discount rate Ecological utility weight Risk sensitivity between time series

Parameters marked with * are selected through optimisation.default values

Value Units Equation

0.005 biomass day1 (1)0.005 biomass day1 (2)0.0025 biomass day1 (3)0.0025 biomass day1 (4)0.73 dimensionless (7)0.001 biomass day1 (16)0.0005 biomass.day1 (1)0.0005 biomass day1 (2)

S 6 2 ( 2 0 0 7 ) 1 0 2 1 1 4biomass (WRt). The effect of tree biomass is included tosimulate competition between grasses and trees for soilwater (Walter, 1971). The intensity of competition is describedby the parameter WG. Grass root biomass decomposes at aconstant rate (dGR). These assumptions yield the growthequation,

GRt1 GRt gGRGSt1GRtaWGWRtdGRGRt: 2

The tree shoot biomass in the next time step, WSt+1, isdetermined by a growth rate parameter gWS, tree root biomass(WRt) and is negatively influenced by the density of tree shootbiomass. Tree shoots decompose at rate dWS and can be

0.00025 biomass day1 (3)0.00025 biomass day1 (4)1 dimensionless (2)1 dimensionless (4)1 dimensionless (1)2 dimensionless (1)0.001 biomass day1 (1)

100 dimensionless (5)20 dimensionless (5)0.9 proportion (1)0.3 fire intensity (1)0.75 dimensionless (1)1 proportion (3)1.5 Fire intensity (3)5 dimensionless (3)

* grass root biomass (19)* dimensionless (19)* grass root biomass (19)* potential purchases (18)* dimensionless (18)* grass shoot biomass (15)

0.3 grass shoot biomass (15)* woody root biomass (15)* dimensionless (15)* proportion (17)

0.1 proportion (17)1 dimensionless (17)

1000 currency (8)1.1 dimensionless (8)0.05 currency (8)0.05 currency (8)0.02 proportion year1 (10)20 dimensionless (13)0.5 dimensionless (21)

and the ecological definition. In the paragraphs that followwe

costs as being time dependent although we treat them asconstants in this paper. Costs are incurred when the farmerbuys livestock and we assume that the buy price differs fromthe selling price by a factor , hence the cost incurred at time tdue to purchasing t livestock is ptt. Costs are also incurredper time step due to the level of technology (examples oftechnology are labour, fencing, boreholes and water pumps)the farmer uses. The costs of technology are simply theproduct of the unit cost of technology at time t (qt) and thetechnology level ( {0, 1}). Costs due to suppressing fires areassumed to be the product of the unit cost of suppressing firesat time t (ct) and the level of fire suppression adopted (|tn|).How the number of animals bought and sold and how thetechnology and fire management levels are determined aredescribed in Section 3.2. The income and cost parameterstogether define t, the economic index at time t as,

M I Cconsumed by fire. Tree shoot consumption by fire is modelledin the same way as grass shoot consumption by fire. That is,we assume that woody shoot consumption increases at athreshold level of fire intensity (a) and that the parameter bdescribes how rapidly woody shoot consumption changes asa is approached; is the maximum shoot biomass removedby a fire. The woody shoot growth equation is thus,

WSt1 WSt gWSWRt1WStf Ft; ag;bggWStdWSWSt: 3

The tree root biomass in the next time step, WRt+1, isdetermined by a growth rate parameter gWR, tree shootbiomass, and is negatively influenced by tree root biomass,aswell as by grass root biomass. The intensity of competition isdescribed by the parameter GW. Tree root biomass decom-poses at a constant rate (dWR). These assumptions yield thegrowth equation,

WRt1 WRt gWRWSt1WRtaGWGRtdWRWRt: 4

Fire intensity (Ft) is assumed to be a function of the fuelconsumed, the energy transfer of ignited fuel and the energyrequired to ignite fuel (Rothermal, 1972). In savannas, grass(GSt) is the primary fuel. The energy required for ignition is afunction of fuel moisture content. We assume that fuel mois-ture content is, in turn, a step function of rainfall, Rt. The stepfunction implies that there is some critical fuel moisture levelrequired for dry fuel to ignite (a). The parameter f is acoefficient that represents the energy transfer flux. Potentialfire intensity, bFt, is then,Ft

qfGStmaxRt;qa

5

Below a certain threshold in potential fire intensity (t) firescannot occur, hence realised fire intensity, Ft, is,

Ft bFt ;bFt > dtFt 0 ;bFt V dt: 6

Weassume that rainfall is the limiting resource in savannasand that plant growth rates are tightly coupled to rainfall.Rainfall varies stochastically over the seasonal cycle andbetween years. Here we define the time dependent versionsof the growthparametersused in Eqs. (1)(4) as linear functionsof monthly rainfall (Rt),

gt gs minRa;Rt; 7

where g is from Eqs. (1)(4) and s is a coefficient that describesthe relationship between rainfall and growth. We assume s isconstant across all growth parameters. Ra is used to specify therainfall at which grass production asymptotes.

In the simulations presented here we use a monthly timestep. Thisallowsus to simulate theseasonality of rainfall typicalof tropical savannas. We use the CRU global climatic database(New et al., 2002) as a source of data for Rt. For the deterministic

E C O L O G I C A L E C O N Osimulations we use the monthly mean rainfall at a site, for thestochastic simulations we use the monthly mean and variancepropose functions for the utility of these different agents.The economic utility is determined by income and costs,

which are a function of the number of animals bought andsold, the costs of technology and the costs of fire manage-ment. The income derived is determined by pt, the price of thelivestock product at time t, and t, the number of livestocksold at time t. Note that for generality we define prices andof rainfall at a site to simulate monthly rainfall as a gammadistributed random number (New et al., 2002). The monthlymean and variance of the rainfall parameters used for the si-mulations are listed in Table 2. The monthly time step additio-nally imposes a 1-month lag between removal of biomass by fireand regrowth.

3. Economic sub-model

3.1. Utility

Given the different definitions of sustainability outlined in theintroduction the agent may either aim to increase income,increase production or increase the value of some ecologicalindicator. A farmer may be interested in the economic defi-nition, the statemay be interested in the production definition

Table 2 Monthly mean and variance in rainfall used inthe simulations (from New et al., 2002)

Month Mean SD

Jan 100 14Feb 78 12Mar 63 10Apr 45 7May 9.0 2Jun 2.3 1Jul 1.2 0Aug 3.2 1Sep 15 2Oct 39 8Nov 58 11Dec 70 13

105S 6 2 ( 2 0 0 7 ) 1 0 2 1 1 4pt ptrtuptvtqtwctjdtdnj: 8

M I CThe economic utility over a planning horizon T,Ueconomic is,

Ueconomic XTt1

p Vt; 9

that is the sum of the discounted economic index (t) over thefinite planning horizon T.

We use a standard exponential discounting function tocalculate t,

p Vt pt1

1 s t

; 10

where is the discount rate. Although different discount rates() are appropriate for different agents we assume =0.02 p.a.for all baseline analyses presented here.

The productionmotivated agent aims tomaximise produc-tion irrespective of any costs that might limit this goal. Hence,the costs of fire management and technology are set to zero.However, this agent remains subject to the price differentialbetween buying and selling animals as described in Eq. (8).When pt is constant (whichwe assume for this paper) the pricedifferential, , ensures that there is always a cost to adjustingherd numbers. Hence, the production index at time t, t, is aspecial case of the economic index (Eq. (8)) where qt and ct arezero,

vt ptrtuptvt: 11

The production utility over a planning horizon T, Uproductionis the sum of the discounted production (t) index over time T,

Uproduction XTt1

v Vt: 12

There aremanypotential indices of ecological integrity, andwe do not claim any single index is appropriate for allquestions.Weassume that the ecological agent aims to protectthe soil from erosion by maintaining a high level of grass rootbiomass (GRt). We therefore assume that an appropriateecological index for a rangeland is the sum of the productionindex at time t (t) and grass root biomass,

gt mGRt vt: 13

The parameter is used to weight the contribution of GRtand t to t. How to appropriately weight these two fundamen-tally different indices (GRt and t) in a single utility index (t) isnot clear. We use =500, this ensures that the ecologicallymotivated agent derives the similar utility from a unit of grassroot biomass (GRt) as from a unit of production (t). Hence, t aswe have defined it is only useful within the context of thisstudy. The ecological utility over a planning horizon T, U-

ecological is the sum of the discounted ecological index (t) overtime T,

Uecological XTt1

g Vt: 14

3.2. Rangeland management

106 E C O L O G I C A L E C O N OIn Section 3.1 we proposed utility functions for agents that aremotivated by economic, production or ecological factors. In allcases the farmer will implement the strategies that aim tomaximise one of these three utilities. Fundamentally, thefarmer's management actions are constrained by rainfall.There is little the farmer can do to influence rainfall, but thefarmer can stimulate grass productivity by manipulating fireregime and livestock numbers. In the paragraphs that followwe describe a series of functions that describe the farmer's fireand livestock management behaviour.

Fire is a major tool in the arsenal of a manager of anextensive rangeland. The farmer can stimulate grass produc-tion by encouraging intense fires that in turn reduce treebiomass. The goal is to ensure that fires only burn once suffi-cient fuel for an intense fire has accumulated and when wea-ther conditions are extreme (Trollope, 1983). Fires alsoconsume grass shoot biomass, hence any benefits of burningneeds to be offset by the cost of lost livestock production due toreduced shoot biomass. However, indefinite fire suppressionleads to excessive accumulation of above ground dead grassbiomass, which shades the living grass shoot biomass andthereby constrains grass shoot production (Trollope, 1989).Therefore, in many savannas burning has a net positive effectbecause it removes dead grassmaterial and thereby stimulatesgrass growth.

The parameter n determines the natural potential fireintensity below which a fire does not occur (cf. Eq. (5)). Hencewe implicitly assume that fuel loads, and not ignitions, limitthe occurrence of fire. The farmer can, through fire suppres-sion, raise thenatural fire spread threshold and thereby ensurethatmore intense fires occur or that fires occur less frequently(cf. Eq. (6)). We assume that the farmer suppresses fires as treebiomass increases. Because tree shoot biomass in the modelfluctuates due to fire, it is not a stable indicator of tree biomass.We therefore use, in the model, tree root biomass (WRt) as anindex of tree biomass (a farmer would probably use stem basalarea as an indicator). The level of fire suppression the farmerapplies at time t (t) is determined using the following function,

dt dn dmaxdnf WRt;ad;bd 15

here n is the natural fire threshold, max is the maximum levelof fire suppression, a is the threshold root biomass atwhich thefarmer suppresses fire, and b describes how rapidly the farmersuppresses fire as the threshold root biomass is approached.

The major task faced by the rangeland farmer is livestockmanagement. At each time point, t, the farmer can purchase tlivestock or sell t livestock, in addition the number of livestockcan grow due to its own density dependent growth process,yielding an equation for Zt+1, the livestock numbers at time t+1,

Zt1 Zt vtrt gZZt 1 ZtKt

16

Here gZ is the biological growth rate of the livestockpopulation and Kt is the farm's carrying capacity at time t. Kt isin turn defined by the consumption rate of livestock (), theavailable forage (GSt) and the farmer's investment in technology,

Kt GStl f w;aw;bw: 17

S 6 2 ( 2 0 0 7 ) 1 0 2 1 1 4Hence,we assume that the technology level a farmer chooses,, determines how close the farmer can get to exploiting the

rt max1f GRt;ar; br;rminZt; 19

(Storn and Price, 1997; Price et al., 2005). Hence, we use thedifferential evolution algorithm to maximise either Ueconomic,Uproduction or Uecological over the planning horizon using thecontrol set,

C ar;br; rmin; av;bv; dmax;ad; bd;w: 20

Note that the algorithm selects one control set for an entiresimulation; that is within a single simulation run over thechosen planning horizon, the parameters remain constant. Inessence this means that the farmers mind-set remains fixed.However, because these parameters summarise the farmers'response to changing conditions, the behaviour of the farmeris not constant. The reader should additionally be aware thatthe solutions found using the simulationoptimisation ap-proach are only optimal given the constraints the manage-ment functions (Eqs. ), (15), (17), (18), (19)) impose.

We choose a planning horizon of 100 years. While this maybe on the upper interval of appropriate time scales, it does incombination with low discount rates (0.02 p.a.) select againstnon-sustainable strategies. We choose initial conditionswhere the relative sizes of tree and grass populations are

107M I C S 6 2 ( 2 0 0 7 ) 1 0 2 1 1 4where t is the number of livestock to be sold at time t and GRtis the grass root biomass at time t. We use grass root biomassbecause this is, in the model, a more stable indicator of thecondition of the grass sward than shoot biomass (shoot bio-mass fluctuates with rainfall, grazing and fire). In reality, thefarmer would probably use grass cover, which is closelyrelated to grass root biomass as an indicator of grass swardcondition. The shape of this response curve is defined by theparameters a and b.

In summary, the farmer is able to manipulate the fireregime by selecting the values of the parameters max, a andb; determine the level of the available forage that can beexploited by selecting a technology level ; determine a stra-tegy for buying and selling livestock by selecting the values ofthe parameters a, b, a, b and min.

4. Optimisation

Themodel is non-linear, stochastic and hasmultiple domainsof attraction; these attributes make it poorly suited to classicoptimisation methods (e.g. optimal control or dynamic pro-gramming). As an alternative, we use a simulationoptimisa-tion approach, that is we combine our process orientatedsimulation model with a numerical optimisation algorithm.Simulationoptimisation methods have been successfullyused to optimise hydrological (Aly and Peralta, 1999, Kout-soyiannis andEconomou, 2003), biochemical (MendesandKell,1998) and ecologicaleconomic systems (Janssen et al., 2004).We experimented with standard non-linear optimisation(quasi-Newton method of Byrd et al., 1995), genetic algorithm(Mebane and Sekhon's 2005 genetic optimisation using deri-vatives algorithm) and differential evolution methods (Stornand Price's, 1997 differential evolution algorithm as imple-mented by Ardia, 2005). Our initial tests suggested thatdifferential evolution was more robust and efficient at findingpotential carrying capacity (GSt/). The parameters a and bdescribe how the technology level chosen influences theproportionof thepotential carryingcapacity thatcanbeexploited.

An important part of a farmers challenge is timing hislivestock sales and purchases. We assume that a farmer buyslivestockwhen the prospects for production are good and sellslivestock when the prospects for production are poor. Speci-fically we assume that the livestock a farmer purchases t is,

vt bZtf bZt; av;bv; 18where bZt is the potential livestock the farmer could purchase(bZt=max(0, KtZt)) and a and b define what proportion ofthese potential purchases the farmer realises. Hence, weassume that the farmer never overstocks the farm throughpurchases. The farmer sells some base proportion of animals,min, in every time step, but sells more livestock when thegrass root biomass is below some critical point. This behaviouris summarised as,

E C O L O G I C A L E C O N Othe global solution than the othermethods, a result confirmedby more systematic comparisons using benchmark problemsFig. 2 Time series showing the development of tree (red),grass (green) and animal (blue) abundance over time. Thepoints indicate the intensity of individual fires. Thesimulations illustrate the bush encroachment problem facedby rangeland farmers in savanna regions. The top panelshows a deterministic simulation, the middle panel usesthe same control strategy as the top panel but occurs understochastic conditions. The lower panel is a stochasticsimulation with low levels of fire management. Plantabundances are the below ground biomasses. (For interpretation

of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

similar and where animal numbers are less than the carryingcapacity K (see Eq. (17) for a definition of K).

In the case of stochastic optimisation it is necessary toconsider the risk sensitivity of the agent. In the deterministiccase we assume that the agent maximises the discountedutility over the planning horizon (cf. Eqs. (9), (12), (14)). For thestochastic case we need to consider the variance betweendifferent stochastic realisations of the same managementstrategy. Hence, for stochastic problems we optimise ,

bU EUevarU 21here describes the level of risk aversion to the variancebetween stochastic evaluations of the same managementstrategy. For this study we assume =0.5 for all benchmarkanalyses and use a sample size of 25 replicate simulations tocalculate . Trial simulations showed that =0.5 effectively

selects against strategies that have a high risk of failure(collapse of the livestock population) when unfavourablesequences of rainfall occur.

We used R (R Development Core Team, 2005) implementa-tions of the optimisation routines. Themodel was written in Cand compiled as an R package allowing us to use R to interfacewith the optimisation routines.

5. Model analysis

5.1. Model behaviour

The system has three attractors that are of ecological interest,grass dominated, tree dominated and savanna (mixed treegrass) states. Scheiter and Higgins (submitted) provide a moreformal discussion of the mathematical conditions necessary

al (in t

108 E C O L O G I C A L E C O N O M I C S 6 2 ( 2 0 0 7 ) 1 0 2 1 1 4Fig. 3 Time series showing grass (green), tree (red) and anim(middle panels) and ecological (bottom panels) optima found

cases. All abundances are the below ground biomasses. (For intereader is referred to the web version of this article.)blue) abundances for the economic (top panels), productionhe deterministic (left panels) and stochastic (right panels)

rpretation of the references to colour in this figure legend, the

for the savanna state. These analyses show that the savannastate is possible when either fire is present or when grasstreecompetitive interactions are weak. The weight of empirical

evidence for savannas, however, supports the notion thatcompetition between grasses and trees is not weak (Scholesand Archer, 1997; Higgins et al., 2000; Bond et al., 2005;

Table 3 Parameter values for the optimal control set selected by numerical optimisation for stochastic and deterministiccases

Deterministic Stochastic

Parameter Economic Production Ecological Economic Production Ecological

a 0.00 0.00 0.09 0.01 0.00 0.00b 8.19 15.00 15.00 8.98 5.67 6.90min 0.02 0.03 0.02 0.02 0.03 0.03a 17.59 17.35 28.70 27.09 30.00 29.65b 30.00 30.0 12.05 28.63 30.00 30.00max 1.32 2.16 1.25 2.07 4.22 4.14a 0.03 0.03 0.07 0.37 0.75 0.49b 0.53 23.07 16.57 9.53 0.01 0.01 1.00 1.00 1.00 0.04 0.26 0.13

109E C O L O G I C A L E C O N O M I C S 6 2 ( 2 0 0 7 ) 1 0 2 1 1 4Fig. 4 Animal sales, purchases and fire suppression strategies s(dashed blue), or ecological (dot dashed green) utility functions. Tare for the stochastic case. (For interpretation of the references toversion of this article.)elected when optimising an economic (solid red), productionhe left panels are for the deterministic case, the right panelscolour in this figure legend, the reader is referred to the web

biomass, which leads to crashes in the animal population. Theanimal population is therefore only briefly at high densities,hence rangeland degradation does not occur. This time series

M I C S 6 2 ( 2 0 0 7 ) 1 0 2 1 1 4Sankaran et al., 2004). Hence, for all simulations presentedhere we assume that grasstree competition is intense(GW=WG=WG=1) and that fire is necessary to preventtrees from dominating grasses. Other parameter values wereselected so as to ensure (1) that grasstree coexistence ispossible despite assuming the competition levels are high, (2)that tree dominance is possible under certain fire and animalstocking regimes and conversely, (3) that certain fire and ani-mal stocking regimes can lead to a grass dominated state.

Fig. 2 shows time series from simulation runs that illustrateimportant properties of the model. Since savanna systems areonly productive rangelands when in grass dominated orsavanna states, the central problem facing rangeland man-agers in savanna regions is to keep the system away from atree dominated state. The upper panel shows a deterministicsimulation initiated with moderate animal abundance. In thissimulation the animal abundance rapidly increases, causinggrass abundance to decrease which in turn allows tree abun-dance to increase. The increasing tree abundance furthersuppresses grass production to a point where fire is lost from

Fig. 5 Technology levels selected when optimising theeconomic, production or ecological utility function. Gray barsare for the deterministic case, black bars are for the stochasticcase.

110 E C O L O G I C A L E C O N Othe system; without fire trees increase further and grass isfurther suppressed. The loss of fire first leads to a pulse inanimal numbers, because animals are no longer competingwith fire for grass consumption. This increase in animal num-bers, however, leads to animals dramatically overshooting thecarrying capacity, and a final crash in animal numbers occurs.Subsequent elimination of animals from the system cannotreturn the system to a productive rangeland. Hence, the treedominated state is stable and the system is, from the per-spective of a livestock farmer, degraded.

The degradation sequence described in the previous para-graph represents a classic rangeland theorist's view of over-grazing. A new rangeland theorist's view is that stochasticvariation in rainfall prevents the animal populations fromreaching high enough densities for long enough timespans todegrade range condition (Ellis and Swift, 1988). This weakcoupling of range condition and animal numbers is illustratedin the second panel in Fig. 2. The management strategyadopted in this simulation is the same as that adopted in thedeterministic simulation (top panel in Fig. 2), yet the stochasticnature of rainfall leads to occasional reductions in grass shoot

Fig. 6 Relationship between potential carrying capacity,technology defined carrying capacity Kt , and stocking ratesZt . The upper panel (showing median, quartiles and range)describes the distribution of the potential carrying capacity ofover a 100-year simulation when optimising either theeconomic, production or ecological utility function underdeterministic (gray bars) or stochastic conditions (black bars).Themiddle and lower panels show the cumulative density ofKtZt for deterministic and stochastic cases respectivelywhen optimising either the economic (red solid lines),production (blue dashed lines) or ecological (green dot-

dashed lines) utility functions. The cumulative densityfunctions are based on a single 100-year simulation for thedeterminstic cases, and on 25100-year simulations for thestochastic case. (For interpretation of the references to colour

in this figure legend, the reader is referred to the web versionof this article.)

represents the situationdescribedbyDesta andCoppock (2002)for Ethiopian rangelands. However, we emphasise that degra-dation is alsopossible in the stochastic case. The lowerpanel inFig. 2 illustrates howdegradation could set in rapidlywhen firemanagement does not encourage intense fires. Moreover, theoptimal rangeland strategies identified in the next section,further emphasise that strategies for optimal rangeland ma-nagement differ from those illustrated in the middle of Fig. 2.

5.2. Optimal range management

The control sets for deterministic and stochastic optimisationof the economic (Eq. (9)), production (Eq. (12)) and ecological(Eq. (14)) utilities are presented in Table 3. Fig. 3 illustrates timeseries of grass, tree and animal numbers generatedusing these

conditions (Fig. 3). Specifically all strategies ensure some levelof grass dominance. However, the modal and peak stockingrates selectedunder stochastic conditions are lower than thoseselected under deterministic conditions (Fig. 3). Hence, eventhough grass biomass often peaks to levels above thoseobserved under deterministic conditions, the optimal strategyis not to aggressively track these changes. This is exemplifiedby the low technology levels adopted by all agents under stoc-hastic conditions (Fig. 5). The technology level defines howclosely the carrying capacity tracks the potential carryingcapacity (see Eq. (17)); hence the strategy adopted by all agentsis to stock well below the potential carrying capacity. In allcases the stocking rate is such that the potential carryingcapacity (Fig. 6, upper panel) is much greater than the stockinglevel (Fig. 3). Effectively this keeps a reserve of grass biomass

111E C O L O G I C A L E C O N O M I C S 6 2 ( 2 0 0 7 ) 1 0 2 1 1 4control sets, Figs. 4 and 5 summarise the control strategiesadopted, and Fig. 6 illustrates how the stocking rates adoptedrelate to the carrying capacity.

The time series show that in all cases the system is ma-naged to ensure some level of grass dominance (Fig. 3). Underdeterministic conditions, the strategies differ in the level ofgrass dominance: the ecological strategy has the highest levelof grass dominance, while grass dominance is only slightunder the production strategy. The ecological strategy a-chieves grass dominance by purchasing animals only whenconditions are extremely favourable. This effectively delaysthe build up of the animal population, and this investment inthe grass layer allows the ecological strategy to achieve thehighest animal numbers in the last 20 years of the planninghorizon.

The economic strategy is cost sensitive (cf. Eq. (9)). It savescosts by opting for a less aggressive fire management strategy(Fig. 4 shows that fire suppression is lowest under theeconomic strategy). The ecological and production utilityfunctions are not influenced by the costs of fire managementor technology; both therefore use high levels of fire suppres-sion (Fig. 4). The production strategy adopts the highest levelof fire suppression (Fig. 4), which allows it to use fire to controltree density even though the fuel loads (grass biomass) arelower than under the other strategies.

The optimal strategies under stochastic conditions quali-tatively reflect the strategies adopted under deterministicFig. 7 Utilities achieved when optimising either the economic, p(gray bars) and stochastic conditions (black bars).that can be used to fuel fires which in turn prevents treedominance.

While the stocking strategy is more conservative understochastic conditions, the fire management strategy is moreaggressive (Fig. 4). The ranking of the intensity of the fire ma-nagement strategies, however, remains as observed underdeterministic conditions, that is the production agent is themost aggressive and the economic agent the least aggressive.The ecological and production agents adopt step functions forfire management, allowing them to implement high levels offire suppression at low tree abundance. The economic agent'sfire suppression function is a more gentle sigmoid function oftree abundance. As was the case under deterministic condi-tions, the economic agent saves costs by being less aggressivein fire management. Hence, in the economic case, tree domi-nance is avoidedby a combinationof firemanagement and lowstocking rates.

All strategies seek to keep Zt less than Kt as this avoidsdensity dependent animalmortality. Under both deterministicand stochastic conditions the probabilities of Zt exceeding Ktare low (

although the stocking rate adopted is lower in absolute terms,it is higher relative to Kt.

Animals are only purchased when the potential number ofpurchases is high and even then only a small proportion of thepotential purchase is realised (Fig. 4). Under deterministicconditions the ecological strategy is the most conservative;and the production and economic purchasing strategies arevery similar. Under stochastic conditions purchasing onlyoccurs when the potential purchase is higher than under de-terministic conditions. In the stochastic scenarios, however,the potential purchase seldom reaches values above 15, mea-ning that animals are almost never purchased. In general, thepurchasing behaviour observed suggests that the agents donot attempt to adopt a seasonal purchasing strategy or to reactto opportunities presented by sequences of high rainfall yearsin the stochastic case.

5.3. Compatibility and sustainability

We explore the compatibility of the different agents' utilityfunctions by examining the utilities achievedwhen optimisingeither the economic, production or ecological utility under

animals die. For instance during a drought, grass production

112 E C O L O G I C A L E C O N O M I Cstochastic and deterministic cases (Fig. 7). We do not presentdata showing that strategies selected under deterministic con-ditions perform poorly under stochastic conditions, as thispoint has been made by Janssen et al. (2004). We find, as didJanssenet al. (2004), that theutilities achievedunder stochasticconditions are lower than those achieved under deterministicconditions (Fig. 7). This result is in agreement with stochasticharvesting theory that suggests that the yield froma stochasticresource system is lower than the yield from a deterministicresource system (Schmitt andWissel, 1985; Lande et al., 1997).

Selecting a low discount rate (2%), a long planning horizon(100 years) and forcing the farmer to have one managementphilosophy for the entire planning horizon ensures that all the

Fig. 8 The sustainability of the optimal solutions selectedusing the economic utility function as a function of thediscount rate under deterministic (open circles) and sto-chastic (solid circles) conditions. The sustainability index issimply the normalised difference between the utility derivedin the first half of the simulation relative to the utility derived

in the second half of the simulation. Sustainable strategieshave positive indices.stops, yet the herd's nutritional requirements must still bemet, hence some animals suffer density dependent mortality.This view is in agreement with evidence from savannas thatsuggests that animals die from hunger and not from thirstduring droughts (Walker et al., 1987), and we are unaware ofcontrary evidence. While this view is consistent with classicrangeland ecology, it is contrary to the new rangeland ecologywhich views drought mortality as a density independent pro-cess (Sullivan and Rohde, 2002).

The optimal strategy under stochastic conditions is to stockwell below the potential carrying capacity at any given timeoptimal strategies we identified were sustainable. For ourpurposes, sustainability iswhen theutility in the secondhalf ofthe planning horizon is not less than the utility in the first halfof the planning horizon. The sensitivity of this sustainabilityindex to the discount rate was explored by seeking optimaleconomic strategies under a variety of discount rates for boththe deterministic and stochastic case (Fig. 8). This analysisshows that the deterministic strategies are sustainable even atdiscount rates as high as 10%, but that under stochastic con-ditions sustainability is lost at discount rates greater than 8%.

6. Discussion

The new rangeland ecology criticises classic rangeland ecolo-gy's recommendation of constant stocking on the basis thatthis represents a lost opportunity to rangeland farmers, pre-venting them from taking advantage of bumper grass produc-tion years (Behnke and Scoones, 1993). One alternative toconstant stocking is to attempt to force animal numbers totrack variation in grass production. Although intuitively ap-pealing, the feasibility of the tracking strategies suggested bythe new rangeland ecology has been difficult to demonstrate(Illius et al., 1998). Our analysis further adds weight to theevidence that tracking strategies are not feasible. Solutions wepresent suggest that elements of tracking strategies, such asrestocking after crashes in livestock numbers, or preemptivelyselling more animals when indicators of poor range conditionare apparent are not optimal. More notably, our solutionssuggest that it is optimal to stock at ratesmuch lower than thepotential carrying capacity.

Ourmodel results are the consequences of our assumptionsregarding the relationship between grass biomass, carryingcapacity, animal numbers and mortality. Since imprecise for-mulation of these relationships has fueled recent debates inrangeland ecology (Vetter, 2005), we feel it prudent to stateexplicitly what our assumptions in this regard are and howthey relate to classic rangeland ecology and the so-called newrangeland ecology.

Our key assumption relates to carrying capacity. Carryingcapacity is defined in the ecological literature as the densitybeyond which growth ceases to be positive and becomes ne-gative. In other words, carrying capacity is the density beyondwhich mortality exceeds natality. In our model carrying ca-pacity is defined by grass shoot biomass. When grass shootbiomass cannot support the herd's nutritional requirements,

S 6 2 ( 2 0 0 7 ) 1 0 2 1 1 4(Fig. 6). The stocking level is therefore not defined as a functionof peak grass shoot biomass, but rather as aminimumbiomass

of sufficient intensity to damage trees. Seasonal variations in

reduce the buffering capacity of the system and henceincrease the risk of a crash in animal numbers, which are ex-

indicate that although the definition of the utility function

M I Cgrass biomass ensure that carrying capacity varies with time;hence animal numbers tend to approach the carrying capacityduring the dry season, but retract from it during thewet season.Animal numbers are, therefore as argued by Illius andO'Connor(1999), only loosely coupled to grass biomass in the wet season,but coupled to it in the dry season.

To understand why tracking strategies are not optimal weneed to consider the risks associated with tracking strategies.When aggressively tracking available forage the animal num-bers at time t (Zt) are close to the carrying capacity at time t(Kt). In deterministic yet seasonally dry systems and especiallyin stochastically variable systems there is a chance that Ztexceeds Kt; this probability is greater when the system is ma-naged so that Zt is close to Kt. When Zt exceeds Kt densitydependent mortality occurs; should Zt greatly exceed Kt thenover-compensating density dependent mortality occurswhich leads to reductions in Zt to levels well below Kt. Oursolutions show that for both deterministic and stochasticcases Zt almost never exceeds Kt (Fig. 6). Hence, stocking atconservative levels provides the farmer with a buffer andprevents the over-compensating density dependent mortalitythat occurs when Zt greatly exceeds Kt. Density dependentmortality is therefore, contrary to the view expressed by thenew rangeland ecologists (Ellis and Swift, 1988; Behnke andScoones, 1993; Sullivan and Rohde, 2002), more importantunder stochastic conditions.

Our model suggests that degradation (e.g. transition to atree dominated state) is avoided by stocking at low levels, afinding which agrees with classic rangeland theory. In con-trast, the new rangeland ecologists propose that stochasticityprevents a coupling between animals and the resource baseand thereby prevents density dependent processes from shif-ting the system into a degraded state. While we illustrate theplausibility of this idea (see Fig. 2), our analyses show thatsuch strategies are not optimal.

Our analysis raises several questions for future study,whilesome of these questions could be addressed with a modifiedversion of this model, others call for empirical research tojustify model parameters we used and assumptions wemade.For instance, we considered only grazing animals, it would beinteresting to explore howmixed feeders influence the results.Moreover, while our results suggest that tracking strategies arenot optimal it would be interesting to seek the conditionsunder which tracking strategies might be optimal. Finally, weassume, as did Perrings and Walker (1997) and Janssen et al.(2004) that fire can be used to control tree densities. Whileempirical data from some savannas support this assumption(e.g. Trollope, 1983) the extent to which this assumption isappropriate to all savanna rangelands is not clear.

7. Conclusions

In this paper we analyse a dynamic model that enables us toneeded to buffer the system against environmental variability.This buffer needs to be large enough to both carry animalsthrough the dry season and to ensure that there is fuel for fires

E C O L O G I C A L E C O N Oanalyse a rangeland system under stochastic conditions. Themodel illustrates how both deterministic and stochastic pro-influences the optimal strategy some generalisations emerge.Namely, itwasoptimal to select a strategy thatwas sustainableand led to grass dominance; furthermore all agents weremoreconservative under stochastic conditions.

Identifying management strategies that are, in principle,sustainable is a necessary first step for achieving sustainabil-ity; additional steps are to (1) explore why savannas are oftentree dominated even though our analysis suggests that it isrational to avoid tree dominance, and to (2) design policyinstruments that encourage the adoption of the sustainablemanagement practices.

Acknowledgements

We thank the Robert Bosch Foundation for financial supportand the reviewers for valuable comments.

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Sustainable management of extensively managed savanna rangelandsIntroductionEcological sub-modelEconomic sub-modelUtilityRangeland management

OptimisationModel analysisModel behaviourOptimal range managementCompatibility and sustainability

DiscussionConclusionsAcknowledgementsReferences