demographic matrix models for structured populations dominique allaine

Demographic matrix models for Demographic matrix models for structured populationsstructured populations

Dominique ALLAINE

Vital rates describe the development of individuals through their life cycle (Caswell 1989)

The response of these rates to the environment determines:

population dynamics in ecological time

the evolution of life histories in evolutionary time

Vital rates are : birth, growth, development, reproductive, mortality rates

Structured populationsStructured populations

Structured populationsStructured populations

Obviously, vital rates differ between individuals. These vital rates may differ according to age, to sex, to size category …

Each individual is then described by a state at a given time. For example,an individual may be in the state « yearling female » at time t.

Populations are made of individuals that are different. So, we can consider the structure of the population according to sex, to age, to size category, to status …

The structure of a population at a given time is then characterised by a distribution function expliciting the number of individuals in each state

Structured-population modelsStructured-population models

Models based on the global size of a population does not take into account the differences between individuals.

It is important to analyse the response of vital rates in structured populations to take into account differences between individuals

Structured-population models aim at including the differences between individuals in vital rates. These models are applied tostructured populations.

General characteristics

Structured-population modelsStructured-population models

We will assume that all individuals in a same state experience thesame environment and then respond in the same way. In other words,all individuals in a same state will have the same values of vital rates.

Structured-population models are mathematical rules that allow tocalculate the change over time of the distribution function of thenumber of individuals in each state.

Three types of mathematical approaches:

• matrix models• delayed differential equation models• partial differential equation models

Matrix population modelsMatrix population models

The different states are discrete

This means that individuals are classified into categories. For example,individuals are either males or females, are eitheir juveniles oryearlings or two-years old … When considering size, discrete categories have to be identified.

We thus have to recognise categories and at a given time, each individual may be associated unambiguously to a category.



The change in the population structure is in discrete time

This means that the change of the distribution of the number ofindividuals in each state is given in discrete time. If we use, for example, an annual time scale, the number of individuals in each stateis given for each year.



First developped by P.H. Leslie during the 1940’s

N(t+1) = M N(t) (n,1) (n,n) (n,1)

The number of individuals in each state at time t is given by a vector N(t)

This vector is projected from time t to time t+1 by a population projection matrix M




Different steps:

• To determine the projection interval (time scale)

• To determine the states of importance

• To identify the vital rates necessary

• To establish the life-cycle graph


The projection interval is usually the year in analyses on vertebrates

States of importance are often age, sex

Vital rates refer to:

• proportion p of reproductive females per age class

• fecundity f per age class

• survival rate s per age and sex classes

Populations structured in age

Life cycle in the case of a population structured in age


1 2 3 4

1S 3S

2F3F

1F

4F

2S 4S


)(

)(

)(

)(

00

000

000

)1(

)1(

)1(

)1(

4

3

2

1

43

2

1

4321

4

3

2

1

tN

tN

tN

tN

SS

S

S

FFFF

tN

tN

tN

tN


The matrix model corresponding to the previous life-cycle graph is:


Leslie matrix


Populations structured in stage

Life cycle in the case of a population structured in stage

E J Ad

1S 3S

F

2S

R1 R2

)(

)(

)(

0

0

0

)1(

)1(

)1(

32

21

1

tA

tJ

tE

SR

SR

FS

tA

tJ

tE

The matrix model corresponding to the previous life-cycle graph is:

Lefkovitch matrix


Populations structured in stage

N(t+1) = M N(t) (n,1) (n,n) (n,1)

tt N

N

ss

sfpsfp

N

N

2

1

21

022011

12

1

Constant linear models

In this case, the matrix M has constant coefficients

Example: a two age-classes model



The Perron-Frobenius theorem

A non negative, irreductible, primitive matrix has 3 properties:

1. the first eigenvalue 1 is unique, real and positive

2. the right eigenvector w1 corresponding to 1 is strictly positive

3. the left eigenvector v1 corresponding to 1 is strictly positive



A reductible matrix has some stages that make no contribution to some other stages

Example:

1 2 3 4

1S 3S

2F3F

1F2S 4S



A matrix is primitive if the greatest common divisor of the lengths of the loops in the life-cycle graph is 1 Example:

1 2 3 4

1S3S

4F

2S


Imprimitive

1 2 3 4

1S 3S

2F3F

1F

4F

2S 4S

Primitive


Leslie matrices are often irreductible and primitive

N(t+1) = M N(t)


N(t+1) = M N(t) = Mt N(0)



Diagonalisation of the matrix M

M = WW-1 M2 = WW-1 WW-1

M2 = W2W-1 Mt = WtW-1

Let an eigenvalue of the matrix M

Let N(t) the right eigenvector of the matrix M associated to



Mt = WtW-1 Mt = WtV*

00

00

00

and

00

00

00

2

1

2

1

t

t

t

t

s s



Remember: N(t) = Mt N(0)

0*

0

*

NvwN

NVWN

iii

tit

tt

wi is a column vector and vi* is raw vector so their product is a matrix



0*1

10

*22

1

20

*11

1

1t1

0*1

ts0

*22

t20

*11

t1

NvwNvwNvwN

NvwNvwNvwN

ss

t

s

tt

t

sst

It follows that:


Constant linear models Asymptotic results

And consequently:

0*11

t10

*11t

1

0*1

10

*22

1

20

*11

1

1t1

lim lim

limlim

NvwNNvwN

NvwNvwNvwN

tt

t

t

ss

t

s

tt

t

t

t

The dynamic is driven by the dominant eigenvalue and associated eigenvectors



N(t+1) = M N(t) = N(t)

It results that asymptotically:



Asymptotically, the right eigenvector associated to the dominant eigenvalue gives the stable state structure

M W = W

Asymptotically, the dominant eigenvalue corresponds to the annual multiplication rate

n(t+1) = n(t)

= er where r is the population growth rate


The annual multiplication rate, the stable state structure and the reproductive values depend on the values of vital rates but are independent of initial conditions (ergodicity)

Asymptotically, the left eigenvector associated to the dominant eigenvalue gives the stable reproductive value, i.e. the contribution of each state tothe population size


VM = V



The damping ratio expresses the rate of convergence to the stable population structure.

In other words, the convergence will be more rapid when the dominanteigenvalue is large relative to the other eigenvalues.

2

1

Stochastic linear models


N(t+1) = Mt N(t) (n,1) (n,n) (n,1)

ttt

tttttt

tN

N

ss

sfpsfp

N

N

2

1

21

022011

12

1

In this case, the matrix M has coefficients varying randomly with time

Example: a two age-classes model




Growth rate

011 NMMMN ttt

The population size at time t is:

ti

tit NNn 011 NMMMn ttt

The simple estimation of the multiplication rate, the stable state structure and the reproductive value calculated from eigenvalues and eigenvectors are no longer valid in stochastic linear models



Growth rate

It has been shown (Furstenberg & Kesten 1960) that:

stt

LnnLnt

)(1

lim

where Lns is the stochastic growth rate

011lim)(1

lim NMMMLnnLnt tt

tt

t



Growth rate

stt

LnnLnt

)(1

lim

st LnNLnt

E )(1

st LnNLnE t )( 2 t )( tNLnV

),(tLn ~ )( 2s tNNLn t

),(Ln ~ )(1 2

s tNNLn

t t



Growth rate

The analytical calculation of Ln s is not easy

The stochastic growth rate can be found by

1. Simulation

2. Approximation



Growth rate

1. Simulation

)()( 0

t

nLnNLnLn t

s

The stochastic growth rate can be estimated from the average growth rateover a long simulation from the maximum likelihood estimator :

An estimation of the stochastic growth rate from a simulation is:

i

it

s Lnt

nLnnLnLn

t

1

)()( 0



Growth rate

1. Simulation

Usually, the stochastic growth rate is calculated as the average ofestimations obtained from several simulations

)()(1

1

0

M

j

jtjs t

nLnnLn

MLn

tt N

N

tsts

tstftptstftp

N

N

2

1

21

022011

12

1

)()(

)()()()()()(

B1(s0,s0)

B1(s2,s2)B1(s1,s1)

P(1)B1(p1,p1)

1. Simulation: environmental stochasticity



Growth rate



Growth rate

1. Simulation: demographic stochasticity

),(),(

),(

2211)1(2

021)1(1

sNsNN

spfpfN

t

t



Growth rate

1. Simulation: demographic stochasticity

),(

),(

2)(22

1)(11

pNnf

pNnf

t

t

Number of reproductive females :

2)2(

1)(

112

111

nf

i

nf

i

fpf

fpf

Number of young produced :



The stochastic growth rate can be analytically approximated when vital ratesvary in a small way that is, their coefficients of variation are much less than 1

2. Approximation

If we assume that all distributions are identical and independent, then, the stochastic growth rate is approximated by:

)var2

12

(LnLn s

where is the mean growth rate

Growth rate


Perturbation analysis

From a biological point of view, it is important to know which factor has thegreatest effect on

Perturbation analyses allow to predict the consequences of changes in thevalue of one (or more) vital rate on the value of

Two concepts:

1. Sensitivity

2. Elasticity


1. Sensitivity

The sensitivity sij indicates how the value of is impacted by a modification of the value of the parameter aij

ijij a

s

wv

wvs ji

ij ,

The sensitivity is dependent on the metric of the parameter aij



2. Elasticity

The elasticity eij indicates the relative impact on of a modification of the value of the parameter aij

ijij

ij

ijij s

λ

a

a

λ

λ

ae

The elasticity is independent on the metric of the parameter aij

and 1 ij

ije



3. Prospective analyses


The prospective analyses are done by calculating sensitivities or elasticities


4. Retrospective analyses

The objective of a retrospective analysis is to quantify the contribution of each vital rate to the variation in

The variability of may be due to variations in vital rates in space (between populations) or in time (between years) for example

The impact of a given component aij of the projection matrix on the variation of depends on both the variation of the component and the sensitivity of to this component



4. Retrospective analyses

A component aij will have a small contribution to the variation in if this component does not change much or if is not very sensitive to aij or if bothare true.

The contribution of aij to the variation in involves the products of sij and the observed variation in aij

Two approaches are possible:

1. random effects

2. fixed effects



4. Retrospective analysesa. Random effects

klijij kj

klij ssaaV ),cov( )(

Covariances between parameters are calculated directly from observed matrices

Sensitivities are calculated from the mean matrix


We want to identify the contribution of vital rates to the variance in during a time period for example (where time is considered as a random factor)



klijkj

klij ssaa ),cov( ij

The contribution of the vital rate aij can be measured as:

It is the sum of contributions including aij





Tuljapurkar showed that:

klijklijij kj

aa eeCVCVVklij ,

2 )(

klijklijkj

aa eeCVCVklij ,

2ij

The contribution of the component aij can be measured as:




These formula can also be rewritten as:

klijij kj klij

klij eeaa

aaV

),cov( )( 2

klijkj klij

klij eeaa

aa

),cov( 2

ij

The contribution of the component aij can be measured as:


4. Retrospective analyses b. Fixed effects


We want to identify the contribution of vital rates to the difference in multiplication rates between traitments.

To simplify, consider two populations in different environments. These populations then vary according to the values of vital rates and, consequently in projection matrices.

To these projection matrices A1 and A2 correspond two multiplication rates respectively equal to 1 and 2




with ij ijij

saij

a

sij

It can be shown that:

2/)A(A A where) - 21*

A

2121

*

ij

ijij

ij aa(a




The contribution of the vital rate aij to the difference in is given by:

*A

21ij )

ijij

ijij a

a(ac

This may be generalised to more than two traitments




This may be generalised to more than two traitments

rmm AAD where r is a reference traitment

*A

mm SDC where Cm is the vector of contributions

Cm gives the contributions of vital rates to the difference between the traitment m and the reference traitment r

MetapopulationMetapopulation

Concept of Metapopulation


A metapopulation is a population of local populations

that are susceptible to extinction and that are

inter connected by migration.

The persistance of the metapopulation depends on

a stochastic balance between local extinctions

and recolonisation of empty sites.

Definition

Population of populations: diffrent types

« island/mainland »

« source/sink »


Metapopulation

Isolated

Extinction

• Stochastic causes

Demographic stochasticity

Genetic stochasticity

Environmental stochasticity

• Deterministic causes


Extinction


General message

The risk of local extinction depends on :

the patch size

the mean growth rate

the variability of the growth rate

Migration - Colonisation

Two processes:

• Migration (dispersal)

• Colonisation




General message

The success of dispersal depends on :

the distance between patches

the realistion of particular conditions (corridors)



Effect of the presence of corridors

Effet of the distance of dispersal



General message

The success of colonisation depends on :

the number of immigrants

Metapopulation dynamicsMetapopulation dynamics

The Levins model (continuous time)

t1 ti tn

p̂ip1p

ePPcPdt

dp )1(

Analogy with the logistic model

)1(1)(

c

eP

Pecdt

dp

K

NrN

dt

dN1

0dt

dp

c

ep 1ˆ

The metapopulation will persist if c > e


The Levins model (continuous time)

The island/mainland model ePPcdt

dp )1(

t1 ti tn

p̂ip1p


The island/mainland model


0dt

dpec

cp

ˆ

The metapopulation will persist as long as

colonisation occurs

Spatially explicit models

iiiii peptC

dt

dp )1)((

t

1-pi

pi

ei

t+1

Ci(t)



Spatially explicit models

iipp ˆˆ

1ˆ

ˆˆ

ii

iii

AC

ACp

R

ijjj

di AtpectC ij )()(

xii aAe

1 ii Ae

Life cycle in the case of several populations

1 2 3 4jjS1jjS2

jjS3j

jjF2

jjF3

jjF1

jjF4Sites

kkS2 kkS3

jkS2

1 2 3 4kkS1

jkS1jkS3

kjS1kjS2

kjS3

k kkF1

kkF2 kkF3 kkF4

Structured populations: constant models


jkifnumber of newborn females in site j born to a female of age i in site k

ppi

jji

i

ppi

pi

jpi

jji

ji

pi

jii

i

f

f

f

ff

fff

fff

F

...0...0

:

0...0

:

0...0...

.........

:

...

:

...... 11

1

1

1111


Fragmented populations: deterministic models

jkis

proportion of surviving females in site j from those of age i-1 in site k

ppi

pi

jpi

jji

ji

pi

jii

i

ss

sss

sss

S

.........

:

...

:

......

1

1

1111



ppp

p

SS

SS

11

1

11

111

ppF

F

1

111

00

0

0

00

)(

)(

)(

1

1

11

tN

tN

tN

p

j

)(

)(

)(

)(

0......0

0

0

0.........0

......

)1(

)1(

)1(

)1(

2

1

1

1

3

2

112111

2

1

tN

tN

tN

tN

SS

S

S

S

FnSFSFSFS

tN

tN

tN

tN

n

i

nn

i

i

n

i



N*(t+1) = M N*(t) (np,1) (np,np) (np,1)

Det (M – I) = 0 annual multiplication rate of the metapopulation

N*(t) right eigenvector stable age structure per site

