The Evolution of Dispersal in Random Environment

REGULAR A RTI CLE

The Evolution of Dispersal in Random Environment

Mohamed Khaladi • Jean-Dominique Lebreton •

Abdelaziz Khermjioui

Received: 16 November 2011 / Accepted: 2 December 2011 / Published online: 16 December 2011

� Springer Science+Business Media B.V. 2011

Abstract In this paper we introduce a stochastic model for a population living and

migrating between s sites without distinction in the states between residents and

immigrants. The evolutionary stable strategies (ESS) is characterized by the max-

imization of a stochastic growth rate. We obtain that the expectation of reproductive

values, normalized by some random quantity, are constant on all sites and that the

expectation of the normalized vector population structure is proportional to eigen-

vector of the dispersion matrix associated to eigenvalue one, which are, in some

way, analogous to the results obtained in the deterministic case.

Keywords Stochastic discrete model � Stochastic growth rate � Dispersion �Evolutionary stable strategies (ESS)

1 Introduction

The heterogeneity of the environment over time and space is though to be a major

factor moulding dispersal patterns (see Levin et al. 1984; Johnson and Gaines 1990

and references therein).The exact nature of dispersal and its theoretical treatment are

coming more and more into question, however, due to technical difficulties, models

for the evolution of dispersal have in general considered constant environment, i.e.

deterministic models (Huston et al. 2003). In particular, using a model with two

sites and no age structure, McPeek and Holt (1992) by simulation and Doebeli

M. Khaladi (&) � A. Khermjioui

Department of Mathematics, Faculty of Sciences, LMPDP and UMI UMMISCO, IRD-UPMC,

40001 Marrakesh, Morocco

e-mail: [email protected]

J.-D. Lebreton

CEFE/CNRS, 1919 Route de Mende, 34293 Montpellier cedex 5, France

e-mail: [email protected]

123

Acta Biotheor (2012) 60:155–165

DOI 10.1007/s10441-011-9142-0

(1995) in a formal approach showed that balanced dispersal was evolutionarily

stable, i.e. could not be invaded successfully by another dispersal strategy. A formal

analysis (Lebreton et al. 2000; Khaladi et al. 2000) extended these results to any

number of age classes and sites. This analysis was based on general sensitivity

results in matrix models (Caswell 2001) from which the equality of reproductive

values over sites, weighted possibly by dispersal costs, could be deduced. In turn,

the equality of reproductive values induced balanced dispersal. The purpose of this

paper is to develop similar analyses for dispersal in a random environment. The

approach uses also sensitivity results, developed for the random environment case

by (Tuljapurkar 1990; Tuljapurkar et al. 2003).

We introduce a stochastic model for a population living and migrating between

s sites without distinction in the states between residents and immigrants. The

Evolutionary Stable Strategies (ESS) is characterized by the maximization of a

stochastic growth rate. We obtain that the expectation of reproductive values,

normalized by some random quantity, are constant on all sites and that the

expectation of the normalized vector population structure is proportional to

eigenvector of the dispersion matrix associated to eigenvalue one, which are, in

some way, analogous to the results obtained in the deterministic case.

After presenting the model and formal results, we develop some numerical

illustrations for two sites with no age-structure.

2 A Multisite Random Environment Model

Consider a population living and dispersing on s connected regions, we assume that

the recruitment depends on local population densities, the vital rates at each time

depend on the state of a random environment and that dispersal is deterministic not

costly for the population i.e. the vital rates are not affected by the transient and

settlement phases.

Let xi tð Þ; i ¼ 1; . . .; s be the number of individuals in the patch i at time t after

dispersal and before growth. As in (Lebreton et al. 2000) we denote as

XðtÞ ¼ ðx1ðtÞ; . . .xsðtÞÞT the population vector after dispersal and before growth at

time t, and ZðtÞ ¼ ðz1ðtÞ; . . .zsðtÞÞT the population vector after growth and before

dispersal at time t, but here X(t) and Z(t) are vectors of random variables. The local

growth is supposed to follow a discrete time stochastic model on the form:

Zðt þ 1Þ ¼ Gðt þ 1ÞXðtÞ

where

Gðt þ 1Þ ¼

w1 t þ 1ð Þ 0 � � � 0

0 w2ðt þ 1Þ � � � 0

� � � � � � � � � � � �0 0 � � � ws t þ 1ð Þ

2664

3775;

is an s 9 s diagonal matrix of random nonnegative vital rates. We assume that there

is a finite number of such matrices.

156 M. Khaladi et al.

123

After growth, dispersal is supposed to take place according to:

Xðt þ 1Þ ¼ DYðt þ 1Þ

where

D ¼

d11 d12 � � � d1s

d21 d22 � � � d2s

� � � � � � � � � � � �ds1 ds2 � � � dss

2664

3775;

dij is the rate of migration from patch j to patch i. We assume that no individual can

be lost during the dispersal phase, the only changes in the overall number of

individuals take place during the growth phase. So, D is a nonnegative deterministic

column-stochastic matrix, i.e.,

0� dij� 1; 8i; j ¼ 1; . . .; s andXs

i¼1

dij ¼ 1; 8j ¼ 1; . . .; s

The p elements of D (p = s2) are the dispersal parameters, we group them

together in a vector h of Rp.

The dynamics of population X(t) are given by a matrix equation on the form:

Xðt þ 1Þ ¼ Mðt þ 1ÞXðtÞ; ð1Þ

here, the projection matrix M(t ? 1) is given by:

Mðt þ 1Þ ¼ DGðt þ 1Þ;

3 Evolutionarily Stable Strategy

The ESS is characterized by the maximization of a well chosen fitness function or

selective value which balances the trade-offs between alternative strategies (Hines

1987; Parker and Smith 1990; Diekmann 1997). The choice of the function to be

maximized is a matter of debate (see Mylius and Diekmann 1995 for instance), various

measures of fitness can be used both in deterministic (see Charnov 1993; Stearns and

Koella 1986; Stearns 1992) or in stochastic case (see Orzack and Tuljapurkar 1989).

The most used is, in general, the intrinsic growth rate. There are many alternative

models for defining fitness random rates (see Caswell 2001 for instance). Here we

adopt an analog of the asymptotic growth rate of population (Furstenberg an Kesten

1960; Tuljapurkar and Orzack 1980). Before defining this rate called stochastic growth

rate we begin by giving some general assumptions (see Tuljapurkar 1990 for instance).

We assume that:

1. The random process generating the vital rates converges towards an ergodic

stationary state, and in what follows the random process will be assumed in the

stationary state.

2. We have demographic weak ergodicity. This happens if the product matrix

M(t)M(t - 1)…M(1) ends up having all entries positive for large t.

The Evolution of Dispersal in Random Environment 157

123

Demographic weak ergodicity guarantees that the dynamics of (1) are stable in

the following sense: let YðtÞ ¼ 11T XðtÞXðtÞ; 1T ¼ ð1; . . .; 1Þ; be the structure

population vector, it satisfies

Yðt þ 1Þ ¼ Mðt þ 1Þ 1

1T Aðt þ 1ÞYðtÞYðtÞ

� �: ð2Þ

Apply the random sequence (3) to b(0) and c(0), two distinct initial structures, then the

ergodicity condition implies that there is some sequence of structures YðtÞ such that

both the vectors b(t) and c(t), resulting from b(0) and c(0) respectively, approach YðtÞ:3. The logarithmic moment of vital rates is bounded,

Eðlogþ jjMð1ÞjjÞ\1;

where E is an expectation, ||.|| is any matrix norm, and log?(x) = max{0,log x}.

Under these assumptions, the long run growth rate of the logarithm of population

is almost surely given by a number a independent of the initial population vector:

a ¼ limt!þ1

1

tlogð MðtÞMðt � 1Þ. . .Mð1Þk kÞ

¼ limt!þ1

1

tlogðcT XðtÞÞ

where c = 0 is any vector of nonnegative numbers (cT ¼ ðc1; . . .csÞ;ci� 0; i ¼ 1; . . .; s). Note that for c = 1, cTX(t) is the total population at time t.

We assume that the stationary state is reached and in the calculation to follow the

various quantities refer to the statistically stationary state.

Let U(t) be the random sequence defined by

Uðt þ 1Þ ¼ 1

kðt þ 1ÞMðt þ 1ÞUðtÞ; ð3Þ

with kðt þ 1Þ ¼ 1|Mðt þ 1ÞUðtÞ; and 1| ¼ ð1; . . .; 1Þ; so we have 1|UðtÞ ¼ 1:UðtÞis called population structure.

V(t) is the random sequence defined by

VðtÞ| ¼ 1

gðt þ 1ÞVðt þ 1Þ|Mðt þ 1Þ; ð4Þ

with gðt þ 1Þ ¼ Vðt þ 1Þ|Mðt þ 1Þ1; we have 1|VðtÞ ¼ 1: This identifies the sto-

chastic analog of a reproductive value.

The stochastic growth rate may be computed as (see Tuljapurkar 1990)

a ¼ limt!1

1

tlog aðtÞ; ð5Þ

with aðtÞ ¼ VðtÞ|MðtÞMðt � 1Þ. . .Mð1ÞUð0Þ:Let us now precise what ESS means. A population obeying the model (1) under a

given value of the dispersal parameters h� have an evolutionary stable strategy

(ESS) if it cannot be invaded by a small number of individuals, of initial vector Z0,


123

having another strategy, i.e. another value of the dispersal parameters. The process

Z(t), representing the invader individuals is a density independent process in

random environment subjected to a sequence of random matrices M(t), with a rate

a(h). As M(t) belong to an ergodic set, so Z(t) benefit from the weak demographic

ergodicity. Recall that set of the individual states are called ergodic states if no

matter where the process starts out, it will soon end up in the ergodic set.

A necessary and sufficient condition for h� to be an ESS of dispersal is

aðhÞ� aðh�Þ;

for h in a neighborhood of h�: Then Z(t), which already consists of few individuals,

will decrease and the alternative strategy will rapidly go extinct.

So, and because of the relations satisfied by the coefficients dij of the dispersal

matrix D; a h�ð Þ is local constrained maximum of the problem

max aðhÞunder

Psi¼1 dij ¼ 1; 8j ¼ 1; . . .; s

�

Let l ¼ l1; � � � ; lsð Þ; a Lagrange multiplier vector. Using the Lagrangian function

L h;lð Þ ¼ a hð Þ �Xs

j¼1

lj

Xs

i¼1

dij � 1

!;

and the optimality conditions we have that h� is a solution of the system:

oL h;lð Þodij =h¼h�

¼ 0; i; j ¼ 1; � � � ; s;oL h;lð Þ

olj =h¼h�¼ 0; j ¼ 1; � � � ; s;

8<:

i.e.,oa hð Þodij =h¼h�

¼ lj; i; j ¼ 1; � � � ; s;Psi¼1 dij ¼ 1; j ¼ 1; � � � ; s:

(

4 Characterization of the ESS in Terms of Reproductive Values

Now, using the expression of a given in (5) we have

oa

odij¼ lim

t!1

1

t

1

aðtÞoaðtÞodij

;

¼ limt!1

1

t

Xt

p¼1

VðtÞ|MðtÞ. . .Mðpþ 1Þ oMðpÞodij

� �Mðp� 1Þ. . .Mð1ÞUð0Þ

VðtÞ|MðtÞ. . .Mðpþ 1ÞMðpÞMðp� 1Þ. . .Mð1ÞUð0Þ

¼ limt!1

1

t

Xt

p¼1

g tð Þ. . .gðpþ 1ÞVðpÞ| oMðpÞodij

� �Uðp� 1Þkðp� 1Þ. . .kð1Þ

g tð Þ. . .gðpþ 1ÞVðpÞ|MðpÞUðp� 1Þkðp� 1Þ. . .kð1Þ :

By the structure of the matrix M(t) it is easily seen that the only nonzero component

of the matrixoMðpÞodij

is the element on the line i and row j whose value is wj, then


123

oa

odij¼ lim

t!1

1

t

Xt

p¼1

VðpÞ|eiwjðtÞe|j Uðp� 1ÞVðpÞ|MðpÞUðp� 1Þ ;

¼ limt!1

1

t

Xt

p¼1

viðpÞwjðpÞujðp� 1ÞVðpÞ|MðpÞUðp� 1Þ :

¼ limt!1

1

t

Xt

p¼1

viðpÞwjðpÞujðp� 1ÞVðpÞ|kðpÞUðpÞ

As all quantities refer to the stationary state, the time average is equal to the

expectation, and then

oa

odij¼ E

viðtÞwjðtÞujðt � 1ÞkðtÞVðtÞ|UðtÞ

� �:

So, the conditions for h� to be an ESS are:

EviðtÞwjðtÞujðt � 1Þ

kðtÞVðtÞ|UðtÞ

� �¼ lj; 8i; j ¼ 1; . . .; s; ð6Þ

conditions depending on the Lagrange multipliers lj; j ¼ 1; . . .; s: This leads us to look

for a characterization of the Lagrange multipliers in terms of the known data of the

model.

4.1 Characterization of the Lagrange Multipliers

Multiplying the both sides of (6) by dij and summing on j, we obtain:

EviðtÞ

Psj¼1 dijwjðtÞujðt � 1Þ

kðtÞVðtÞ|UðtÞ

� �¼Xs

j¼1

dijlj; 8i ¼ 1; . . .; s;

but by (3) we have

Xs

j¼1

dijwjðtÞujðt � 1Þ ¼ kðtÞuiðtÞ:

So

EviðtÞuiðtÞVðtÞ|UðtÞ

� �¼Xs

j¼1

dijlj; 8i ¼ 1; . . .; s: ð7Þ

In a same way we obtain from (6) that

E

Psi¼1 dijviðtÞ

� wjðtÞujðt � 1Þ

kðtÞVðtÞ|UðtÞ

� �¼ lj

Xs

i¼1

dij

¼ lj; 8i ¼ 1; . . .; s;

and using the fact that from (4) we have

VðtÞ|MðtÞ ¼ gðtÞVðt � 1Þ|;


123

so

kðtÞVðtÞ|UðtÞ ¼ VðtÞ|MðtÞUðt � 1Þ;¼ gðtÞVðt � 1Þ|Uðt � 1Þ:

we obtain that

Evjðt � 1Þujðt � 1ÞVðt � 1Þ|Uðt � 1Þ

� �¼ lj; 8j ¼ 1; . . .; s;

that is, by stationarity on t:

EvjðtÞujðtÞVðtÞ|UðtÞ

� �¼ lj; 8j ¼ 1; . . .; s: ð8Þ

From (7) and (8) one derive

Dl ¼ l:

The Lagrange multipliers are, as in the deterministic case (Lebreton et al. 2000),

an eigenvector of D associated to eigenvalue one.

Moreover, by summation on j in (8) we obtain:

EVðtÞ|UðtÞVðtÞ|UðtÞ

� �¼Xs

i¼1

li;

and so

1|l ¼ 1:

4.2 Characterization of the Population Structure and Reproductive Values

From (6), summing in i we have:

EwjðtÞujðt � 1ÞkðtÞVðtÞ|UðtÞ

� �¼ slj; 8j ¼ 1; . . .; s:

So, by the fact thatP

j=1s dijlj = li and

Pj=1s dij wj(t)uj(t - 1) = k(t)ui(t), for

i ¼ 1; . . .; s; we obtain:

EuiðtÞ

VðtÞ|UðtÞ

� �¼ sli; 8i ¼ 1; . . .; s;

or in vector notation

E1

VðtÞ|UðtÞUðtÞ� �

¼ sl: ð9Þ

In the ESS, the expectation of the normalized population structure is proportional

to the eigenvector l of the dispersion matrix associated to the eigenvalue 1,

satisfying 1|l ¼ 1:In the same way, we obtain:


123

EviðtÞurðtÞVðtÞ|UðtÞ

� �¼ lr; 8i; r ¼ 1; . . .; s;

and sinceP

r=1s ur(t) = 1 almost surely, then:

EviðtÞ

VðtÞ|UðtÞ

� �¼ 1; 8i ¼ 1; . . .; s;

that is

E1

VðtÞ|UðtÞVðtÞ� �

¼ 1: ð10Þ

In the ESS, the expectation of the normalized stochastic reproductive values are

constant and equal on all sites.

We see from the analytical results obtained here that, as in the deterministic case

(Lebreton et al. 2000), the spatial reproductive values play a central role in the

determination of the ESS conditions.

5 Numerical Illustration

In this section we use numerical simulation for a stochastic model with two sites

(s = 2). The ESS parameters obtained from the criteria (9), (10) are the same as

those obtained from the maximization of the stochastic growth rate a. They are

slightly close to the parameters given by the ESS conditions for the corresponding

deterministic model (see Figs. 1, 2).

The model we will use is given by

Xðt þ 1Þ ¼ Mðt þ 1ÞXðtÞ;

X(t) = (x1(t),x2(t))T and M(t) = DG(t) with

D ¼1� d1 d2

d1 1� d2

� �; GðtÞ ¼

p1 0

0 p2

� �x1 x1ð Þ 0

0 x2 x2ð Þ

� �

d1, d2 are the dispersal parameters, xi xið Þ ¼ exp ai � bixið Þ; i ¼ 1; 2 growth density

dependent functions, ai, bi, i = 1, 2 constant parameters of the environment and p1,

p2 two stationary random processes given by

piðtÞ ¼ 1þ qðpiðt � 1Þ � 1Þ þ ei; i ¼ 1; 2

with, E eið Þ ¼ 0 (so, E pið Þ =1), var eið Þ ¼ r2; i ¼ 1; 2: q is the correlation coefficient

of the environment.

For pi = 1, i = 1, 2 (the deterministic case), the equilibrium point without

migration is X� ¼ a1

b1; a2

b2

� �and the ESS is obtained for any ðd�1; d�2Þ satisfying:

d�2d�1¼ a1b2

a2b1

; ð11Þ

so, for a given d�1 there is only one d�2 ¼ a1b2

a2b1d�1 such that ðd�1; d�2Þ is an ESS strategy.


123

In the stochastic case , for a fixed d�1 we will use the criterion 1:

EuiðtÞ

VðtÞ|UðtÞ

� �� sli ¼ 0; 8i ¼ 1; . . .; s;

and the criterion 2:

EviðtÞ

VðtÞ|UðtÞ

� �� 1 ¼ 0; 8i ¼ 1; . . .; s;

to obtain d2*.

Fig. 1 Determination of optimal dispersal parmeter d�2 using a characterization of the ESS by two

criteria. For r2 = 0.2; q = 0 and fixed d�1 ¼ 0:4 we obtain d�2 ¼ 0:156 with both criteria (instead of

0.1250 in fixed environment), crit1.JPG (a) criterion 1 for ESS: E uiðtÞVðtÞ|UðtÞ

n o� sli ¼ 0;8i ¼ 1; . . .; s,

crit2.JPG (b) criterion 2 for ESS: E viðtÞVðtÞ|UðtÞ

n o� 1 ¼ 0; 8i ¼ 1; . . .; s


123

We plot the graph of the stochastic growth rate a in a neighborhood of d�2 and see

that the strategy ðd�1; d�2Þ cannot be invaded: for d1 ¼ d�1 ; the growth rate of

alternative strategies is less than zero and reaches 0 only near d2 ¼ d�2 : Figures (1, 2)

show the results obtained for a1 = 0.5, a2 = 0.8, b1 = 0.01, b2 = 0.005,

r2 = 0.2, q = 0 and d�1 ¼ 0:4:

6 Discussion

Our results extend to stochastic case results available in the deterministic one. For a

population dispersing in a stochastic environment, we obtain, as in the deterministic

case (Lebreton et al. 2000), a characterization of the ESS by two analytic criteria (9)

and (10). The first one (9) can be interpreted as balanced dispersal: the expectation

of the total number of migrants to and from a site should be balanced.

The second criterion (10) confirms the role played by the reproductive values in

dispersal strategies: there is no advantage to remove in fitness by adopting an

alternative strategy when the equality over sites on the expectation of normalized

stochastic spatial reproductive values is reached.

However, In the deterministic case, we showed (Lebreton et al. 2000; Khaladi

et al. 2000) that the set of all ESS strategies is a subspace of the space of all possible

dispersal strategies. The equations defining the subspace are explicitly obtained. In

the stochastic case it is not easy to obtain similar results and we cannot say if the set

of ESS strategies is a linear subspace.

References

Caswell H (2001) Matrix population models, 2nd edn. Sinauer Associates, Sunderland

Charnov EL (1993) Life history invariants: some explorations of symmetry in evolutionary ecology.

Oxford series in ecology and evolution. Oxford University Press, Oxford

Fig. 2 Change in the growth rate a of alternative strategies; for d1 ¼ d�1 ¼ 0:4; a\0 and reaches 0 only

near d2 ¼ d�2 : The resulting strategy d�1 ¼ 0:4; d�2 ¼ 0:156 cannot be invaded


123

Diekmann O (1997) The many facets of evolutionarily dynamics. J Biol Syst 5:325–339

Doebeli M (1995) Dispersal and dynamics. Theor Pop Biol 14:82–106

Furstenberg H, Kesten H (1960) Products of random matrices. Ann Math Stat 104:457–469

Hines WGS (1987) Evolutionary stable strategies: a review of basic theory. Theor Pop Biol 31:195–272

Huston V, Martnez S, Mischaikow K, Vikers GT (2003) The evolution of dispersal. J Math Biol

47:483–517

Johnson ML, Gaines MS (1990) Evolution of dispersal: theoretical models and empirical tests using birds

and mammals. Annu Rev Ecol Syst 21:449–490

Khaladi M, Grosbois V, Lebreton JD (2000) An explicit approach to evolutionarily stable strategies with

a cost of dispersal. Nonlinear Anal Real World Appl 1:137–144

Lebreton JD, Khaladi M, Grosbois V (2000) An explicit approach to evolutionarily stable strategies no

cost of dispersal. Math Biosci 165:163–176

Levin SA, Cohen D, Hastings A (1984) Dipersal strategies in patchy environments. Theor Popul Biol

26:165–191

McPeek MA, Holtz RD (1992) The evolution of dispersal in spatially and temporally varying

environments. Am Nat 140:1010–1027

Mylius SD, Diekmann O (1995) On evolutionarily stable strategies, optimization and the need to be

specific about density dependence. Oikos 74:218–224

Orzack SH, Tuljapurkar S (1989) Population dynamics in variable environments VII. The demography

and evolution of iteroparity. Am Nat 133:901–923

Parker GA, Maynard Smith J (1990) Optimality theory in evolutionary biology. Nat Biotechnol

348:27–43

Stearns SC, Koella J (1986) The evolution of phenotypic plasticity in life-history traits: predictions for

norms of reaction for age and size at maturity. Evol Appl 40:893–913

Stearns SC (1992) The evolution of life histories. Oxford University Press, Oxford

Tuljapurkar S, Orzack SH (1980) Population dynamics in variable environments, I. Long-run growth rates

and extinction. Theor Popul Biol 18:314–342

Tuljapurkar S (1990) Population dynamics in variable environments. Springer, New York

Tuljapurkar S, Horvitz CC, Pascrarella JB (2003) The many growth rates and elasticities of populations in

random environments. Am Nat 162:489–502


123

The Evolution of Dispersal in Random Environment

Documents

Transcript of The Evolution of Dispersal in Random Environment