The Evolution of Dispersal in Random Environment
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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
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(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.
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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.
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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,
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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
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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|;
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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:
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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.
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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
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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.
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