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Mathematical Biosciences 165 (2000) 163±176
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An explicit approach to evolutionarily stable dispersal
strategies: no cost of dispersal
Jean-Dominique Lebreton a,*, Mohamed Khaladi b, Vladimir Grosbois a
a

b

CEFE/CNRS, 1919 Route de Mende, 34 293 Montpellier cedex 5, France
D
epartement de Math
ematiques, Universit
e Caddi Ayyad, Marrakech, Morocco

Received 5 May 1999; received in revised form 9 November 1999; accepted 21 March 2000

Abstract
The evolution of dispersal is examined by looking at evolutionarily stable strategies (ESS) for dispersal
parameters in discrete time multisite models without any cost of dispersal. ESS are investigated analytically,

based on explicit results on sensitivity analysis of matrix models. The basic model considers an arbitrary
number of sites and a single age class. An ESS for dispersal parameters is obtained when the spatial reproductive values, calculated at the density-dependent population equilibrium, are equal across sites. From
this basic formulation, one derives equivalently that all local populations should be at equilibrium in the
absence of migration, and that dispersal between sites should be balanced, i.e., the numbers of individuals
arriving to and leaving a site are equal. These results are then generalized to a model with several age
classes. Equal age-speci®c reproductive values do not however imply balanced dispersal in this case. Our
results generalize to any number of sites and age classes those available [M. Doebeli, Dispersal and dynamics, Theoret. Popul. Biol. 47 (1995) 82] for two sites and one age class. Ó 2000 Elsevier Science Inc. All
rights reserved.
Keywords: Population dynamics; Dispersal; Evolution; Game theory; Matrix models

1. Introduction
While the role of area and space in the regulation of populations was already explicitly recognized in the 19th century [1], it is only recently that dispersal became considered as an integral
component of demographic processes. After the early di€usion models proposed by Fisher [2] and

*

Corresponding author. Tel.: +33-4 67 61 32 05; fax: +33-4 67 41 21 38.
E-mail addresses: lebreton@cefe.cnrs-mop.fr (J.-D. Lebreton), khaladi@ucam.ac.ma (M. Khaladi), grosbois@
cefe.cnrs-mop.fr (V. Grosbois).
0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.

PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 1 6 - X

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J.-D. Lebreton et al. / Mathematical Biosciences 165 (2000) 163±176

Skellam [3], models considering simultaneously dispersal and regulation appeared progressively
(e.g., [4]) while dispersal became the subject of a strong interest in population biology, particularly
of vertebrates [5,6]. As a demographic process, dispersal is submitted to evolutionary pressures,
but remains dicult to study: ``dispersal is not a simple trait like color or weight, it is often
dicult to qualify or quantify it, let alone analyze it genetically'' [7]. Following works devoted to
the molding of survival and fecundity by selective pressures (e.g., [8]), various authors used simple
models [9,10] to investigate the existence of evolutionarily stable strategies (ESS) for dispersal. An
ESS for dispersal consists of a set of dispersal parameters such that individuals presenting this set
of parameters cannot be invaded by individuals presenting another set of parameters. One of the
simplest models considers two sites with no age structure and no cost of dispersal. McPeek and
Holt [9] concluded by simulation that the ESS for dispersal should in this model correspond to
balanced dispersal between the two sites, i.e., to equal number of migrants in each direction (see
also [10]). Doebeli [11] took advantage of the availability of an explicit expression of the dominant
eigenvalue of a 2  2 matrix to give an explicit mathematical treatment of ESS in this model. He

con®rmed that balanced dispersal is a necessary condition for an ESS and proved further that
dispersal parameters leading to an ESS constitute a one-dimensional subspace of the overall twodimensional dispersal parameter space. Reciprocally, Hastings [12] proved that dispersal was
selected against in presence of spatial heterogeneity. Meanwhile, developments in multisite demographic theory [13±15] tended to make of the spatial reproductive value a central concept
[16,17], while it did not appear naturally in the ESS models, although Morris [18] introduced
reproductive value directly in a formula giving the eventual genetic contribution of individuals in
di€erent habitats.
The purpose of this paper is to obtain explicitly ESS for dispersal parameters for an arbitrary
number of sites and age classes, i.e., to generalize DoebeliÕs results [11] to any number of sites and
age classes. Our analytical approach to the analysis of dispersal ESS relies on the following bases:
· use of discrete sites, age classes and time scale, i.e., of multisite density-dependent Leslie
matrices;
· proof that an ESS corresponds to a supremum of the dominant eigenvalue of such a Leslie matrix in the dispersal parameter space. As a consequence of the broad use of simulation, the very
concept of ESS is frequently presented in an unclear way in the literature [19];
· explicit research of conditions for a set of dispersal parameters to be an ESS, i.e., to correspond
to a local maximum of the dominant eigenvalue, based on classical sensitivity results [20].
The spatial reproductive values in the population at equilibrium appears naturally in the expression of ESS, from which various results, including balanced dispersal, are derived in a
straightforward way. In this paper, we restrict our attention to the case of no dispersal cost but
further work [21] indicate that the spatial reproductive value play the same central role as in the
absence of dispersal cost. Similar results have been obtained in a slightly di€erent context by
Rousset [22].

2. A growth-dispersal model with s sites and no-age structure
Our growth-dispersal model is a natural generalization of the two site model of Doebeli [11] to
an arbitrary number of sites noted s. We consider ®rst a single age class in each site. A gener-

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165

alization to several age classes is considered later. Hence, in this section, population vectors, with s
coordinates, are made up of the population sizes at a given time in each of the s sites. We denote
as x the population vector before growth and after dispersal, and y the population vector after
growth and before dispersal, i.e., the population vector changes over time as x…0†; y…1†;
x…1†; y…2†; x…2†; . . .
The local growth in site i is supposed to follow a discrete time density-dependent model with
®tness function wi …xi …t††
yi …t ‡ 1† ˆ wi …xi …t††xi …t†;

i ˆ 1; . . . ; s

or, in matrix notation
2
w1 …x1 …t††
0
6
0
w
…x
2
2 …t††
y…t ‡ 1† ˆ 6
4







0

















0

3
0
7
0
7x…t† ˆ G…x…t†† x…t†:



5
ws …xs …t††

After growth, dispersal is supposed to take place according to
3
2
d11 d12




d1s
6 d21 d22
d2s 7
7
x…t ‡ 1† ˆ 6
4








dij


5y…t ‡ 1† ˆ Dy…t ‡ 1†:

ds1







dss

The matrix D is (column-) stochastic, i.e., has all its column sums equal to 1:
s
X
dij ˆ 1; j ˆ 1; . . . ; s:
iˆ1

Thus, the only changes in the overall number of individuals take place during the growth phase. In
relation with the assumption of no cost of dispersal, no individual can be lost during the dispersal
phase. As a direct consequence, denoting the transpose of a matrix by 0 , 10s ˆ …1 . . . 1† is the left
eigenvector of D associated with its leading eigenvalue 1 (e.g., [23, p. 49])
10s D ˆ 10s :
Moreover, to avoid problems of multiplicity of eigenvalues, D is supposed to be irreducible and
aperiodic. Aperiodicity will hold in particular for populations with an annual birth pulse followed
by dispersal. Irreducibility assumes that all sites can communicate and seems also biologically
relevant. The overall growth dispersal model obeys the density-dependent matrix equation
2

3







d1s ws …xs …t††
d11 w1 …x1 …t††
6
7
















7x…t† ˆ DG…x…t††x…t† ˆ M…x…t††x…t†:
x…t ‡ 1† ˆ 6
4
5








dij wj …xj …t††



ds1 w1 …x1 …t††







dss ws …xs …t††

We assume that there is a unique stable equilibrium x , corresponding to M…x †x ˆ x . Conditions for the existence and uniqueness of a stable equilibrium are discussed by Beddington [24]; see
also [25]. The dominant eigenvalue of M…x †, denoted as k…M…x ††, is thus equal to 1 and x is the
corresponding positive eigenvector. The dispersal matrix D is assumed to depend continuously

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upon p dispersal parameters grouped in a vector h, belonging to a subset of admissible values H of
Rp. In most cases, h will consist simply of the p ˆ s2 elements of D. Account should then be taken
for the linear dependency between the elements of D resulting from D being stochastic, as will be
seen later.
h is submitted to evolutionary pressures, and one wishes to determine if the population obeying
the model under a given value of the dispersal parameters, noted h , can be invaded or not by
individuals having another `strategy', i.e., another value h 2 H of the dispersal parameters. If not,
the strategy h will be declared an ESS, in accordance with the usual de®nition [19]. It is worth
noting that one implicitly assumes in general that the h strategy di€ers moderately from the h
strategy, i.e., h is in a neighborhood of h .
Among the x individuals distributed over the s sites, let us assume that z is a vector of numbers

of individuals with the alternative strategy h. This alternative strategy is supposed to be that of
mutants and one assumes thus that z is negligible with respect to x . The matrix that applies to
individuals with strategy h is M…x ; h† of which the leading eigenvalue is 1, while that which
applies to the potential invaders is M…x ; h†. Hence, a necessary and sucient condition for h to
be an ESS is that the leading eigenvalue of M…x ; h† is smaller than or equal to 1 for h in a
neighborhood of h noted N …h †
k…M…x ; h†† 6 1 ˆ k…M…x ; h ††;

h 2 N …h †  H:

In the simplest case, h corresponds to a strict maximum, i.e.,
k…M…x ; h†† < 1 ˆ k…M…x ; h ††;

h 2 N …h †  H:

Then z, which already consists of few individuals, will decrease and the alternative strategy will
rapidly go extinct. k…M…x ; h†† being continuous over N …h †, following standard di€erential geometry arguments, a necessary and sucient condition for the leading eigenvalue (equal to 1) of
M…x ; h † to be a local maximum is
ok…M…x ; h††
hˆh ˆ 0;

ohk

k ˆ 1; . . . ; p

…1†

and the matrix with term
o2 k…M…x ; h††
hˆh
ohk oh1

…2†

is semi-de®nite negative.
We will see that it is sometimes possible to prove that k…M…x ; h†† < 1 ˆ k…M…x ; h )) by a direct
approach avoiding the calculations inherent in condition (2). When k…M…x ; h†† ˆ 1 ˆ
k…M…x ; h ††, the relative proportions of individuals with the two strategies will vary in an neutral
way. If the number of individuals with the alternative strategy is negligible, the alternative strategy
will most likely go to extinction by random drift. The assumption that the numbers of potential
invaders z is negligible with respect to x is then critical to conclude that the alternative strategy
will fail to invade the population.
This approach is fairly classical, but remains often poorly formalized [26]. In general, the nullity
of the derivatives is explored by numerical simulation in a neighborhood of the studied strategy

J.-D. Lebreton et al. / Mathematical Biosciences 165 (2000) 163±176

167

[9]. As mentioned in Section 1, an analytical approach has only been used for s ˆ 2 [11], based on
the availability of an explicit expression for k…M…x ; h†† as the root of a polynomial with degree 2.
When the dispersal parameters are the elements of D themselves, the fact that D is columnstochastic introduces s linear constraints between the dispersal parameters
s
X
dij ˆ 1;

j ˆ 1; . . . ; s:

iˆ1

The above equations have to be adapted to this case using s Lagrange multipliers denoted as
9).
lj …j ˆ 1; . . . ; s† (see e.g., [27], Chapter
P
P Then, at a local maximum with respect to h and
lj …j ˆ 1; . . . ; s† of k…M…x ; h†† ÿ sjˆ1 lj … siˆ1 dij ÿ 1†, one will have
Ps dispersal parameters simultaneously maximizing k…M…x ; h†† and obeying the constraints iˆ1 dij ˆ 1. The counterpart
of Eq. (1) is then:

ÿ Ps


P
o k…M…x ; h†† ÿ sjˆ1 lj
iˆ1 dij ÿ 1
i; j ˆ 1; . . . ; s;
…3†
hˆh ˆ 0;
odij

ÿ Ps


P
o k…M…x ; h†† ÿ sjˆ1 lj
d
ÿ
1
ij
iˆ1
olj

hˆh

ˆ 0;

j ˆ 1; . . . ; s:

0

…3 †

3. Explicit search of ESS
Formal results for the sensitivity analysis of Leslie matrices [20,28] can be readily extended to
multisite models [17]. In particular, the derivative of the dominant eigenvalue of M with respect to
mij , the term in row i and column j of M, is
ok…M…x ; h††
hˆh ˆ Ui Vj ;
omij

…4†

where U and V are, respectively, the left and right eigenvectors of M…x ; h † associated to its
dominant eigenvalue, scaled to ensure U 0 V ˆ 1. V is obviously proportional to x . U consist of
site-dependent reproductive values [16]. As such, the reproductive values characterize the relative
contributions of individuals in each site to the future growth of the overall population. Since the
overall growth-dispersal matrix is the product of a diagonal growth matrix G and a dispersal
matrix D, as M…x ; h † ˆ D…h †G…x †, the generic term mij of M…x ; h† is dij …h†wj …xj †.
Expressions (3) and (4) can then be linked via the chain rule (e.g., [29, p. 90])

ÿ Ps


P
o k…M…x ; h†† ÿ sjˆ1 lj
d
ÿ
1
iˆ1 ij

i; j ˆ 1; . . . ; s:
hˆh ˆ Ui Vj w…xj † ÿ lj ;
odij
In an even more straightforward way:

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J.-D. Lebreton et al. / Mathematical Biosciences 165 (2000) 163±176


ÿ Ps


P
o k…M…x ; h†† ÿ sjˆ1 lj
d
ÿ
1
iˆ1 ij
olj

hˆh



ˆÿ

X


dij ÿ 1 ;

j ˆ 1; . . . ; s:

Equating these derivatives to zero leads to the system of equations:
Ui Vj wj …xj † ÿ lj ˆ 0;
X

dij ÿ 1 ˆ 0;

i; j ˆ 1; . . . ; s;

j ˆ 1; . . . ; s:

While …50 † just expresses the constraints on the column sums of D, (5) leads to
lj
; i; j ˆ 1; . . . ; s:
Ui ˆ
Vj wj …xj †

…5†
0

…5 †

…6†

The right term in these equations varies with j, and the left with i. Ui must thus be independent of
i, i.e., to be an ESS, the dispersal strategy must ensure reproductive values equal across sites. U ˆ 1s ,
up to a multiplicative constant, is thus the left eigenvector of M…x ; h †
10s ˆ 10s D…h †G…x †;
D…h † being stochastic, this equation simpli®es to
10s ˆ 10s G…x †
or
1 ˆ wj …xj †;

j ˆ 1; . . . ; s:

G is thus the identity matrix, i.e., all sites are at local equilibrium in the absence of migration.
DGx ˆ x leads to Dx ˆ x , i.e., the equilibrium population distribution is also the stationary
distribution P
of the stochastic migration matrix.P
At equilibrium, the overall number of individuals
individuals arleaving j is Pi dij xj ÿ djj xj ˆ xj ÿ djj xj , since i dij ˆ 1. The overall number ofP
riving in j is k djk xk ÿ djj xj , which is equal to xj ÿ djj xj , since Dx ˆ x implies k djk xk ˆ xj .
As a consequence, the numbers of individuals migrating to a site and from this same site in one
time step are equal: dispersal is balanced. However, there is no reason for dispersal between pairs
of sites (dij xj ˆ dji xi ) to be equal for all pairs of sites.
The nature of the maximum k…M…x ; h †† ˆ k…D…h †G…x †† can then be checked directly without
resorting to second-order derivatives.
For an alternative dispersal parameter value h, D being column-stochastic
10s ˆ 10s D…h†:
By virtue of the local equilibrium property, when the population is at the ESS
10s ˆ 10s G…x †:
Hence for the vector z of individuals with the alternative strategy h immersed in a population
following the ESS:
10s ˆ 10s D…h†G…x † ˆ 10s M…x ; h†:
Hence, M…x ; h† admits 1 as its dominant eigenvalue. The corresponding left eigenvector is 10s ,
while the right eigenvector has no reason to be equal to x as long as h 6ˆ h . Let us assume that the

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J.-D. Lebreton et al. / Mathematical Biosciences 165 (2000) 163±176

population reaches an equilibrium with a total population vector x , made up of a vector x ÿ z of
individuals with strategy h and of a vector z of individuals with the alternative strategy h. At this
equilibrium x ÿ z ˆ M…x ; h †…x ÿ z † and z ˆ M…x ; h†z .
The uniqueness of the equilibrium assumed in the population model implies that x is the
unique vector leading to a dominant eigenvalue equal to 1. Since the ®rst equation above implies
k…M…x ; h †† ˆ 1, then x ˆ x . From the uniqueness of the associated eigenvector up to a multiplicative constant: x ÿ z ˆ cx . Hence the equilibrium vector z for the alternative strategy can
only be proportional to x , and in turn M…x ; h† ˆ x . If D…h† is such that D…h†x ˆ x , the number
of invaders z will change in a neutral way in a population at demographic equilibrium x . This
may happen only when h is another ESS. In all other cases the only possibility left is z ˆ 0, i.e.,
the extinction of the alternative strategy. This result means that in a population with even reproductive values there is no way for a mutant to get a selective advantage by moving to any
particular site because all sites are saturated by density-dependence.
For a given growth model, the general form of dispersal matrices D that constitute an ESS may
then be obtained after some algebra. Any matrix D of the form
D ˆ I ‡ …I ÿ sÿ1 J †H …I ÿ x‡ x‡0 †;

…7†
1s 10s , H is
‡0 ‡

‡

where I is the s  s identity matrix, J ˆ
an arbitrary s  s matrix and x is a normalized
 ‡
 ÿ1 
version of x …x ˆ kx k x , which implies x x ˆ 1), constitute an ESS for dispersal, provided its
terms are positive. The pre-multiplication by I ÿ sÿ1 J ensures that the second term has column
sums equal to zero and makes D column-stochastic. The post-multiplication by (I ÿ x‡ x‡0 ) ensures that Dx ˆ Ix ˆ x , i.e., that x is the eigenvector of D associated with the eigenvalue 1. The
latter condition Dx ˆ x ensures in turn that dispersal leaves x unchanged, i.e., dispersal is
balanced. As seen above, when several such strategies are present in the population, their relative
proportions will change in a neutral way. Then, the rarest ones will tend to get extinct by random
drift, as in the classical Markov chain model for two alleles [30].
An interesting feature of (7) is that it is a linear formula. Dispersal ESS correspond thus to a
portion of a linear subspace among s  s matrices. The dimension of this subspace is
…s ÿ 1†…s ÿ 1† while that of column-stochastic matrices is …s ÿ 1†s. This clearly shows by di€erence
that s ÿ 1 constraints are needed among the dij to obtain an evolutionarily stable dispersal
strategy, expressed in a most straightforward way by the s ÿ 1 equalities
U1 ˆ


ˆ Ui ˆ


ˆ Us :
The application of (7) with s ˆ 2 provides the one-dimensional space for the dispersal parameters already obtained by Doebeli [11].

4. Several age classes
The discrete time dynamics over s sites of a population structured in n age classes can be
represented by a multisite Leslie matrix M, with s  n rows and columns [14±17]. In this matrix,
the usual scalar fecundity of individuals aged k is replaced by a diagonal fecundity matrix Fk , with
generic diagonal element fii (k). Considering as above that dispersal takes place after survival, the
usual scalar survival probability of individuals from age k ÿ 1 to k is replaced by a s  s transition

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J.-D. Lebreton et al. / Mathematical Biosciences 165 (2000) 163±176

matrix Dk Sk : Dk is a stochastic dispersal matrix with generic element dij (k). Sk is a diagonal survival matrix with generic element sii (k). The population vector x consists of the numbers of individuals in each site j…j ˆ 1; . . . ; s† within each age class i…i ˆ 1; . . . ; n†
x ˆ …x11 x12


x1s x21






xnÿ1s xn1 xn2


xns †0 :

The multisite Leslie matrix M can then be written as M ˆ DG, with, in block matrix notation (see,
e.g., [31]):
3
2
0




0
D1
6 0
D2



0 7
7
Dˆ6
4











5
0




0
Dn
and

2

S1 F1
6 S2
Gˆ6
4



0

S1 F2 ;
0









jˆ1

iˆ1













Sn

3
S1 Fn
0 7
7:



5
Sn‡1

To adapt this framework to the search of dispersal ESS, one considers that G varies with population size x, with in general local density-dependence, i.e., dependence of parameters for individuals in site i over the numbers on individuals …x1i ; . . . ; xni † in this site. The column-stochastic
dispersal matrix D, depends as above on a set of dispersal parameters h. M is then a densitydependent multisite Leslie matrix M…x; h† ˆ D…h†G…x† with dispersal parameters h and equilibrium
vector of numbers x . The left eigenvector of M, U 0 , consists of the reproductive values Uki by age
k and site i [16]. The right eigenvector, V, consists of the stable relative numbers Vki by age k and
site i.
Let us consider dispersal between age k ÿ 1 and k. For the sake of simplicity, the dependence of
the demographic parameters on population size is dropped from the notation, the demographic
parameter values at the equilibrium population x being considered in all what follows. The
approach used with the one age-class model can be applied to M ˆ DG, with h being formed of the
elements dij (k) of Dk . The criterion to be maximized is then
!
s
s
X
X
k…M† ÿ
dij …k† ÿ 1 ;
lj
which leads as previously to

ÿ Ps


P
o k…M† ÿ sjˆ1 lj
iˆ1 dij …k† ÿ 1
odij …k†

hˆh

ˆ Uki Vkj hkj ÿ lj ;

The term hkj depends on the age k considered.
For the ®rst age class: h1j ˆ sjj …1†…fjj …1† ‡


‡ fjj …n††:
For further age classes …1 < k < n† : hkj ˆ sjj …k†:
For the last age class: hnj ˆ sjj …n† ‡ sjj …n ‡ 1†.

i; j ˆ 1; . . . ; s:

J.-D. Lebreton et al. / Mathematical Biosciences 165 (2000) 163±176

171

Equating these derivatives to zero leads to the system of equations
Uki Vkj hkj ÿ lj ˆ 0;

i; j ˆ 1; . . . ; s:

…5†

The derivatives with respect to the Lagrange multipliers lj ensures as previously that Dk is column-stochastic
X
0
…5 †
dij …k† ÿ 1 ˆ 0; j ˆ 1; . . . ; s:
From (5), Uki ˆ lj =Vkj hkj implies as previously that the reproductive values at age k cannot vary
across sites. As a consequence, when considering dispersal at all ages simultaneously, an ESS
requires equal reproductive values are over the s sites at each age k. There is however no reason
for reproductive values to be equal across age classes. Hence we note hereafter Uk ˆ Uki , the
reproductive value at age k in site i. From U 0 DG ˆ U 0 and U 0 ˆ …U1


U1 U2


U2


Un


Un †
one has, since D is stochastic by block
3
2
S1 F1 S1 F2


S1 Fn
6 S
0




0 7
7
6 2
‰ U1


U1 U2


U2


Un


Un Š6
7
4













5
0

ˆ ‰ U1


U1

U2


U2











Sn

Sn‡1

Un


Un Š:

All the S1 Fk and Sk matrices being diagonal, with generic element sii (1)fii (k) and sii (k),
respectively, this equation reduces to s equations
3
2
sii …1†fii …1† sii …1†fii …2†


sii …1†fii …n†
7
6 sii …2†
0




0
7 ˆ ‰ U1 U2 . . . Un Š:
‰ U1 U2


Un Š6
5
4
















0




sii …n† sii …n ‡ 1†

This indicates that at population equilibrium under an ESS for dispersal, the s one-site Leslie
matrices in the absence of migration admit 1 as their dominant eigenvalue with associated left
eigenvector …U1 U2


Un †. Hence, as in the model with no age structure, all sites are at local
equilibrium in the absence of migration.
However, contrary to the previous situation, there is no reason for dispersal at a given age, or
even summed over the n age classes, to be balanced. One may easily check that it is the amount of
reproductive value leaving a site that should equal to the amount of reproductive value arriving in
this site. Because of the age structure, G(x ) is a collection of Leslie matrices with dominant eigenvalue equal to 1 but is no more the identity matrix. In general one will have x ˆ DGx with
y  ˆ G…x †x 6ˆ x and x ˆ Dy  , as in the following numerical example:
d11
6 1 ÿ d11
Dˆ6
4 0
0
2

1 ÿ d22
d22
0
0

0
0
0
d11
0
1 ÿ d11

3 2
0
0:5
6 0:5
0 7
7 6
0 5 ˆ 4
1 ÿ d22
0
0
d22
0

0:7
0:3
0
0

0
0
0:6
0:4

3
0
0 7
7
0:3 5
0:7

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J.-D. Lebreton et al. / Mathematical Biosciences 165 (2000) 163±176

and
2

0:6000
0
6 0
0:3753
Gˆ6
4 0:5000
0
0
0:7809

0:6400
0
0:2000
0

3
0
0:6247 7
7:
0 5
0:2191

These two matrices have 1 as dominant eigenvalue. The associated right eigenvector of DG
(up to a multiplicative constant) is x ˆ …0:6453 0:4562 0:3775 0:4827†: y  ˆ Gx ˆ
…0:6288 0:4727 0:3981 0:4620†0 is not equal to x . Overall population stability appears as
10 x ˆ 10 y  . The dispersal is an ESS since the reproductive values are equal across sites (row vector
of reproductive values: U 0 ˆ …0:5522 0:5522 0:4417 0:4417††.
In the above model, the dominant eigenvalue of M…x ; h† ˆ D…h†G…x † is lower than 1 as soon as
0
0
; d22
varies independently of the others (Figs. 1±4). However,
any of the four parameters d11 ; d22 ; d11
the change induced in the dominant eigenvalue by any change in dispersal parameters is very
small, i.e., the selective pressure against alternative strategies will be very weak.
Lebreton [17] provides an example of model for a Black-headed Gull population Larus ridibundus over two contrasted sites, where the reproductive values at birth are not far from equality.

5. Discussion
Our analytical results extend to any number of sites and age classes results available only for
two sites and one age class by simulation [9] of analytically [11]. Besides this generalization, our
explicit approach gives a clear support to the intuitive feeling that spatial reproductive values

Fig. 1. Change in dominant eigenvalue k with dispersal parameter d11 . The change is expressed as 106 (1 ÿ k). The ESS
is obtained for d11 ˆ 0:5 (arrow), i.e., k ˆ 1.

J.-D. Lebreton et al. / Mathematical Biosciences 165 (2000) 163±176

173

Fig. 2. Change in dominant eigenvalue k with dispersal parameter d22 . The change is expressed as 106 (1 ÿ k). The ESS
is obtained for d22 ˆ 0:3 (arrow), i.e., k ˆ 1.

0
Fig. 3. Change in dominant eigenvalue k with dispersal parameter d11
. The change is expressed as 106 (1 ÿ k). The ESS
0
is obtained for d11 ˆ 0:6 (arrow), i.e., k ˆ 1.

should play a central role in dispersal strategies [17]. Equal spatial reproductive values over sites
make alternative dispersal strategies uninteresting since there is no site, in the population at
equilibrium, where to gain a ®tness advantage.
When there is a single age class, the two equivalent formulations derived from equal spatial
reproductive values are easily interpreted. The ®rst one is the absence of local growth in the

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J.-D. Lebreton et al. / Mathematical Biosciences 165 (2000) 163±176

0
Fig. 4. Change in dominant eigenvalue k with dispersal parameter d22
. The change is expressed as 106 (1 ÿ k). The ESS
0
is obtained for d22
ˆ 0:7 (arrow), i.e., k ˆ 1.

absence of migration. This shows that source-sink models [32] are evolutionarily unstable in the
absence of dispersal cost (see also [33]). The second formulation, balanced dispersal, means that
migration should not modify the local stability. This intuitive argument as well as the dimension
of the set of ESS show clearly that it is overall dispersal, i.e., the total number of migrants to and
from a site that should be balanced, and not dispersal between all pairs of sites. The two situations
coincide when there are two sites [9], and which generalization was valid for any number of sites
was unclear. Balanced dispersal between all pairs of sites, investigated by Doncaster et al. [33]
receives thus no theoretical support in the present framework at least.
Simultaneous dispersal in several age classes leads similarly to age-speci®c reproductive values
equal over sites and to the absence of local growth in the absence of migration. However balanced
dispersal does not follow from equal reproductive values when there are several age classes. In fact
there cannot be any change in reproductive value, and dispersal can be considered as balanced
only if reproductive value rather than the number of individuals is used as a currency. Depending
on their age, more individuals with a low reproductive value can exchange with fewer individuals
with a high reproductive value.
In the presence of dispersal costs, we obtain also straightforward relationships between reproductive values [21]. We expect also that our approach could be used to investigate the evolution of dispersal in random environment, based, e.g., on a ®rst order approximation of the
asymptotic growth rate [34]. This is fortunate since when an ESS in reached in the above model,
the selective pressure against alternative dispersal strategies appears as very weak. In such a
context, the variability of the environment and di€erences in quality between individuals should
play a central role in molding dispersal. Our approach could also easily be used, with similar
results in density-independent models. Moreover, the results above could also be of interest when,

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175

instead of dispersal between sites, one considers transition between states, such as infected±noninfected in epidemiological models (Pontier and Lebreton in prep.).

Acknowledgements
We thank H. Caswell and an anonymous referee for very helpful comments.

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