An explicit approach to evolutionarily stable dispersal
strategies: no cost of dispersal
Jean-Dominique Lebreton
a,*, Mohamed Khaladi
b, Vladimir Grosbois
a aCEFE/CNRS, 1919 Route de Mende, 34 293 Montpellier cedex 5, France
b
Departement de Mathematiques, UniversiteCaddi 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 re-productive 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 dy-namics, 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 rec-ognized in the 19th century [1], it is only recently that dispersal became considered as an integral component of demographic processes. After the early diusion models proposed by Fisher [2] and www.elsevier.com/locate/mbs
*
Corresponding author. Tel.: +33-4 67 61 32 05; fax: +33-4 67 41 21 38.
E-mail addresses: [email protected] (J.-D. Lebreton), [email protected] (M. Khaladi), grosbois@ cefe.cnrs-mop.fr (V. Grosbois).
0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
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 dicult to study: ``dispersal is not a simple trait like color or weight, it is often dicult 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 22 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 two-dimensional dispersal parameter space. Reciprocally, Hastings [12] proved that dispersal was selected against in presence of spatial heterogeneity. Meanwhile, developments in multisite de-mographic 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 dierent 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
ma-trix 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 ex-pression 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 dierent context by Rousset [22].
2. A growth-dispersal model withs sites and no-age structure
Our growth-dispersal model is a natural generalization of the two site model of Doebeli [11] to
gener-alization to several age classes is considered later. Hence, in this section, population vectors, withs
coordinates, are made up of the population sizes at a given time in each of thessites. We denote
asx 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 functionwi xi t
After growth, dispersal is supposed to take place according to
x t1
The matrixDis (column-) stochastic, i.e., has all its column sums equal to 1:
Xs
i1
dij1; j1;. . .;s:
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 by0, 10
s 1. . .1 is the left
eigenvector ofDassociated with its leading eigenvalue 1 (e.g., [23, p. 49])
10
sD10s:
Moreover, to avoid problems of multiplicity of eigenvalues,Dis 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
x t1
tions for the existence and uniqueness of a stable equilibrium are discussed by Beddington [24]; see
also [25]. The dominant eigenvalue ofM x, denoted ask M x, is thus equal to 1 andx is the
uponpdispersal parameters grouped in a vectorh, belonging to a subset of admissible valuesHof
Rp. In most cases,hwill consist simply of theps2 elements ofD. Account should then be taken
for the linear dependency between the elements ofDresulting fromDbeing stochastic, as will be
seen later.
his 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 valueh2Hof 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 diers moderately from the h
strategy, i.e., h is in a neighborhood ofh.
Among thexindividuals distributed over thessites, let us assume thatzis 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 tox. 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 isM x;h. Hence, a necessary and sucient condition forh 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 ofh notedN h
k M x;h61k M x;h; h2N h H:
In the simplest case,h corresponds to a strict maximum, i.e.,
k M x;h<1k M x;h; h2N 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 dierential
ge-ometry arguments, a necessary and sucient condition for the leading eigenvalue (equal to 1) of
M x;hto be a local maximum is
ok M x;h
ohk hh 0; k 1;. . .;p 1
and the matrix with term
o2k M x;h
ohkoh1
hh 2
is semi-de®nite negative.
We will see that it is sometimes possible to prove thatk M x;h<1k M x;h)) by a direct
approach avoiding the calculations inherent in condition (2). When k M x;h 1
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
invadersz is negligible with respect tox is then critical to conclude that the alternative strategy
will fail to invade the population.
[9]. As mentioned in Section 1, an analytical approach has only been used fors2 [11], based on
the availability of an explicit expression fork M x;has the root of a polynomial with degree 2.
When the dispersal parameters are the elements of D themselves, the fact that D is
column-stochastic introduces s linear constraints between the dispersal parameters
Xs
i1
dij1; j1;. . .;s:
The above equations have to be adapted to this case using s Lagrange multipliers denoted as
lj j1;. . .;s (see e.g., [27], Chapter 9). Then, at a local maximum with respect to h and
lj j1;. . .;s of k M x;h ÿPs j1lj
Ps
i1dijÿ1, one will have dispersal parameters
si-multaneously maximizingk M x;h and obeying the constraints Ps
i1dij1. The counterpart
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 ofMwith respect to
mij, the term in row i and columnj ofM, is
ok M x;h
omij hh UiVj; 4
where U and V are, respectively, the left and right eigenvectors of M x;h
associated to its
dominant eigenvalue, scaled to ensure U0V 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
matrixD, as M x;h D hG x, the generic termm
ij ofM x;h isdij hwj xj.
Expressions (3) and (4) can then be linked via the chain rule (e.g., [29, p. 90])
o k M x;h ÿPs
o k M x;h ÿPs
Equating these derivatives to zero leads to the system of equations:
UiVjwj xj ÿlj 0; i;j1;. . .;s; 5
X
dijÿ10; j1;. . .;s: 5
0
While 50 just expresses the constraints on the column sums ofD, (5) leads to
Ui lj
Vjwj xj
; i;j1;. . .;s: 6
The right term in these equations varies withj, and the left withi.Uimust thus be independent of
i, i.e.,to be an ESS,the dispersal strategy must ensure reproductive values equal across sites.U 1s,
up to a multiplicative constant, is thus the left eigenvector ofM x;h
10
s 10sD h
G x;
D hbeing stochastic, this equation simpli®es to
10s 10sG x
or
1wj xj; j1;. . .;s:
Gis thus the identity matrix, i.e., all sites are at local equilibrium in the absence of migration.
DGxx leads toDx x, i.e., the equilibrium population distribution is also the stationary
distribution of the stochastic migration matrix. At equilibrium, the overall number of individuals
leaving j is P
idijxjÿdjjxjxj ÿdjjxj, since
P
idij1. The overall number of individuals
ar-riving inj isP
kdjkxkÿdjjxj, which is equal to xjÿdjjxj, sinceDx x implies
P
kdjkxk 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 (dijxj djixi) to be equal for all pairs of sites.
The nature of the maximumk M x;h k D hG xcan then be checked directly without
resorting to second-order derivatives.
For an alternative dispersal parameter valueh, D being column-stochastic
10s 10sD h:
By virtue of the local equilibrium property, when the population is at the ESS
10s 10sG x:
Hence for the vector z of individuals with the alternative strategy h immersed in a population
following the ESS:
10s 10sD hG x 10 sM x
;h:
Hence, M x;h admits 1 as its dominant eigenvalue. The corresponding left eigenvector is 10
s,
population reaches an equilibrium with a total population vectorx, made up of a vectorxÿzof
individuals with strategyhand of a vectorzof individuals with the alternative strategyh. At this
equilibriumxÿz M x;h xÿz andzM x;hz.
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 1, then x x. From the uniqueness of the associated eigenvector up to a
mul-tiplicative constant:xÿzcx. Hence the equilibrium vectorz for the alternative strategy can
only be proportional tox, and in turnM x;h x. IfD his such thatD hx x, the number
of invaders z will change in a neutral way in a population at demographic equilibriumx. This
may happen only whenhis another ESS. In all other cases the only possibility left isz0, i.e.,
the extinction of the alternative strategy. This result means that in a population with even re-productive 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 matricesDthat constitute an ESS may
then be obtained after some algebra. Any matrixDof the form
DI I ÿsÿ1JH Iÿxx0; 7
whereI is thessidentity matrix,J 1s10s,H is an arbitraryssmatrix andxis a normalized
version ofx x kxkÿ1
x, which impliesx0x 1), constitute an ESS for dispersal, provided its
terms are positive. The pre-multiplication byI ÿsÿ1J ensures that the second term has column
sums equal to zero and makes D column-stochastic. The post-multiplication by (I ÿxx0)
en-sures thatDx Ix x, i.e., thatx is the eigenvector ofDassociated with the eigenvalue 1. The
latter condition Dx 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 ss matrices. The dimension of this subspace is
sÿ1 sÿ1while that of column-stochastic matrices is sÿ1s. This clearly shows by dierence
that sÿ1 constraints are needed among the dij to obtain an evolutionarily stable dispersal
strategy, expressed in a most straightforward way by thesÿ1 equalities
U1 Ui Us:
The application of (7) with s2 provides the one-dimensional space for the dispersal
pa-rameters 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 matrixM, with sn rows and columns [14±17]. In this matrix,
the usual scalar fecundity of individuals agedkis replaced by a diagonal fecundity matrixFk, with
generic diagonal elementfii(k). Considering as above that dispersal takes place after survival, the
matrix DkSk: Dk is a stochastic dispersal matrix with generic element dij(k). Sk is a diagonal
sur-vival matrix with generic element sii(k). The population vector x consists of the numbers of
in-dividuals in each site j j1;. . .;s within each age class i i1;. . .;n
x x11 x12 x1s x21 xnÿ1s xn1 xn2 xns0:
The multisite Leslie matrixMcan then be written asMDG, with, in block matrix notation (see,
e.g., [31]):
To adapt this framework to the search of dispersal ESS, one considers that G varies with
pop-ulation size x, with in general local density-dependence, i.e., dependence of parameters for
indi-viduals 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
density-dependent multisite Leslie matrixM x;h D hG xwith dispersal parametershand equilibrium
vector of numbersx. The left eigenvector ofM,U0, consists of the reproductive valuesU
kiby age
kand sitei[16]. The right eigenvector,V, consists of the stable relative numbersVkiby agekand
site i.
Let us consider dispersal between agekÿ1 andk. 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 toMDG, withhbeing formed of the
elements dij(k) of Dk. The criterion to be maximized is then
k M ÿX
which leads as previously to
o k M ÿPs
The term hkj depends on the agek considered.
For the ®rst age class:h1jsjj 1 fjj 1 fjj n:
For further age classes 1<k <n:hkjsjj k:
Equating these derivatives to zero leads to the system of equations
UkiVkjhkjÿlj 0; i;j1;. . .;s: 5
The derivatives with respect to the Lagrange multipliers lj ensures as previously thatDk is
col-umn-stochastic
X
dij k ÿ10; j1;. . .;s: 5 0
From (5),Ukilj=Vkjhkj implies as previously that the reproductive values at age kcannot vary
across sites. As a consequence, when considering dispersal at all ages simultaneously, an ESS
requires equal reproductive values are over thes sites at each agek. There is however no reason
for reproductive values to be equal across age classes. Hence we note hereafter UkUki, the
reproductive value at agek in sitei. From U0DGU0 and U0 U
1 U1U2 U2 Un Un
one has, since Dis stochastic by block
U1 U1 U2 U2 Un Un
respectively, this equation reduces to sequations
U1 U2 Un
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 U1U2 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 thenage 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
ei-genvalue equal to 1 but is no more the identity matrix. In general one will havex DGx with
and
G
0:6000 0 0:6400 0
0 0:3753 0 0:6247
0:5000 0 0:2000 0
0 0:7809 0 0:2191
2
6 6 4
3
7 7 5
:
These two matrices have 1 as dominant eigenvalue. The associated right eigenvector of DG
(up to a multiplicative constant) is x 0:6453 0:4562 0:3775 0:4827: y Gx
0:6288 0:4727 0:3981 0:46200 is not equal to x. Overall population stability appears as
10x 10y. The dispersal is an ESS since the reproductive values are equal across sites (row vector
of reproductive values: U0 0:5522 0:5522 0:4417 0:4417.
In the above model, the dominant eigenvalue ofM x;h D hG xis lower than 1 as soon as
any of the four parametersd11;d22;d110 ;d220 varies independently of the others (Figs. 1±4). However,
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
ridi-bundusover 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 eigenvaluekwith dispersal parameterd11. The change is expressed as 106(1ÿk). The ESS
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
Fig. 3. Change in dominant eigenvaluekwith dispersal parameterd0
11. The change is expressed as 106(1ÿk). The ESS
is obtained ford0
110:6 (arrow), i.e.,k1.
Fig. 2. Change in dominant eigenvaluekwith dispersal parameterd22. The change is expressed as 106(1ÿk). The ESS
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 re-productive values [21]. We expect also that our approach could be used to investigate the evo-lution 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 dierences 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,
Fig. 4. Change in dominant eigenvaluekwith dispersal parameterd0
22. The change is expressed as 10
6(1ÿk). The ESS
instead of dispersal between sites, one considers transition between states, such as infected±non-infected in epidemiological models (Pontier and Lebreton in prep.).
Acknowledgements
We thank H. Caswell and an anonymous referee for very helpful comments.
References
[1] J.-D. Lebreton, L'espace en dynamique des populations, in: F. Blasco (Ed.), Tendances nouvelles en modelisation de l'environnement, Elsevier, Paris, 1997, p. 3.
[2] R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937) 355. [3] J.G. Skellam, Random dispersal in theoretical populations, Biometrika 38 (1951) 196. [4] M. Kot, W.M. Schaer, Discrete-time growth-dispersal models, Math. Biosci. 80 (1986) 109.
[5] M.L. Johnson, M.S. Gaines, Evolution of dispersal: theoretical models and empirical tests using birds and mammals, Annu. Rev. Ecol. Syst. 21 (1990) 449.
[6] P.J. Greenwood, P.H. Harvey, The natal and breeding dispersal of birds, Annu. Rev. Ecol. Syst. 13 (1982). [7] R. Korona, Evolution of dispersal: a humble but important example, Trends Evol. Ecol. 10 (2) (1995) 53. [8] R. Levins, Evolution in Changing Environments, Princeton University, Princeton, 1968.
[9] M.A. McPeek, R.D. Holt, The evolution of dispersal in spatially and temporally varying environments, Am. Nat. 140 (1992) 1010.
[10] J.-Y. Lemel, et al., The evolution of dispersal in a two-patch system: some consequences of dierences between migrants and residents, Evol. Ecol. 11 (1997) 613.
[11] M. Doebeli, Dispersal and dynamics, Theoret. Popul. Biol. 47 (1995) 82.
[12] A. Hastings, Can spatial variation alone lead to selection for dispersal? Theoret. Popul. Biol. 24 (1983) 244. [13] H. LeBras, Equilibre et croissance de populations soumisesa migration, Theoret. Popul. Biol. 2 (1971) 100. [14] A. Rogers, The multiregional growth operator and the stable interregional age structure, Demography 3 (1966)
537.
[15] A. Rogers, The multiregional net maternity function and multiregional stable growth, Demography 11 (1974) 473. [16] A. Rogers, F. Willekens, The spatial reproductive value and the spatial momentum of zero population growth,
Environ. Planning A 10 (1978) 503.
[17] J.D. Lebreton, Demographic models for subdivided populations: the renewal equation approach, Theoret. Popul. Biol. 49 (3) (1996) 291.
[18] D.W. Morris, On the evolutionary stability of dispersal to sink habitats, Am. Nat. 137 (6) (1991) 907. [19] W.G. Hines, Evolutionary stable strategies: a review of basic theory, Theoret. Popul. Biol. 31 (1987) 195. [20] H. Caswell, A general formula for the sensitivity of population growth rate to changes in life history parameters,
Theoret. Popul. Biol. 14 (1978) 215.
[21] M. Khaladi, V. Grosbois, J.D. Lebreton, An explicit approach to evolutionarily stable dispersal stategies with a cost of dispersal, J. Nonlinear Anal. Ser. B, in press.
[22] F. Rousset, Reproductive value vs sources and sinks, Oikos 86 (1999) 591.
[23] N.T.J. Bailey, The elements of stochastic processes with application to the natural sciences, Wiley, New York, 1964.
[24] J.R. Beddington, Age distribution and the stability of simple discrete time models, J. Theoret. Biol. 47 (1974) 65. [25] J.A. Silva, T.G. Hallam, Compensation and stability in nonlinear matrix models, Math. Biosci. 11a (1992) 67. [26] S.D. Mylius, O. Diekmann, On evolutionarily stable strategies, optimization and the need to be speci®c about
density dependence, Oikos 74 (1995) 218.
[28] H. Caswell, Matrix Population Models, Sinauer, Sunderland, MA., 1989, p. 328.
[29] W. Rudin, Principles of Mathematical Analysis, 2nd Ed., McGraw-Hill, New York, 1964, p. 270.
[30] G. Malecot, Sur un probleme de probabilites en cha^õne que pose la genetique, C. R. Acad. Sci. Paris 219 (1944) 379. [31] F.A. Graybill, Matrices with Applications in Statistics, Wadsworth, Belmont, CA, 1969.
[32] R.H. Pulliam, Sources, sinks, and population regulation, Am. Nat. 132 (1988) 652.
[33] C.P. Doncaster, et al., Balanced dispersal between spatially varying local populations: an alternative to the source-sink model, Am. Nat. 150 (1997) 425.
