Inclusive fitness for traits affecting metapopulation demography

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Theoretical Population Biology 65 (2004) 127–141

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Inclusive fitness for traits affecting metapopulation demography Fran@ois Rousset
and Ophe´lie Ronce Laboratoire Ge´ne´tique et Environnement, CNRS UMR 5554, Universite´ de Montpellier II, Institut des Sciences de l’E´volution, CC065, USTL, Place Euge`ne Bataillon, FR-34095 Montpellier Cedex 05, France Received 30 July 2002

Abstract Defining computable analytical measures of the effects of selection in populations with demographic and environmental stochasticity is a long-standing problem. We derive an analytical measure which takes in account all consequences of the discrete nature of deme size. Expressions of this measure are detailed for infinite island models of population structure. As an illustration we consider the evolution of dispersal in populations made of small demes with environmental and demographic stochasticity. We confirm some results obtained from the analysis of models based on deterministic approximations. In particular, when there is an Allee effect, we show that evolution of the dispersal rate may lead the metapopulation to extinction. Thus, selection on the dispersal rate could restrict the distribution of species subject to Allee effects. This selection-driven extinction is prevented by kin selection when the environmental extinction rate is small. r 2003 Elsevier Inc. All rights reserved. Keywords: Dynamics; Allee effects; Selection-driven extinction; Kin selection; Dispersal

Demographic and genetic stochasticity represent two major threats for the persistence of small populations in fragmented habitats. Dispersal between habitat patches may counterbalance those forces, by connecting isolated fragments, preventing the extinction of small demes, or allowing the recolonization of empty patches of habitat once extinction has occurred locally (Hanski, 2001; Macdonald and Johnson, 2001). This in turn may affect local adaptation and genetic effective size in metapopulations (e.g. Barton, 2001; Rousset, 2003a; Whitlock, 2001). Given the multiple genetic and demographic consequences of dispersal, there exists an optimal dispersal rate that maximizes the size or the long-term persistence of a metapopulation. However, selection on genetic variation for dispersal may not lead to such an optimal strategy (Olivieri and Gouyon, 1997). Selection on the dispersal rate could even drive the metapopulation to extinction. Genetic variation for dispersal or dispersal adaptations has been documented in many species (Imbert, 2001; Roff and Fairbairn, 2001; Trefilov et al., 2000; Venable and Bu´rquez, 1989) and some evidence


Corresponding author. Fax: +33-4-67-14-36-22. E-mail address: [email protected] (F. Rousset).

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suggest rapid evolution of dispersal phenotypes (Hill et al., 1999a, b). A growing body of theory is concerned with dispersal evolution (e.g. Gandon and Michalakis, 1999; Hamilton and May, 1977; Heino and Hanski, 2001; Holt and McPeek, 1996; Leimar and Nordberg, 1997; Olivieri et al., 1995; Parvinen, 1999). Heino and Hanski (2001) found that evolution of dispersal in a changing landscape could rescue the whole metapopulation from extinction. By contrast, Gyllenberg and Parvinen (2001) and Gyllenberg et al. (2002) found that evolution of dispersal could drive the metapopulation to extinction when there are Allee effects, i.e. when an individual’s expected number of offspring is relatively low when it interacts with few neighbors, and higher when it interacts with some higher numbers of neighbors. Predicting the fate of metapopulations in a changing landscape thus requires that we understand the feedbacks between dispersal evolution and the metapopulation genetic structure and dynamics. Theoretical studies addressing evolution in metapopulations face two difficulties. First, relevant measures of the effects of selection in the context of demographic and genetic stochasticity are hard to compute. Second, this task is even more difficult when the evolving trait affects genetic and demographic processes, such as

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dispersal does. Most theoretical models of dispersal evolution in a metapopulation have focused on the feedback between dispersal evolution and either the genetic structure or the demographic dynamics of the metapopulation. Both approaches miss part of the issue and can lead to contradictory predictions. The former models are exemplified by previous analyses based on inclusive fitness. They generally assume that dispersal does not affect population demography, i.e. that there is a stage in the life cycle where the distribution of deme size in the metapopulation is independent of the dispersal strategy. But they take random fluctuations in allele frequencies within demes into account, by way of the relatedness coefficients of kin selection theory. Alternative approaches generally neglect this consequence of genetic drift. They predict a zero dispersal rate when dispersal is costly in absence of demographic fluctuations (e.g. Holt and McPeek, 1996; Parvinen, 1999). However, a mutant with zero dispersal cannot colonize any deme beyond the one where it appeared, so zero dispersal is not expected to evolve in real metapopulations. In the simple island model without environmental and demographic stochasticity, the kin selection approach predicts nonzero dispersal rates (Taylor, 1988). Methods based on recursion equations for the frequencies of demes with different numbers of a ‘‘mutant’’ allele (Motro, 1982) can take drift into account, and they yield results consistent this the kin selection approach. Such recursion equations may take demographic fluctuations in account. Thus, a fitness measure based on these recursions, such as the one described by Metz and Gyllenberg (2001) for the finite deme size case, can take into account stochastic changes in local allele frequency due to finite deme size. As expected then, this fitness measure predicts a nonzero dispersal rate even in the absence of environmental stochasticity (Parvinen et al., 2003). Without further simplifications, the approach of Metz and Gyllenberg (2001) requires computing the total expected time during which a deme has different given numbers of mutants and nonmutants, after a single mutant entered the deme and until the deme becomes extinct. Metz and Gyllenberg (2001) and Parvinen et al. (2003) partially circumvent this computational burden by considering a continuous time model where only one event can occur within a short time interval. The resulting structure of the equations allows some simplifications in numerical analyses. The inclusive fitness approach of kin selection theory may be seen as an alternative way to obtain analytical simplifications, since the problem of computing the distribution of the number of copies of an allele among demes is reduced to the simpler problem of computing the probability that two genes are identical. Kin selection models generally assume very simple demo-

graphic dynamics such as constant adult deme size (Frank, 1986; Gandon and Michalakis, 1999). The inclusive fitness approach has already been used in a model with demographic stochasticity (Ronce et al., 2000), but not in models where the trait under selection affects the demographic dynamics of the population, such as number of adults per deme, or age structure. Thus, there is a gap between the recent metapopulation models and those considered by kin selection approaches, which this paper will fill. The first result is a condition for convergence to an evolutionarily stable strategy (ESS) under a general population structure, derived from results for the probability of fixation of mutants. We next derive the ‘‘fitness’’ measure for an infinite island model. In the example, we investigate some consequences of population regulation on selected dispersal rates and population persistence. In particular, recent work has shown that selection-driven extinction (‘‘evolutionary suicide’’) could occur under Allee effects (Gyllenberg and Parvinen, 2001; Gyllenberg et al., 2002). However, these models did not take kin selection into account. We will show that selection-driven extinction may occur in the present models, although kin selection effects may prevent it for low extinction rates.

1. Method Consider a two-allele model, with alleles a and A; and some phenotype (e.g. dispersal rate of gametes) associated to these alleles, za and zA ¼ za þ d: A simplified view of selection considers only the first- and secondorder effects in d on the fitness of the A mutant. For mutants of small effect, the first-order effects will usually dominate the second-order effects, hence the direction of selection will be determined by the first-order effects. These effects allow to determine convergence stability (see e.g. Eshel, 1996), i.e. whether a population will converge towards some strategy z
by successive allele replacements. In z
; the first-order effects vanish and fitness is dominated by the second-order effects. These second-order effects allow to determine evolutionary stability, i.e. whether a population with strategy z
is stable against invasion by rare mutants, or alternatively whether rare deviants can invade yet do not go to fixation (‘‘branching’’, Geritz et al., 1998). These computations are well known but rely on some assumptions to be useful. In particular, they imply that branching cannot be found from first-order effects only. In other words, they assume that first-order effects are not frequency dependent, or are frequency dependent in a way that does not result in stable polymorphisms (Rousset and Billiard, 2000). This assumption can be verified in island models, and seems to hold also when dispersal is restricted in space (Rousset, 2004), when all

ARTICLE IN PRESS F. Rousset, O. Ronce / Theoretical Population Biology 65 (2004) 127–141

demes are equivalent in terms of expected demographic dynamics and of selection pressures. Convergence stability has previously been defined in deterministic systems (Christiansen, 1991; Eshel, 1983) where it unequivocally determines the direction of evolution. In real finite metapopulations, unfavored alleles may be fixed by genetic drift. In addition, mutation bias may to some extent keep the metapopulation away from a state predicted from the effects of selection alone. However, the key idea of convergence stability, i.e. that selection drives the metapopulation towards some strategy z
by successive replacements of mutations with small effects, can still be expressed in terms of fixation probabilities (Rousset and Billiard, 2000; for an alternative perspective on fixation probability as a measure of fitness effects, see Proulx and Day, 2001). The probability of fixation pa-A ðzA Þ of a A mutant in a a metapopulation can be written as a Taylor expansion, pa-A ðzA Þ ¼ pa-A ðza Þ þ dfðza Þ þ oðdÞ; where fðza Þ ¼ dpa-A ðzA Þ=ddjd¼0 is the first-order effect of selection on probability of fixation. When fðza Þ40 and d40; the probability of fixation of allele A is higher than the probability of fixation of allele a; hence selection favors some mutants with d40 (conversely, if fo0; selection favors some mutants with do0). Hence, a strategy z
is stable by convergence if fðz
Þ ¼ 0 and dfðza Þ=dza jza ¼z
o0: Effects on fixation probabilities can be computed as follows. For sufficiently weak mutation rates, the stationary probability distribution of allele frequency in a two allele model with symmetrical mutation rate m is determined by the probabilities of fixation pa-A (resp. pA-a ) of a A (resp. a) allele in a metapopulation initially fixed for the other allele. In particular, the expected frequency E½p of A in the metapopulation obeys pa-A lim E½p ¼ : ð1Þ m-0 pa-A þ pA-a Denote E½Dpjpsel the expected change in allele frequency due to selection, given the genetic state of the parental metapopulation characterized by a vector of allele frequencies p in different demes. Then, given convenient assumptions on the order of selection and mutation (Rousset and Billiard, 2000), the total expected change over one generation is the sum of E½Dpjpsel and of an expected change in allele frequency due to mutation, of the form mð1 2ðp þ E½Dpjpsel ÞÞ: At stationary equilibrium, the expected change is null, i.e. E½Dpsel þ mð1 2ðE½p þ E½Dpsel ÞÞ ¼ 0;

ð2Þ

where the expectations are taken over the distribution of allele frequency at stationary equilibrium. Then,
 1 E½Dpsel 1 þ lim lim E½p ¼ : ð3Þ m-0 m-0 2 m First-order effects in d on E½p are then limm-0 dE½Dpsel =2m dd: From Eq. (1) and given

129

dpa-A =dd ¼ dpA-a =dd; they are also dpa-A =2p dd; where p is the fixation probability of neutral (d ¼ 0) mutant (which is the initial frequency of the mutant). By identification of these two results, f

p dE½Dpsel =dd dpa-A ðzA Þ ¼ lim m-0 dd m

ð4Þ

(Rousset and Billiard, 2000). Based on this result, we use the following method to obtain an analytical measure of convergence stability. First, we compute the expected change in allele frequency E½Dpjpsel ; given the allele frequencies p in different groups of individuals in the parental metapopulation. Second, we compute the firstorder effect of selection on this change, dE½Dpjpsel =dd: Third, we compute the expectation dE½Dp=dd of dE½Dpjpsel =dd; i.e. its expectation integrated over the stationary distribution of p under recurrent mutation. Finally, we compute the limit of this expression for low mutation. Although considering recurrent mutation is not strictly necessary, it allows to express f immediately in terms of probability of identity of different pairs of genes in models with recurrent mutation, which provides an immediate link with the algebra of genetic identity, described in many papers (e.g. Male´cot, 1975; Nagylaki, 1983; Rousset, 2004). An alternative argument, not based on a stationary mutation–selection model, relates effects on fixation probabilities to average coalescence times of pairs of gene lineages (Rousset, 2003b). These average coalescence times can be computed directly, or again through the known algebra of identity by descent (Rousset, 2002; Slatkin, 1991). Whatever the exact argument considered, it leads to considering the distribution of allele frequencies in a neutral model (see Eq. (19) for a more detailed argument). This is appropriate only because we want to compute a weak selection effect on the probability of fixation of a mutant. In this specific context, the above arguments provide a justification for previous kin selection approaches making use of relatedness coefficients computed in a neutral model. With several classes of individuals we must consider expected changes in frequency for a weighted average of the frequencies in different classes. The appropriate weights in previous models without trait-dependent demography were reproductive values (Charlesworth, 1994; Leturque and Rousset, 2002; Taylor, 1990), which are defined such that the weighted frequency does not change over generations in a neutral model, i.e. expected weighted frequency in the offspring generation is the actual weighted frequency in the parental generation. This allows to express effects of selection as departures from this zero expected change. We will see how this concept applies in the present context. This paper is organized as follows. (1) We formulate the model. (2) We define the appropriate reproductive

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values. (3) We compute expected changes in weighted allele frequency, first for a given demographic state and given allele frequencies in the parental metapopulation; next we integrate this expectation over the distribution of allele frequencies and the distribution of demographic states in the parental metapopulation under a neutral model. (4) By the methods outlined above, this yields the required convergence stability measure. With respect to previous fitness measures for class structured population, a new term appears, that measures the effects of the trait under selection on the expected reproductive value of offspring of an individual expressing this trait, through its effects on the demography of the demes where its offspring settle. (5) We derive an infinite island approximation for the fitness measure. (6) We apply it to the evolution of the dispersal rate in presence of Allee effects, and provide simulation checks. (7) We discuss some differences between our results and earlier ones.

2. Model 2.1. Formulation We consider models described by a stationary Markov chain conditional on nonextinction of the metapopulation. The metapopulation is made of nd demes. Haploid individuals reproduce and die. Their juveniles disperse and form ‘‘juvenile pools’’ in different demes, competition occurs, and some number Ni of them survive to adulthood in deme i: The state of the metapopulation may be described in terms of the allele frequencies in the different demes, p  ðp1 ; y; pnd Þ; as well as of the ‘‘demographic state’’ D of the different demes. Below, this demographic state is characterized by the number of adults in each deme, D  ðN1 ; y; Nnd Þ: It may also be the time to the most recent extinction of the deme (Olivieri et al., 1995) or the local age-structure (Ronce et al., 2000). D should be defined so that it includes any random variable that affects expected allele frequency at t given allele frequencies at t 1: D should also include any variable required to define a Markov chain on D as required below for the definition of weighted allele frequency. We consider the following life cycle. (i) Each adult produces an independently distributed number of juveniles. (ii) These juveniles independently stay or disperse in other demes. (iii) Each juvenile independently dies or survives during dispersal. (iv) Finally, there is density-dependent survival, such that the number of juveniles that survive to adulthood in a deme is a random variable whose distribution is dependent on the number of competing juveniles, on parental or juvenile phenotypes, and on D:

2.2. Changes in allele frequencies Our method requires computing the first-order effect of selection on an expected change in weighted allele frequency, dE½Dp=dd: The expectation is over replicates of the joint demographic and genetic process defined above. To compute this expectation, we will first compute the expected change for given allele frequencies p in the parental metapopulation, and for given demographic state D: Next we will integrate these conditional expectations over the distribution of allele frequencies xðpjDÞ given D and over the distribution of demographic states PrðDÞ: 2.2.1. Conditional changes in allele frequency given p and D The following generic notation is used: variables are by default considered at some parental generation t 1; let primes 0 denote variables at time t (e.g. p0 ; D0 ). For a given demographic state D and initial vector of allele frequencies p at some time t 1; we express expected changes in allele frequency over one generation as follows. As in previous models (Taylor and Frank, 1996), we consider the fitness of an individual in the parental generation, called the ‘‘focal’’ individual, and assume that the expected number of offspring wij in deme i of this focal parent in deme j; depends on the phenotypes of different individuals in the metapopulation, all called ‘‘actors’’ since they act on the fitness of the focal individual. The actors usually include the focal individual, and the phenotypes to be considered include the phenotype of the focal parent itself, z ; and the average phenotype of adults in the different demes z  ðz1 ; y; znd Þ: We assume that any such variable zc can be expressed as zc ¼ za þ dpc þ Oðd2 Þ  zc ðpÞ; where pc is allele A frequency in the category of ‘‘actors’’ considered. In the present haploid model, these assumptions hold true when only one individual genotype affects the phenotype of an individual (e.g. its own genotype), and when the fitness functions wij are differentiable, in the usual mathematical sense (Courant and John, 1989, p. 41). The assumption of differentiability is implicit in any use of the direct fitness formalism (Taylor and Frank, 1996) and indeed in any computation of firstorder effects. Likewise, we assume that the transition probabilities between different demographic states D and D0 are functions, to first-order in d; of the phenotypes z: This is so in particular if the number of adults in each deme is a random variable whose distribution is determined by z: For simplicity, we have considered only parental control of the trait. Accounting for juvenile control would only require defining additional variables but would not affect the argument otherwise. Under these assumptions, the expected number of offspring in deme i of a parent in deme j; given D and

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D0 ; may be written wij ðD; D0 ; z ; zðpÞÞ; in terms of a fitness function wij (see e.g. Eqs. (31)–(32)). Likewise, the transition probabilities between deme sizes may be written PrðD0 jD; zðpÞÞ: Expected changes in allele frequencies are then described by E½p0 jp; D; D0  ¼ FðD; D0 ; zA ; zðpÞÞp:

ð5Þ

where FðD; D0 ; zA ; zðpÞÞ is the matrix with elements ðfij ðD; D0 ; zA ; zðpÞÞÞ defined as Nj ð6Þ fij ðD; D0 ; zA ; zÞ  0 wij ðD; D0 ; zA ; zÞ: Ni Since there are Nj pj A-bearing parents in deme j; fij ðD; D0 ; zA ; zðpÞÞpj is the probability that a gene in i is a copy of a gene from any of the A parents in j: 2.2.2. Weighted allelic frequencies Let generically denote variables evaluated in the neutral model (zA ¼ za ; d ¼ 0). We define Pnd a weighted allele frequency, with weights ai ; as i¼1 ai pi  a  p; using standard dot notation for the scalar product of vectors. In a model with fixed demography the weights usually considered are reproductive values, defined to give the ultimate contribution of a gene lineage (individual reproductive value), or of all the gene lineages in a class (class reproductive value), to the future composition of the population (Charlesworth, 1994; Taylor, 1990). Thus we may consider the reproductive value of a deme (as in the general analysis), or of all demes with the same demographic characteristics (as in the infinite model developed later). The class reproductive values obey a recursion of the form a ¼ a0 F : This equation expresses the probability that the ancestor of some gene was in class j at time t (element j of a) as the sum over i of the probabilities that the ancestor was in class i at time t þ 1 (element i of a0 ) times the probability that this gene came from j (which is fij ). We generalize this definition as follows. The weight given to deme i at time t will here depend on the demographic state of the metapopulation at time t: We associate to each demographic state D of the metapopulation a row vector aðDÞ of reproductive values. These vectors are defined to give the ultimate contribution, in a neutral model, of a gene lineage considered at time t to the future composition of the metapopulation, given the demographic state at t: In other words, for a gene sampled in any deme, aj ðDÞ is the probability that its ancestral lineage in a distant past t was in deme j given the ancestral demographic state of the metapopulation at t was D: As in the case with fixed demography, aj ðDÞ can be expressed as a sum of reproductive values in the next generation ai ðD0 Þ; weighted by the probability fij ðD; D0 Þ of origin of a gene lineage given D; D0 : The only difference is that we have to integrate over all possible states of the

131

metapopulation at t þ 1: aj ðDÞ ¼

X X i

Pr ðD0 jDÞai ðD0 Þfij ðD; D0 Þ:

ð7Þ

D0

Thus, the vector of reproductive values in a metapopulation with demography D is given by the equilibrium solution of the recursion aðDÞ ¼

X D

Pr ðD0 jDÞaðD0 ÞF ðD; D0 Þ;

ð8Þ

0

where the a’s are normalized such that

P

ai ðDÞ ¼ 1:

2.2.3. Conditional changes in weighted allele frequency given D and p Using Eq. (5) and taking expectations over all possible demographic states D0 at t þ 1; we find that the expected weighted allele frequency obeys X E½aðD0 Þ  p0 jp; D ¼ PrðD0 jD; zðpÞÞaðD0 Þ D0

 FðD; D0 ; zA ; zðpÞÞp:

ð9Þ

From Eqs. (8) and (9), weighted allele frequency does not change in expectation in the neutral model (d ¼ 0): E ½aðD0 Þ  p0 jp; D ¼ aðDÞ  p:

ð10Þ

With selection (da0), we focus on changes in allele frequency over one generation, Dða  pÞ  aðD0 Þ p0 aðDÞ  p: E½Dða  pÞjp; D ¼ E½aðD0 Þ  p0 jp; D aðDÞ  p  dE½aðD0 Þ  p0 jp; D ¼d  þoðdÞ dd d¼0

ð11Þ

ð12Þ

by Taylor expansion near d ¼ 0; and using Eq. (10). Notice that by definition the a values are given by the neutral model and hence are independent of d: We can always consider effects on change in weighted allele frequency for any set of weights independent of d: The only point of the choice of a as weights is that the expected change in allele frequency is null in the neutral model (as a result of Eq. (10)), which simplifies later computations. Using Eq. (9) dE½aðD0 Þ  p0 jp; D X X X ¼ pj ai ðD0 Þ dd 0 j i D 

d ðPrðD0 jD; zðpÞÞ dd

 fij ðD; D0 ; zA ; zðpÞÞÞ:

ð13Þ

ARTICLE IN PRESS F. Rousset, O. Ronce / Theoretical Population Biology 65 (2004) 127–141

132

By definition of fij ðD; D0 ; zA ; zðpÞÞ; dfij ðD; D0 ; zA ; zðpÞÞ dd @fij ðD; D0 ; zA ; zðpÞÞ dzA ¼ @zA dd nd X @fij ðD; D0 ; zA ; zðpÞÞ dzc þ @zc dd c¼1

(17) over the distribution xðpjD; 0Þ; which yields X X X ai ðD0 Þ E½Dða  pÞjD ¼ d D0 n d X

ð15Þ

ð16Þ

E½Dða  pÞjp; D nd X X X X ai ðD0 Þ pj pc ¼d 

i

c¼

The E½pj pc jD; d ¼ 0 factors can be related to standard concepts of the theory of genetic population structure, as follows. They are the probabilities that two genes sampled with replacement, one in a focal individual in class j and the other in an c-actor, are both of allelic type A: In the neutral two-allele model with identical mutation rate for both alleles, this is half the probability QjcjD that two genes sampled with replacement are identical, irrespectively of allele type: QjcjD ¼ 2E½pj pc jD; d ¼ 0: Remember that p  1 2E½pj jD; d ¼ 0 ¼ 1:

¼

XX X dX Pr ðDÞ ai ðD0 Þ 2 D i j D0

nd X @ PrðD0 jD; zÞfij ðD; D0 ; z ; zÞ

@zc

c¼

so

Qj jD ¼

D0

D

i

@zc

c¼

p

ð19Þ

where it is seen that the first-order effect ddx=dd does not contribute a first-order term to this expectation since it comes in factor with an OðdÞ term. Thus, evaluating the above expectation only requires taking the expectation of the products pj pc in expression

QjcjD

þ oðdÞ: ð22Þ

j

nd X @ PrðD0 jD; zÞfij ðD; D0 ; z ; zÞ

ð18Þ

!

Note that first-order effects on PrðDÞ near d ¼ 0 do not matter for the same reason that first-order effects on x did not matter. Hence, in all later computations the different terms need be computed only for the neutral model, i.e. as function of za but not of d: Let S ¼ 2dE½Dða  pÞ=dd: Then, X X X X S¼ PrðDÞ ai ðD0 Þ

E½Dða  pÞjD X ¼ E½Dða  pÞjp; DðxðpjD; 0Þ

X dE½Dða  pÞjp; D xðpjD; 0Þ þ oðdÞ; ¼d dd p

(Eq. (15)),

D

ð17Þ

2.2.4. Conditional changes in weighted allele frequency given D We obtain them by summing conditional changes given D and p overPthe distribution xðpjDÞ of p given D: E½Dða  pÞjD ¼ p E½Dða  pÞjp; DxðpjDÞ: The distribution of allele frequency depends on the phenotypic effect d of the A allele. So we can write xðpjDÞ as a function of d; and we expand it near d ¼ 0 as xðpjD; dÞ ¼ xðpjD; 0Þ þ ddxðpjD; dÞ=dd þ oðdÞ: Then using Eq. (12),

þ ddxðpjD; dÞ=ddÞ þ oðdÞ

ð21Þ

E½Dða  pÞ X ¼ PrðDÞE½Dða  pÞjD þ oðdÞ

j

@ ðPrðD0 jD; zÞfij ðD; D0 ; zÞÞ þ oðdÞ: @zc

ð20Þ

2.2.5. Unconditional changes in allele frequencies We now integrate over the distribution of D: From Eqs. (20) and (21), we have

Eq. (12) then takes the form

D0

0

 E½pj pc jD; d ¼ 0 þ oðdÞ:

ð14Þ

where Pnd p ¼ 1 for a focal individual of genotype A; and c¼ is a shorthand for considering all variables z ; z1 ; y; znd : All derivatives are evaluated in d ¼ 0: Likewise, nd d PrðD0 jD; zðpÞÞ X @ PrðD0 jD; zÞ ¼ pc : dd @zc c¼1

j

@ PrðD jD; zÞfij ðD; D0 ; z ; zÞ @zc c¼

and using the notation z for the phenotype of the focal individual, this may be written nd dfij ðD; D0 ; zA ; zðpÞÞ X @fij ðD; D0 ; z ; zÞ ¼ pc ; dd @zc c¼

i

¼

X D nd X

PrðDÞ

X

PrðD0 jDÞ

D0 0

X

QjcjD

ai ðD0 Þ

i

X j

@fij ðD; D ; z ; zÞ QjcjD @zc c¼ X X X X PrðDÞ ai ðD0 Þ fij ðD; D0 Þ þ D0

D

nd X @ PrðD0 jD; zÞ c¼

@zc

i

QjcjD

ð23Þ

j

ð24Þ

ARTICLE IN PRESS F. Rousset, O. Ronce / Theoretical Population Biology 65 (2004) 127–141

 Sf þ SPr ;

ð25Þ

where Sf is made of the @f =@zc terms, and SPr is made of the @ Pr=@zc terms. The expression for S weights the effects of c-actors on the number of offspring, in deme i in a metapopulation with demography D0 ; of the focal individual in deme j in a metapopulation with demography D: The weights are (i) the reproductive value of such offspring; and (ii) the probability QjcjD that a random c-actor bears the same allele as the focal individual. Sf has the same form as some previous inclusive fitness measures for class-structured populations (e.g. Leturque and Rousset, 2002; Taylor, 1990; Taylor and Frank, 1996). SPr describes the additional selective effects, due to the effects on an individual’s offspring which result from changes in the probabilities of different future demographic states of the metapopulation. As shown in Eq. (4), a ‘‘fitness’’ measure is derived from S as f  limm-0 Sp =ð2mÞ: Exact computation of f requires exact computation of p ; which is a weighted average frequency over the different demes (Leturque and Rousset, 2002) and demographic states of the population in which a mutant appears.

3. Infinite island model We now give a practical way of measuring the effect of selection in an ‘‘island’’ model of dispersal, with an infinite number of demes (nd -N). A detailed derivation is given in the Appendix. It is based on the fact that each deme becomes independent from any other one, in two respects. First, genes in different demes may be considered ‘‘unrelated’’, as follows. First-order effects of mutation on probabilities of identity do not depend on the mutation model (Rousset, 2004, Eq. (4.17)). Hence, the value of limm-0 Sp =ð2mÞ does not depend on the mutation model, so the value of f is unchanged if we substitute identity by descent to identity in state in the expression for S; i.e. if we substitute identity in a model where each mutation produces a new allele to identity in the two-allele model considered above. In the infinite island model, genes sampled in different demes are unrelated in the sense that their identity by descent is null (e.g. Crow and Kimura, 1970, Eq. (6.6.2); and Appendix). Below, we apply this simple rule to the computation of S: Computing S is sufficient to obtain the sign of f: Second, we assume that the demographic state of the metapopulation converges to some stationary equilibrium as the number of demes increases. We may then neglect fluctuations of D between generations. The frequency of demes of a given size converges to its

133

expectation when nd -N: Then the average deme size also converges to its expectation. Notations and definitions are thus altered compared to the general framework presented previously. In particular, we now group demes with the same size in the same class. The class of an individual is now the size of its deme, and we index classes according to this size. Let aðnÞ be the total reproductive value of demes of size n: Let PrðnÞ be the frequency of demes of size n: Let uðn0 jnÞ  PrðNi0 ¼ n0 jNi ¼ nÞ be the forward transition probabilities of demes from class n to class n0 : Let vðnjn0 Þ  PrðNi ¼ njNi0 ¼ n0 Þ be the corresponding backward transition probabilities. Let Qðn; cÞ be the probability of identity of a focal gene in a deme of size n with a gene in a c-actor (c ¼ for the focal individual itself, and c ¼ 0 for actors in its deme; note that Qðn; Þ ¼ 1). For any deme i; we express fii ðD; D0 ; z ; zÞ; to firstorder in d; in terms of a single function, fp ðn; n0 ; z ; z0 ; z%Þ; evaluated in n ¼ Ni ; n0 ¼ Ni0 (the size of deme i at times t 1 and t), and z0 ¼ zi (the average trait value in the deme of the focal individual). fp (where ‘‘p’’ stands for philopatric) is also function of the average phenotype z% in the metapopulation, and of average deme size N% (see e.g. Eq. (33) for the dispersal model; a general method to construct such functions is detailed in the Appendix). Likewise we can gather all fij terms for all demes i of some size n at t 1 and n0 at t; and for all demes jai of some size m at t 1; into a single term expressed as PrðmÞfd ðn; n0 ; m; z ; z0 ; z%Þ (see e.g. Eq. (33)), where ‘‘d’’ stands for dispersed. Finally, the transition probabilities of a deme between different sizes can be expressed as a function of variables z0 and z% with the same meaning as above (see e.g. Eq. (35)). The shorthand notations fp ðn; n0 Þ and fd ðn; n0 ; mÞ will be used for brevity, fp and fd being always functions of phenotypes. In the infinite island model, gathering the @f terms yields Sf ¼

X

X @fp ðn; n0 Þ Qðn; cÞ @zc n n0 c¼ ;0 ! X X @fd ðn; n0 ; mÞ þ PrðmÞ Qðm; cÞ : @zc m c¼ ;0 aðn0 Þ

X

vðnjn0 Þ

ð26Þ

In the last sum m is the size of the deme of the focal individual and n is the previous size of a deme in which juveniles compete. A similar result has already been used (Ronce et al., 2000). The remaining terms, contributed by the @ PrðD0 jDÞ’s, account for the selection on the trait considered through its effects on demography. They take the form X X @vðnjn0 Þ SPr ¼ aðn0 Þ fp ðn; n0 ÞQðn; 0Þ @z0 n n0 X aðn0 Þ X @uðn0 jnÞ ¼ PrðnÞfp ðn; n0 ÞQðn; 0Þ ð27Þ 0 Prðn Þ n @z0 n0

ARTICLE IN PRESS F. Rousset, O. Ronce / Theoretical Population Biology 65 (2004) 127–141

134

(see the Appendix for details). This result shows that in the infinite island model, one does not need to consider the effects of individuals on the demography of other demes. In this case, one needs only to consider the effects on the fitness of the focal individual through the variations @uðn0 jnÞ=@z0 in the probability that the focal deme will be of size n0 : These changes are weighted by the contribution fp ðn; n0 Þ of the focal individual to its deme and by the same reproductive value weights as above. The result that we can ignore effects on other demes relies on the assumption that, if an individual interacts with an increasing number of demes as nd -N; the effect on each deme must decrease. For example, if an individual affects the size of different demes through dispersal of its juveniles, the effect on the size of each deme decreases because, for a fixed effect on total dispersal rate, the effect on the number of juveniles emigrating to a given deme decreases. To compute S we need: 1. The stationary probabilities of the deme states, PrðnÞ; in the stationary Markov chain model described by the matrix forward transition probabilities with n0 ; nth element uðn0 jnÞ (the forward transition probabilities of demes from class n to class n0 ). The stationary probabilities PrðnÞ are a right eigenvector of this matrix, as usual (e.g. Taylor, 1990); 2. The total reproductive values in demes of size n0 ; aðn0 Þ: These values are determined by the backward transition probabilities ln0 m of a gene lineage between classes of size n0 and m: The ln0 m ’s are functions of the probabilities that a deme of size n0 derives from a deme of size n (the backward transition probabilities vðnjn0 Þ): ln0 m ¼ vðmjn0 Þfp ðm; n0 ; za ; za ; za Þ X vðnjn0 Þfd ðn; n0 ; m; za ; za ; za Þ: þ PrðmÞ

ð28Þ

n

The fp term in this expression is for philopatric genes in a deme that was of size m: The remaining terms are for immigrant genes from a size m deme into a deme formerly of size n: The class reproductive values aðn0 Þ are a left eigenvector of this ðlij Þ matrix, as usual (e.g. Taylor, 1990); 3. Finally, we need the probabilities of identity. D Qðn; 0Þ ¼ 1=n þ ð1 1=nÞQD n where Qn is the probability of identity ‘‘by descent’’ of different individuals in a deme of size n; given by the following equation (e.g. Rousset, 1999): QD ¼ AðQD þ cÞ

ð29Þ

where QD is the column vector ðQD n Þ; c is a vector with elements ð1 QD Þ=n; and A is a matrix with n elements an0 ;n obtained as the probabilities vðnjn0 Þ that a deme of size n0 derives from a deme of size n; times

the probability that both genes were from philopatric juveniles given n and n0 : In the example given below, the probability of origin of two genes is the square of the probabilities for each gene, i.e. fp ðn; n0 Þ2 : More generally, more complex expressions may be required (e.g. Ronce et al., 2000).

4. Example: extinction and dispersal We here analyze a model for the evolution of dispersal under demographic and environmental stochasticity, and investigate the possibility of selection-driven extinction under Allee effects (Gyllenberg et al., 2002). 4.1. The model We first consider a finite island model where the deme sizes Ni fluctuate randomly in each deme. Following previous basic models we assume that there is a cost C of dispersal to a different deme, and no cost for philopatric juveniles. 4.1.1. Population regulation We assume the following demography. The numbers of juveniles produced by each parent are independent and Poisson-distributed, with expectation written r: Let m be the dispersal probability of juveniles of an adult. Then the numbers Ji of juveniles of each adult that settle in deme i are independent and Poisson-distributed, with means rð1 mÞ if the adult is in deme i and ð1 CÞrm=ðnd 1Þ otherwise; and the total number of juveniles that compete for deme i is also Poisson distributed. We assume the following density dependence (step (iv) of the life cycle). If a deme contains j juveniles after dispersal, each of them is assumed to survive independently with probability sð jÞ ¼ ss j as 1 e ks j

ð30Þ

with suitably chosen parameter values, such that sð jÞp1: This form yields a Ricker model (Ricker, 1954) for juvenile survival when as ¼ 1 and ks 40; and an Allee effect (low survival at low density) when as 41 and ks 40: Conditional on j; the number of adults at t will follow a Binomial distribution with parameters j and sð jÞ; with probabilities noted bðNi0 ; j; sð jÞÞ: Some demes become extinct by this process (demographic stochasticity). We also consider an extrinsic cause of extinction (environmental stochasticity), with probability e; according to which PrðNi0 ¼ 0j jÞ ¼ e þ ð1 eÞbð0; j; sð jÞÞ

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and PrðNi0 ¼ n0 j jÞ ¼ ð1 eÞbðn0 ; j; sð jÞÞ for n0 40: A ceiling effect is further imposed, such that the number of adults is set to some value K whenever a number of adults n0 4K is obtained under the above probability distribution. This ceiling is required for numerical computation, and K should be chosen so that the probability that n0 4K is as small as possible given computational constraints. In this example the parameters that affect the transition probabilities between deme sizes, r; as ; ks ; ss ; e; and C; are independent of time t: 4.1.2. Fitness functions and fij ’s The fitness functions for this model are wii ðD; D0 ; z ; zÞ ¼

Ni0 ð1 z Þ P ; Ni ð1 zi Þ þ ð1 CÞ kai Nk zk =ðnd 1Þ

ð31Þ

wij ðD; D0 ; z ; zÞ ¼

ð1 CÞNi0 z =ðnd 1Þ P Ni ð1 zi Þ þ ð1 CÞ kai Nk zk =ðnd 1Þ ð32Þ

for jai: 0

The fij ’s (Eq. (6)) are then independent of D : In the infinite island model, their denominator may be written as Ni ð1 z0 Þ þ ð1 CÞN% z%; where N% is the average deme size in the metapopulation. For each deme i with Ni ¼ n; the term for ‘‘philopatric’’ offspring is then fp ðnÞ  fp ðn; n0 ; z ; z0 ; z%Þ ¼

nð1 z Þ : nð1 z0 Þ þ ð1 CÞN% z%

ð33Þ

Likewise the probability that a gene, presently in a deme that was previously of size n; descends through dispersal from a parent in a deme of size m one generation before, can be written PrðmÞfd ðn; mÞ; where fd ðn; mÞ  fd ðn; n0 ; m; z ; z%Þ ¼

ð1 CÞmz : ð34Þ nð1 z%Þ þ ð1 CÞN% z%

Only z% appears in the denominator because each immigrant juvenile comes in competition with juveniles not from its own birth deme. 4.1.3. Transition probabilities between deme sizes For any n0 40; they can be written as N X PrðNi0 ¼ n0 jJ ¼ jÞ Prð jjNi ¼ nÞ uðn0 jnÞ ¼

ð35Þ

j¼0

(where J is the number of juveniles in competition for the deme) N X Mj : ð36Þ bðn0 ; j; sð jÞÞe M ¼ ð1 eÞ j! j¼0

135

In this expression the binomial term bðn0 ; j; sð jÞÞ was described above (see Population regulation), and the next terms are the Poisson terms for the probability of j juveniles competing, as function of its expectation M  nð1 z0 Þ þ ð1 CÞN% z%: In these expressions, the assumption of Poisson-distributed fecundity allows an arbitrary large number j of juveniles with some nonzero probability, even if the number of adults is finite. In numerical p computations a finite ceiling was assumed: ffiffiffiffiffiffi 1 þ M þ 6 M ; where M is the mean value of the Poisson distribution as function of the resident strategy: M ¼ nð1 z%Þ þ ð1 CÞN% z%: In this way, the fraction of the Poisson distribution neglected is less than 10 6 (see Feller, 1968, p. 245). 4.1.4. Probabilities of identity They are computed using Eq. (29), where the probability that two settled offspring are philopatric given Ni and Ni0 is the square of the probability that one individual is philopatric. The latter result follows from using properties of multivariate Poisson distributions: the probability of event P E ‘philopatry of two individuals’ is PrðEjNi0 Þ ¼ jX2 PrðEj J ¼ jÞ Prð jjNi0 Þ; where J is the number of juveniles in competition for the deme. Given J ¼ j; the number of philopatric juveniles is binomially distributed with parameters j and fp ðn; n0 Þ independent of j: Then, sampling two juveniles, PrðEj j Þ ¼ PrðEjNi0 ¼ n0 Þ ¼ fp ðn; n0 Þ2 : 4.2. Analysis Selected dispersal rates were determined by seeking z such that SðzÞ ¼ 0 numerically by the secant method (the sign of SðzÞ being used to check convergence stability). Fig. 1 shows the convergence stable dispersal rates z
and average deme size at z
as a function of the environmental extinction rate e; as well as some simulations checks, in two cases. We focus on selection-driven extinction, which occurs when there is an Allee effect on juvenile survival and large extinction rate e: In this case selection brings the metapopulation outside the range of dispersal rates allowing global persistence of the metapopulation. This appears in Fig. 1a for a local extinction rate e40:11: By contrast, selection-driven extinction does not happen without an Allee effect (Fig. 1b). Similar results were obtained for other parameter values investigated. Selection-driven extinction was confirmed by individual-based simulations (Fig. 2): the mean dispersal rate in the metapopulation decreases until it reaches a value (predicted: 0.449) where metapopulation size crashes rapidly. By contrast a metapopulation placed in identical conditions except that the dispersal rate was fixed (no mutation) at 0.459 did not get extinct in 105 generations.

ARTICLE IN PRESS F. Rousset, O. Ronce / Theoretical Population Biology 65 (2004) 127–141

Dispersal rate

0.8 0.6 0.4

0.05

0.1

0.15

0.2

0.25

0.3

0.6 0.4

100

0.35

1 0.8 0.6 0.4 0.2 0.1

(b)

0.8

0.2

0.2

(a)

Dispersal rate and saturation

1

1

Metapopulation size

Dispersal rate and saturation

136

0.2

0.3

0.4

0.5

0.6

0.7

Extrinsic extinction rate


Fig. 1. Evolution of the mean dispersal rate as function of the extinction rate e: In both cases C ¼ 0:7; K ¼ 20 and r ¼ 4: (a) Allee effect, described by as ¼ 2:8; ss ¼ 0:05; and ks ¼ 0:13 in Eq. (30). With such values the survival rate of juveniles was at most 0:937; when there are 14 juveniles, and was 0.36 when there are only four juveniles. (b) No Allee effect (i.e. as ¼ 1; other parameter values ss ¼ 1 and ks ¼ 0). The grey area is the set of parameters not allowing infinite persistence of the metapopulation in the infinite island model. It was obtained numerically from the computation of stable equilibria for average deme size. Filled triangles are selected dispersal rates, and empty % triangles are levels of metapopulation saturation, N=K; at these rates. Crosses ðÞ are average dispersal rates observed in simulations of at least 200,000 generations of evolution of a metapopulation of 1000 demes, with mutation rate 10 3 and maximum effect of mutation 0.1 on dispersal rate. The arrow shows the predicted evolution of a metapopulation for the case considered in Fig. 2.

5. Discussion The general result (23) holds in principle for a variety of population structures. However, such expressions remain complicated. Applying the general method to a finite number of demes is problematic for numerical reasons. The number of possible demographic states D increases rapidly with the number of demes and the maximal number of individuals per deme. Simplification is possible only in specific cases such as the infinite island model. Eqs. (26)–(29) provide a practically computable measure of the effects of selection in this case. This measure should be proportional to the firstorder effects on the fitness measure considered by Metz and Gyllenberg (2001). In practice results for finite island models are well approximated by the infinite island model. The simulations confirm that this measure accurately predicts the evolution of the trait considered.

200

300

400

500

10000 8000 6000 4000 2000 100

200

300

400

500

Generation Fig. 2. Mean dispersal and metapopulation sizes in 10 replicate simulations of the model with Allee effect. e ¼ 0:2; and other demographic and mutation parameter values are as in Fig 1a. A metapopulation of 1000 patches evolves from an initial dispersal rate z ¼ 0:7: The metapopulation declines in size slowly as dispersal rate decreases, then rapidly gets extinct when dispersal goes below the predicted extinction threshold, z ¼ 0:449 (shown as the dashed line). The metapopulation would persist much longer at dispersal rates above this threshold (see text).

The general model suggests a way to compute convergence stability in ‘‘stepping stone’’ or ‘‘isolation by distance’’ models. The probabilities of identity can be computed in such models without demographic fluctuations (e.g. Male´cot, 1951, 1975; Nagylaki, 1983) and such results have been used to find candidate ESSs under isolation by distance (Gandon and Rousset, 1999; Rousset and Gandon, 2002). However, it remains an outstanding problem to compute the probabilities of identity of pairs of genes in populations with both this pattern of dispersal, and demographic fluctuations, so for the moment the present method can only be readily applied to the infinite island case. In general, it is necessary to consider only a finite number of demographic states. Thus, we may consider arbitrarily large deme sizes, and use approximations that bin demes with large sizes in a single class. However, this is likely to introduce subsequent biases in the analysis, unless these large deme sizes have low probabilities of occurrence. But then, it is simpler to impose a maximum number of adults as we did and as done in related works (e.g. Parvinen et al., 2003). When metapopulation extinction is possible, it is worth considering long-term evolution if the average time to extinction is much longer than the time scale of

ARTICLE IN PRESS F. Rousset, O. Ronce / Theoretical Population Biology 65 (2004) 127–141

allelic substitutions. In large metapopulations with Allee effects, there is a sharp boundary between cases where there is such a separation of time scales, and cases where the metapopulation quickly go extinct. Our analysis does not exactly describe what occurs at the boundary, but it tells whether the metapopulation will approach the boundary, and in a finite metapopulation, approaching the boundary implies that extinction is likely to occur rapidly.

5.1. Modelling approaches Previous analyses of fluctuating demography have usually assumed deterministic allele frequency dynamics (that is, p0 is considered a function of p), or modelled deme size as a continuous variable (e.g. Dieckmann and Law, 1996; Ferrie`re and Gatto, 1995; Gyllenberg and Parvinen, 2001; Rand et al., 1994; but see Metz and Gyllenberg, 2001). Here we have both considered a discrete stochastic model for deme size, and used probabilities of identity to take into account differences between p0 and E½p0  (covariances in allele frequencies within and between different demes are measured by the probabilities of identity). The fitness measure obtained takes interactions between relatives in account. The approach taken here yields a computable fitness measure in terms of widely used concepts such as relatedness and reproductive value, and specific results can be understood by the analysis of the different terms of the fitness measure. Expression (25) for the fitness measure is a generalization of previous ‘‘direct fitness’’ measures. In particular, the term Sf is analogous to previous expressions (Ronce et al., 2000; Taylor and Frank, 1996). The additional term SPr measures the effects of neighbors on the size of the deme, and weights these effects by the reproductive value of offspring as function of deme size. Hence it measures the effects of neighbors of an individual on the reproductive value of its offspring in its deme. Because there are relatedness coefficients in any terms of SPr ; and because these coefficients decrease as deme size increases, this additional term vanishes for high deme sizes. Our results give the simplest exact analytical expressions for determining candidate ESS, to which approximations may be compared. No a priori assumption about the relative magnitude of direct and inclusive fitness effects has been made, and it may often be useful to ignore inclusive effects in order to obtain simple approximations and/or to understand better the importance of competition between relatives. Ignoring the inclusive fitness terms in the formula for S directly give such approximations. However, unless unrealistic models of metapopulation regulation are considered, this will usually not yield a simple closed expression for the candidate ESS.

137

5.2. Evolution under Allee effects Various social interactions result in an Allee effect (Courchamp et al., 1999). For example, vertebrate cooperative breeders often show an Allee effect. Such species, which are also often characterized by small deme size and low dispersal rates, are particularly prone to extinction (Courchamp et al., 1999). For this and other reasons (Alexander et al., 1991), a stable environment is important for the evolution of sociality. The present model does not directly apply to a particular social species, but the deme size and dispersal costs in the numerical example are not unrealistic (compare to some social meerkat populations, Clutton-Brock et al., 2001). Our results therefore suggest another reason for the importance of a stable environment. With an Allee effect and environmental stochasticity, selection on the dispersal rate may select against any dispersal rate that would allow persistence of the metapopulation. Such selection-driven extinction was not observed without an Allee effect. Dispersal may not be the only trait which evolves in such a way. Similar ‘‘evolutionary suicide’’, through the evolution of dispersal in the presence of an Allee effect, has recently been found by Gyllenberg and collaborators (Gyllenberg and Parvinen, 2001; Gyllenberg et al., 2002). They assumed continuously varying populations sizes and thus neglected kin selection. This suggests that evolutionary suicide has little to do with kin selection. The only qualitative discrepancy between our results and theirs is that they also predict evolutionary suicide for very low extinction rates (Gyllenberg et al., 2002), which is not the case in our model. This occurs in their deterministic model because the selected dispersal rate goes to zero when the disturbance rate goes to zero. However, in any model where genetic drift occurs, some level of dispersal is selected in the absence of local extinctions. When all effects of discrete deme size are taken into account, kin competition is shown to occur, and relatively high dispersal rates are selected for in perfectly stable habitats. Kin selection therefore prevents the occurrence of evolutionary suicide for low extinction rates. In summary, we have defined a ‘‘fitness’’ measure which correctly predicts the results of individual-based simulations in models where the trait under selection interacts with the demography of the metapopulation. We have used these general results to demonstrate that a metapopulation may be driven to extinction through evolution of the dispersal rate in a model where genetic drift is taken into account. Virtually any trait affects the demography of natural populations, and the general results may be useful to understand some general trends. In particular, we may ask to what extent it matters that the behavior of the mutant affects the local demography, and conclude from our analysis that it affects

ARTICLE IN PRESS 138

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selection only through terms vanishing at high deme size. Further progress is restrained by the need for numerical analysis of demographic models in subdivided populations, but the general results may suggest further analytical approximations to be investigated in more specific models.

Acknowledgments We thank S. Billiard, S. Gandon, Y. Michalakis, I. Olivieri, K. Parvinen, M. Raymond, and the reviewers for comments. This is publication ISEM 03-069. O.R. acknowledges support from the program Plant Dispersal, contract EVK2-CT1999-00246 allocated to Ben Vosman.

Appendix A. Derivation of the infinite island result The sign of f is obtained as the sign of S: We will summarize the argument that leads to limit expressions for S when nd -N; i.e. in the infinite island model. The main result is the expression for SPr : The expression for Sf is a previously recognized and used result (Ronce et al., 2000), which derivation is given for comparison. A.1. Probabilities of identity In the infinite island model, the fitness measure f is derived in the limit case nd -N; m-0: Previous kin selection analyses have used the convenient simplification that probabilities of identity between genes in different demes vanish in the limit case mnd -N (Crow and Kimura, 1970, Eq. (6.6.2)). This simplification extends to the present models, as the following general argument shows (see Rousset, 2002 for further discussion). Two genes are identical if no mutation occurred since their coalescence, and a fortiori before their ancestral lineages first join in the same deme (looking backwards in time). This joining event has probability Oð1=nd Þ in each generation. Hence, when mnd -N; mutations always occur before two ancestral lineages first join in the same deme. A.2. Convergence to the infinite island model We note PrðnjDÞ the frequency of demes of size n in a metapopulation with demographic state D: Likewise, considering a deme with successive sizes n; n0 ; we note Prðn; n0 jD; D0 Þ the frequency of such demes in a metapopulation with demographic states D; D0 in successive generations. Considering a pair of demes, one with successive sizes n; n0 and the other with size m; we note Prðn; n0 ; mjD; D0 Þ their frequency.

We use the following results. First, as the number of demes increases, the frequency PrðnjDÞ of demes of size n in a metapopulation at state D converges in probability to a limit value PrðnÞ (e.g. Chesson, 1981). In other words, as the number of demes increases, for any fixed Z40; demographic states D such that jPrðnjDÞ PrðnÞj4Z become increasingly unlikely. Using a standard notation, we may write this as PrðnjDÞ ¼ PrðnÞ þ op ð1Þ

ðA:1Þ

where op ð1Þ is a random term which converges in probability to zero as nd -N: This result extends to PrðSjD; D0 Þ for S ¼ n; n0 or n; n0 ; m: Likewise, the total reproductive value in demes of size n; written aðnjDÞ; converges in probability to limit values aðnÞ: aðnjDÞ ¼ aðnÞ þ op ð1Þ:

ðA:2Þ

Hence the reproductive value of a deme of size n is aðnÞ=ðnd PrðnÞÞ þ op ð1=nd Þ: Finally the probabilities of identity between adults in demes of size n; written Qðn; 0jDÞ; converge in probability to a limit value Qðn; 0Þ: Qðn; 0jDÞ ¼ Qðn; 0Þ þ op ð1Þ:

ðA:3Þ

For convenience, we define Qðn; jDÞ  1 for any n: Below we freely use standard results for convergence in probability of sums and products of random variables (e.g. Lehmann, 1994, Section 5.1). A.3. Effects on the fij We first consider the term Sf ; contributed by the functions fij : X X Sf ¼ PrðDÞ PrðD0 jDÞ D0

D



X

ai ðD0 Þ

i

¼

X D nd X c¼

PrðDÞ

X D0

nd X X @fij ðD; D0 Þ QjcjD @zc c¼ j

PrðD0 jDÞ

X

ðA:4Þ

ai ðD0 Þ

i

! X @fij @fii QicjD þ QjcjD : @zc @zc jai

ðA:5Þ

Let ni ; n0i ; and nj be realized values of the random variables Ni ; Ni0 ; and Nj ; respectively. We will express Sf in terms of the functions fp and fd considered in the main text. Their definition is not unique. We only need a function fp such that fii ðD; D0 ; z ; zÞ ¼ fp ðni ; n0i ; z ; z0 ; z%Þ þ Oðd2 Þ þ op ð1Þ: Such a function can

ARTICLE IN PRESS F. Rousset, O. Ronce / Theoretical Population Biology 65 (2004) 127–141

always be obtained as the limit when nd -N of

A.4. Effects on deme sizes

@fii @z X @fii @fii þ ðz0 za Þ þ ðz% za Þ : @zi @zc cai

fii ðD; D0 ; za ; za ; y; za Þ þ ðz za Þ

ðA:6Þ

The last term is not really necessary as the derivative of fp ðnÞ with respect to z% comes in factor with a weighted average probability of identity of genes in different demes. Using identity by descent to compute S as explained in the main text, this probability vanishes in the infinite island model. The same considerations hold for fd : Using Eq. (A.3), Eq. (A.5) may then be written X X X aðn0 ; D0 Þ i Sf ¼ PrðDÞ PrðD0 jDÞ 0 jD0 Þ n Prðn d 0 i i D D X X @fp ðni ; n0 Þ i Qðni ; cÞ þ  @zc j c¼ ;0 ! X @fd ðni ; n0 ; nj Þ i Qðnj ; cÞ þ op ð1Þ: @z c c¼ ;0 ¼

X

PrðDÞ

X D0

D

X



n

þ

X X n

SPr ¼

X

PrðDÞ



nd X X X i

D

j

QjcjD

c¼1

X @ PrðD0 jDÞ fij ðD; D0 Þai ðD0 Þ: @z c 0 D

ðA:11Þ

Let D0k be the set of variables defining the demographic state D0 ; with Nk0 excluded, so that PrðD0 jDÞ ¼ PrðD0k jD; Nk0 Þ PrðNk0 jDÞ: Then SPr ¼

X



PrðDÞ

nd X X X i

j

QjcjD

c¼1

nd X X @ PrðNk0 ¼ n0k jDÞ @zc k¼1 n0 k

ðA:7Þ



X

PrðD0k jD; n0k Þfij ðD; D0 Þ

D0k

aðn0i ; D0 Þ : ðA:12Þ nd Prðn0i jD0 Þ

Consider the terms of this sum for i ¼ j ¼ k: Using Eq. (A.3), they are X X X @ Prðn0 jDÞ i PrðDÞ Qðni ; 0Þ @zc i D n0

X @fp ðn; n0 Þ Qðn; cÞ @zc c¼ ;0

i

Prðn0 ; n; mjD0 ; DÞ

m

X @fd ðn; n ; mÞ Qðm; cÞ @zc c¼ ;0 0



!

X

PrðD0i jD; n0i Þfij ðD; D0 Þ

D0i

þ op ð1Þ:

ðA:8Þ

From Eq. (A.2) this is X X X aðn0 Þ Sf ¼ PrðDÞ Prðn0 Þ D n0 D0

PrðD0i jD; n0i Þfij ðD; D0 Þ

D0i

¼

X aðn0 Þ X X @fp ðn; n0 Þ ¼ PrðnÞ uðn0 jnÞ Qðn; cÞ 0 Prðn Þ n @zc n n0 c¼ ;0 ! X X X @fd ðn; n0 ; mÞ 0 þ Prðn ; mjnÞ Qðm; cÞ @zc n m c¼ ;0

which yields Eq. (26) in the main text.

X



ðA:9Þ

X

þ op ð1Þ;

ðA:13Þ

aðn0i ; D0 Þ nd Prðn0i jD0 Þ

PrðD0i jD; n0i Þfij ðni ; n0i Þ

D0i

m

! X @fd ðn; n0 ; mÞ Qðm; cÞ þ op ð1Þ; @zc c¼ ;0

aðn0i ; D0 Þ : nd Prðn0i jD0 Þ

As for the fij ’s, we consider that Prðn0i jDÞ; which is function of all z phenotypes, can be written as Prðn0i jni Þ þ Oðd2 Þ þ op ð1Þ; where Prðn0i jni Þ is a function of the zi only. Since X

X @fp ðn; n0 Þ Prðn0 ; n; D0 jDÞ Qðn; cÞ  @zc n c¼ ;0 X X þ Prðn0 ; n; m; D0 jDÞ X

n

Similar arguments are used to evaluate the remaining term SPr of the fitness measure, contributed by the derivatives of transition probabilities between demographic states. These terms are

D

X aðn0 ; D0 Þ PrðD0 jDÞ Prðn0 jD0 Þ n0

Prðn0 ; njD0 ; DÞ

139

aðn0i Þ þ op ð1Þ nd Prðn0i Þ

¼ fij ðni ; n0i Þ

aðn0i Þ þ op ð1Þ; nd Prðn0i Þ

ðA:14Þ

ðA:15Þ

expression (A.13) simplifies to X X @ Prðn0 jnÞ PrðnÞQðn; 0Þ fp ðn; n0 Þ @zc n n0

ðA:10Þ 

aðn0 Þ þ op ð1Þ: Prðn0 Þ

ðA:16Þ

ARTICLE IN PRESS F. Rousset, O. Ronce / Theoretical Population Biology 65 (2004) 127–141

140

This is expression (27) in the main text. Hence it remains to show that all terms of Eq. (A.12), other than those for i ¼ j ¼ k; vanish in the limit. Consider the effects on the reproductive value of nondispersing offspring of the focal individual through effects on the size of other demes (k not the deme of the focal individual; kaj ¼ i in Eq. (A.12)). By the same argument leading to Eq. (A.15), these terms are X

PrðDÞ

X

Qðni ; 0Þ

i

D



X

k

fp ðni ; n0i Þ

n0i

X X @ Prðn0 jDÞ k @z c 0 kai n

aðn0i Þ þ op ð1Þ: nd Prðn0i Þ

ðA:17Þ

P Since n0 @ Prðn0k jDÞ=@zc ¼ @1=@zc ¼ 0; the main sum is k null, so the whole expression is op ð1Þ: Finally consider the terms for dispersing offspring, jai: For caj in Eq. (A.12), these terms vanish since QjcjD is Oð1=nd Þ: For c ¼ j; if kai; we can use P 0 n0k @ Prðnk jDÞ=@zc ¼ @1=@zc ¼ 0 to show that these terms are op ð1Þ (as in Eq. (A.17)). Finally, for k ¼ i the terms are X

PrðDÞ

X X i

D



X

jai

QjjjD

X @ PrðN 0 ¼ n0 jDÞ i i @z j 0 n i

PrðD0i jDÞfij ðD; D0 Þ

D0i

aðn0i ; D0 Þ : nd Prðn0i jD0 Þ

ðA:18Þ

We assume that when nd -N; each individual cannot have a fixed (i.e. Oð1Þ) effect on the demography of an increasing number of demes. Hence for jai (the only difference with expression (A.13)), @ Prðn0i jDÞ=@zj ¼ op ð1Þ; and the whole expression is op ð1Þ:

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