Kin Selection and Natal Dispersal in an Age ... - ScienceDirect.com

Theoretical Population Biology 58, 143159 (2000) doi:10.1006tpbi.2000.1476, available online at http:www.idealibrary.com on

Kin Selection and Natal Dispersal in an Age-Structured Population Ophelie Ronce Institut des Sciences de l'Evolution, CC 65, Universite Montpellier II, Place Eugene Bataillon, 34095 Montpellier Cedex 5, France; and Section of Integrative Biology C0930 University of Texas, Austin, Texas 78712

Sylvain Gandon Laboratoire d'Ecologie, CNRS-UMR 7625, CC 237, Universite Pierre et Marie Curie, 7, quai Saint Bernard, 75252 Paris Cedex 05, France; and Centre d ' Etudes sur le Polymorphisme des Microorganismes, UMR CNRS-IRD 9925, IRD, 911, Avenue Agropolis, 34032 Montpellier Cedex 1, France

and Francois Rousset Institut des Sciences de l'Evolution, CC 65, Universite Montpellier II, Place Eugene Bataillon, 34095 Montpellier Cedex 5, France Received July 19, 1999

We examine the effect of iteroparity on the evolution of dispersal for a species living in a stable but fragmented habitat. We use a kin selection model that incorporates the effects of demographic stochasticity on the local age structure and age-specific genetic identities. We consider two cases: when the juvenile dispersal rate is allowed to change with maternal age and when it is not. In the latter case, we find that the unconditional evolutionarily stable dispersal rate increases when the adult survival rate increases. Two antagonistic forces act upon the evolution of age-specific dispersal rates. First, when the local age structure varies between patches of habitat, the intensity of competition between adults and juveniles in the natal patch is, on average, lower for offspring born to older senescent mothers. This selects for decreasing dispersal with maternal age. Second, offspring born to older parents are on average more related to other juveniles in the same patch and they experience a higher intensity of kin competition, which selects for increasing dispersal with maternal age. We show that the evolutionary outcome results from a balance between these two opposing forces, which depends on the amount of variance in age structure among sub-populations. ] 2000 Academic Press

Key Wordsy ESS; dispersal; kin selection; senescence; age structure; demographic stochasticity.

habitat, at least half of the progeny of a given mother should disperse out of the maternal site. This result crystallized the idea that dispersal could be viewed as an altruistic trait and that its evolution could be understood in the light of kin selection theory. Since then, several theoretical elaborations of their model have generalized

INTRODUCTION The importance of kin competition as a force driving the evolution of dispersal in stable habitats was first established by Hamilton and May (1977). In a simple model, they found that, in a persistent and uniform

143

0040-580900 K35.00 Copyright ] 2000 by Academic Press All rights of reproduction in any form reserved.

144 the study of kin selection processes and dispersal to different situations with more complicated habitat structure and population dynamics (Comins et al., 1980; Comins, 1982; Motro, 1982, 1983; Frank, 1986; Taylor, 1988; Crespi and Taylor, 1990; Ozaki, 1995; Ezoe and Iwasa, 1997; Gandon and Michalakis, 1999; Gandon and Rousset, 1999). Some quantitative predictions generated by these models have been tested empirically in colonial thrips (Crespi and Taylor, 1990) and gall aphids (Ozaki, 1995). Almost all these models however considered only semelparous organisms with an annual life cycle. For iteroparous species with more complex life cycles, the robustness and accuracy of the previous predictions have been little explored. We can identify two reasons that might explain the apparent lack of interest for complex life histories in previous theoretical studies of dispersal and kin selection. First, preliminary results in simple cases have suggested that the same dispersal rates evolved in semelparous and iteroparous species (Hamilton and May, 1977; Comins, 1982). In their seminal paper, Hamilton and May (1977) considered a perennial life cycle and showed that evolution of dispersal was unaffected by the adult survival rate as long as survival probabilities did not change with age. The same conclusion was reached by Comins (1982) using a stepping stone model of migration. As Hamilton and May (1977), Comins assumed a single adult per site and that adult survival rates were constant. In both models, adults were resident and competitively superior to juveniles. The latter could establish only in living sites freed by the death of adults. Because of the site retention by adults, the probability of successful establishment of a philopatric juvenile in its natal site is decreased by a factor corresponding to the survival probability of its mother. However, as all adults in the population have the same survival probability, the probability of recruitment for any migrant juvenile is decreased by the same factor. The habitat is then completely homogenous with respect to recruitment probability. As a result, adult survival cancels out of the analysis, A first challenge to the robustness of these results is when adult survival rates change with age. Senescence then introduces some spatial heterogeneity in a formerly uniform habitat, since the probability of recruitment differs in sites occupied by young and old adults. If dispersal strategies cannot change with maternal age and there is only one mother per site, this has no consequence however (Ronce et al., 1998). On the other hand, if individuals are able to express a plastic dispersal behavior conditional on maternal age, then the offspring dispersal rate is predicted to decline when the mother grows old (Hamilton and

Ronce, Gandon, and Rousset

May, 1977; Ronce et al., 1998). This prediction was confirmed in the common lizard, Lacerta vivipara, where old mothers produce a greater fraction of philopatric offspring than do young ones (Ronce et al., 1998). Therefore preliminary conclusions of these studies are that, first, when dispersal is not conditional on maternal age adult survival has no influence on the outcome of evolution. Second, when dispersal changes with maternal age, evolution of dispersal depends on age-specific survival probabilities. When adult survival rates decline with aging, dispersal is then predicted to decrease with maternal age. However, these results were derived in the particular case where a single adult occupies each living site or territory. In numerous species, individual territories overlap so that several adults share, at least in part, the same home range and the same resources. Natal dispersal can also occur between groups of individuals or sub-populations rather than between individual living sites. The robustness of the previous predictions is questionable in these cases. First, the effect of adult survival on relatedness within such groups of individuals is unknown. Second, the probability of local recruitment for a philopatric juvenile now depends not only on the probability that its parent dies but also on the probability that any adult in the group dies. Maternal age may then convey little information about the expected recruitment success in the local sub-population. The second reason why iteroparous life histories have been little studied in a kin selection context is a technical one. Incorporation of iteroparous life histories in kin selection models of dispersal evolution involves considering populations that are structured both genetically and in age classes. Theoretical treatment of kin selection in populations structured in finite groups with several classes of individuals is even more complicated because the finite size of local groups has stochastic consequences on both the distribution of genes and the local age structure. This second source of stochasticity (demographic stochasticity), which generates variance in the local age structure of local sub-populations, has not been yet considered explicitly in existing theoretical frameworks. In this paper, we use recent theoretical developments (Taylor and Frank, 1996; Rousset, 1999) to study the evolution of dispersal in an iteroparous species living in a spatially structured population. We assume that the habitat is stable, adults live in persistent discrete groups containing a finite number of individuals, but they can now survive several years. Because of demographic stochasticity, local sub-populations vary in their age structure. We also contrast this situation with a situation in which all sub-populations have the same age structure. We consider both the cases where dispersal is a fixed

145

Natal Dispersal and Maternal Age

character (no maternal age effect) and where it varies plastically as a function of maternal age (age-specific dispersal). Our questions are the following. Does the evolutionarily stable (ES) dispersal rate change with adult survival when dispersal is a fixed unconditional strategy? What are the patterns of variation in dispersal when it is conditional on maternal age? We proceed in a two-step argument. First, we give general conditions for evolutionary stability of the different dispersal strategies. ES dispersal rates appear to depend on various probabilities of genetic identity. A general method for computing these identities is given. Second, we explore numerically a simple case with two age classes. Results concerning genetic identities when dispersal is fixed at the ESS value for a semelparous life cycle are presented for their heuristic value. Then we let both genetic identities and dispersal rates evolve jointly. We find that previous predictions derived in the particular case when there is a single adult per site are not entirely robust. When unconditional, the ES dispersal rate increases slightly with the adult survival rate. When dispersal is conditional on maternal age, the prediction that dispersal rates decline with maternal age still holds for a large range of the parameter space. But the reverse trend can also occur. In particular, in the absence of variance in local age structure, old mothers always disperse a larger fraction of their progeny than do young ones. This may occur also when there is variance in local age structure. The effects of adult survival on genetic structure and in particular the higher level of kin competition experienced by offspring born to old parents explain these non-trivial results.

ASSUMPTIONS OF THE MODEL The Life Cycle The life cycle can be partitioned into four steps: (1) reproduction, (2) dispersal, (3) adult survival, and (4) competition among juveniles and recruitment. The life cycle is shown in Fig. 1 for a simple case where adults survive only 2 years. Reproduction. Reproductive maturity is attained after 1 year. All adults have the same fecundity f. For the sake of simplicity, we assume that individuals are haploid females that reproduce asexually. Dispersal. Only juveniles disperse between sub-populations. Individuals are born with a certain probability to disperse out of their natal site to attempt to establish

FIG. 1. Schematic representation of the life cycle in the simple case where there are only two age classes. At time t, the local sub-population is initially of type i with n1i adults aged 1 and n 2i adults aged 2. After reproduction, juveniles are present in the sub-population. They are issued from adults aged 1 and 2. Some fraction of these juveniles emigrate while other juveniles immigrate. All adults aged 2 die. A fraction of the adults aged 1 survive to become adults with age 2. A number of juveniles corresponding to the number of vacant sites freed by the death of adults then survive and reach the age of 1. The local subpopulation is now of type k, with n 1k young adults and n 2k old adults.

somewhere else. This predisposition may depend on the genotype of the individual, on the maternal phenotype (the age of the mother), and their interaction. Dispersal entails a cost c, which is expressed through the survival of dispersed offspring (1&c) relative to the survival of a resident offspring during the period between birth and the settling phase. We assume that migrants are distributed evenly over space (an island model of migration). Adult Survival. The probability of surviving from age a to age a+1 is s(a). Surviving adults die eventually at the maximal age a max . The number of survivors in each age class is drawn randomly according to the age-specific survival probability of the class. Competition and Establishment of Juveniles. We assume that adults are competitively superior to juveniles and that a juvenile can successfully establish at one site only if the adult that previously occupied this site has died. Juveniles compete for each site freed by the death of an adult, and one offspring is chosen at random among the young present locally to establish at this site. Juveniles that fail to establish die. The Habitat We consider a fragmented habitat inhabited by a population made of an infinite number of sub-populations. Within each sub-population, there are n living sites and adults produce a large number of offspring, so that no

146 living site is ever vacant. Each year, every sub-population contains exactly n reproductive adults. However, because of the finite size of the sub-populations, the number of adults that survive within a subpopulation varies stochastically. Thus, as a result of demographic stochasticity, local sub-populations vary in their age structure (but not in their size). Local sub-populations that have the same age structure are said to be of type k, described by the vector n k =(n 1k , n 2k , ..., n ak , ...), where n ak is the number of individuals with age a in the sub-population. So to each sub-population type corresponds a different age structure.

EVOLUTIONARILY STABLE DISPERSAL RATES In the following we give necessary conditions for evolutionarily stable dispersal strategies (ESS, Maynard-Smith, 1982). First, we give a general condition for evolutionary stability of a reaction norm in a class-structured population. These conditions require the computation of several fitness components. Second, we explain definitions used in this computation and derive the expression of the fitness components under the assumptions of our model. Finally, we give a necessary condition for the ES dispersal rates. Different subcases are considered depending on the occurrence of maternal age effects and on the existence of variance in age structure between local subpopulations. By comparing the case with and without variance in local age structure, we aim at identifying the selective forces associated with this form of stochasticity. Evolutionary Stability in a Class-Structured Population We first assume that the dispersal rate of an offspring may change with its mother's age. We are interested in the evolution of age-specific dispersal rates and look for the evolutionarily stable reaction norm for dispersal z=(z 1 , z 2 , ..., z l , ...), where z l is the age-specific dispersal rate of offspring born to a parent of age l. The evolutionarily stable (ES) reaction norm is such that the dispersal rate z l at each age l is evolutionarily stable. In the present system, there exist several classes of individuals. Each individual is characterized by its age, a, and the type of sub-population to which it belongs, k. In a class-structured population, relevant measures of fitness cannot be simply computed as the expected number of offspring per individual. Instead, predicting change in allelic frequency entails considering the


different classes of individuals, their respective frequency, how they contribute to other classes of individuals, and what is the reproductive value of those new individuals (Taylor, 1990). Following Taylor and Frank (1996), a condition for evolutionary stability of z l in a population structured with different classes of individuals is W w ak Â ai =: : : : v a, k u a, i =0, z l z l k a$ i a

(1)

where w a$k Â ai is the contribution of individuals with age a inhabiting sub-populations of type i to the production of individuals with age a$ in sub-populations of type k. Those are weighted by the reproductive value v a$, k of an individual with age a$ living in a sub-population of type k. Reproductive values are computed as the terms of the left eigenvector associated with the largest eigenvalue of the demographic projection matrix M=(w a$k Â ai ). Different classes of individuals are also weighted by their frequency u a, i . The frequency in the whole population of individuals that have age a and live in a sub-population of type i can also be written u a, i =u v i

n ai , n

(2)

with u vi the frequency of sub-populations with type i among all sub-populations in the population, and n ai is the number of individuals with age a in a sub-population of type i among the n total reproductive adults in this sub-population. Note that Eq. (1) and the following results in the present paper give necessary but not sufficient conditions for evolutionary stability. Furthermore those conditions do not guarantee that the ESS is convergent stable. We now give an expression for the components of fitness w a$k Â ai , the elements of M. Components of the Fitness Function Reproduction. We first compute the contribution of individuals with age a in sub-populations i to the production of new individuals in sub-populations of type k (transitions corresponding to reproduction events). This contribution is written w 1k Â ai =(1&z a )

n 1k n(1&z +(1&c) z v v) p

iÄk a vi

+z a (1&c) : u v j j

p j Ä k n 1k . n(1&z vj +(1&c) z v v) (3)

147


All other transitions between classes of individuals have a null probability. Maternal Effecty Evolutionarily Stable Age-Specific Dispersal Rates

FIG. 2. Focal individuals and different neighbors in the metapopulation. z%ak is the expected phenotype of a focal juvenile born to a mother of age a in a sub-population of type k. z abk is the expected dispersal phenotype of a juvenile born to a mother of age b neighboring the previous focal individual. z avk is the average dispersal phenotype of neighbors of this focal individual, averaging over all classes of juveniles. z v j and z v v are the average dispersal phenotypes of juveniles respectively in a sub-population j and in the whole population.

The first term on the right-hand side of Eq. (3) corresponds to philopatric offspring born to individuals aged a in a sub-population of type i, where z a is the dispersal rate of an offspring born to such parents. The second term in Eq. (3) corresponds to surviving dispersed offspring, where c is the dispersal cost. Dispersed offspring immigrate into a type j sub-population with probability u v j . Philopatric and dispersed offspring compete with other juveniles, whose total number is, respectively, nf (1&z av i )+nf (1&c) z vv for the philopatric offspring, and nf(1&z vj )+nf(1&c) z vv for the dispersed offspring immigrating in a type j sub-population. The dispersal rate z avi stands for the average dispersal rate of offspring born in the same patch as our focal juvenile, while z v i and z v v are respectively the average phenotype of juveniles in all sub-populations of type i and in the whole population (see Fig. 2). The probability that a type i subpopulation becomes a type k sub-population after adult survival is noted p i Ä k . Finally, once a sub-population has become of type k, the number of sites available for juvenile recruitment is n 1k . Survival. We now compute the contribution of individuals aged a in sub-populations of type i to individuals aged a+1 in sub-populations of type k (transitions corresponding to survival events). The expected number of surviving (a+1, k) individuals per (a, i) individual is w a+1k Â ai = p i Ä k

n a+1k , n ai

(4)

Variance in Local Age Structure. As shown previously, the survival and fecundity of a focal individual may depend on its own dispersal strategy, as well as on the dispersal behavior of its neighbors. Small sub-population sizes are likely to generate strong genetic correlation among neighbors in the same sub-population. Because of genetic similarity, the dispersal behavior of a focal individual is likely to be correlated to the dispersal behavior of its neighbors. The effect of change in dispersal behavior at age l on the fitness components depends on the change in the dispersal of each focal individual but it also depends on the correlated change in the dispersal of its related neighbors, as in Taylor and Frank (1996). The full proof for the ESS condition can be obtained from the authors upon request. Using Eq. (1) and the expression for the fitness components, the ESS condition can be written as W =nl v (1&c) g v &: u vi n li gi 1&hi : .ai Q$ali =0, zl i a

\

+

(5) where n l v = i u v i n li is the average number of individuals aged l per sub-population, averaged over all sub-population types; h i =(1&z v i )(1&z v i +(1&c) z vv) is the probability that a given juvenile competing in a sub-population of type i is native to the site where it now attempts to establish; . ai =(1&z a ) n ai (1&z vi ) n is the probability that a philopatric juvenile was born to a parent aged a in the sub-population of type i; and Q$ali is the probability that genes in two juveniles born in the same sub-population of type i, respectively to a parent aged a and l, are identical in state. Probabilities of identity among juveniles are computed in the Appendix. The variables g i and g v are defined as follows. Let F i = k p i Ä k n 1k v 1, k be the expected number of new recruits knowing that a sub-population is of type i at the moment of reproduction, weighted by the reproductive value of such new recruits. Then g i =F i n(1&z v i + (1&c) z vv) is the expected contribution of each juvenile present in the sub-population of type i to these new recruits weighted by their reproductive value (after competition). Similarly, g v = i u v i g i is the average

148


contribution of a juvenile to new recruits, averaged over all sub-population types. The ESS condition for x can be rearranged and is equivalent to for all l, : u vi i

n li g 1&h i : . ai Q$ali =(1&c) g v . nl v i a

\

+

(6)

The left-hand term in Eq. (6) corresponds to the expected gain in inclusive fitness conveyed by the production of an extra philopatric juvenile born to a parent aged l. The right-hand term in (6) corresponds to the expected gain in inclusive fitness conveyed by the production of an extra migrant juvenile. The marginal gain for philopatric offspring depends on the distribution of individuals with age l across different sub-population types and on the expected probability of establishment and expected reproductive value in each type of sub-population. But this gain is devalued by the intensity of kin competition for offspring born to a parent aged l in each type of subpopulation. For migrants, the gain depends simply on the cost of dispersal and on the average probability of establishment, weighted by reproductive values. Note that the marginal gain through the production of migrants is the same for all classes of parents. No Variance. We also consider the case when there is no variance in the age structure of local sub-populations. By contrasting our results in the presence and absence of variance, we will be able to better understand the selection pressures acting on age-specific dispersal rates. In the absence of variance, the number of individuals in each age class is fixed (for instance, every year in every sub-population there are exactly four individuals aged 1 and two individuals aged 2). This implies that the survivals of different individuals are now not independent. The survival rate is now the frequency of surviving individuals in each sub-population, not simply its expectation (in the previous example the survival rate from age 1 to age 2 is in fact a survival frequency, which necessarily equals 0.5 in each sub-population). In the absence of variance among populations, the general condition for evolutionary stability reduces to for all l,

: a

na v (Q$al &c 2 ) z a =Q$l v &c, n

(7)

where Q$al is the probability of identity between two juveniles born in the same sub-population, respectively to a parent aged a and l, and Q$l v = a (n a v n) Q$al is

the probability of identity between a juvenile born to a parent aged l and another random juvenile born in the same sub-population, before migration. Thus in the absence of local variation in age structure, age-specific dispersal rates depend on the age-specific probabilities of genetic identity and on the dispersal cost, but no more on the local probability of establishment and reproductive values. No Maternal Effecty Evolutionarily Stable Dispersal Rate We now consider a case where there is no maternal effect on dispersal. In this case, juveniles of a given genotype express the same dispersal phenotype whatever the age of their mother: for all l, z l =z. The condition for evolutionary stability in the case with no maternal effects can be obtained simply by summing the partial derivatives of fitness with respect to age-specific dispersal rates: : l

dW =0. dz l

(8)

Using Eqs. (5) and (8), we found that the ES dispersal rate was z=

\&c \&c 2

with \=

i u v i F i Q$vi v , i u vi Fi

(9)

where Q$vi v= a b (n ai n)(n bi n) Q$abi is the average identity between two juveniles drawn at random among those born in a sub-population of type i before dispersal. In the case where there is no variance in age structure this expression reduces to z=

Q$v v&c , Q$v v&c 2

(10)

where Q$v v= a b (n a v n)(n b v n) Q$ab . Note that the above condition for evolutionary stability generalizes the classical result obtained by Frank (1986) for semelparous species: z=(R&c)(R&c 2 ), with R=Q$v v the coefficient of relatedness between two offspring born in the same sub-population (here it reduces to the probability of identity between two such offspring). When local populations vary in their age structure, Eq. (9) indicates that the coefficients of relatedness must be weighted by the reproductive values of offspring born in different types of sub-populations. The previous analytical results suggest that the agespecific ES dispersal rates will depend on the age-specific probabilities of identity among juveniles. However, the

149


probabilities of identity depend themselves on the agespecific dispersal rates in the resident population, so that dispersal rates and coefficients of genetic identity evolve jointly. The same argument applies for the reproductive values of the different types of juveniles, which evolve with the age-specific dispersal rates. A simple explicit analytical expression for the ES dispersal rates cannot be derived in general. Instead, we use numerical methods to obtain quantitative predictions for the candidate ESS. Starting from an initial dispersal strategy, we compute the probabilities of identity among the different classes of juveniles as indicated in Appendix. The reproductive values of each type of juvenile are computed numerically as the terms of the left eigenvector of the demographic projection matrix M. We then replace the identities and reproductive values in the ESS condition by their computed value and we solve numerically the corresponding system of equations. Candidate ESS values for the agespecific dispersal rates are then obtained. Probabilities of identity and reproductive values are computed again for these new dispersal rates. This process is repeated until age-specific dispersal rates converge to a constant value. For some parameter sets, we compared the predicted age-specific ES dispersal rates to the results of an individual-based simulation model, simulating explicitly the processes of drift and mutation. Standard errors of estimates were obtained by the batching method for Markov chains (Hastings, 1970). Computation time increases rapidly with the size of sub-populations and the number of age classes. For instance, when there are 20 adults per sub-population and individuals survive for only 2 years, one must compute 63 different identities at each step of the calculation, and this number rises to 1386 when individuals survive for only 3 years. For this reason we will restrict our numerical analysis to a simple case with only two age classes.

A SIMPLE CASE WITH TWO AGE CLASSES In the following, we assume that adults survive only 2 years and die after reproduction at the end of their second year. Individuals with age 1 have a probability s(1)=s of surviving to age 2, while the probability of survival of individuals aged 2 is zero. Both adults aged 1 and 2 reproduce. Though extremely simple, this life cycle presents the two properties, iteroparity and senescence, whose consequences we want to investigate for the evolution of dispersal.

We adopt the following convention: a sub-population of type i contains exactly i old individuals aged 2 after recruitment and n&i young recruits aged 1. The probability that a sub-population of type i becomes the following year a sub-population of type k, with k old individuals, corresponds to the probability that k individuals among the n&i adults aged 1 survive to age 2. This probability is given by the binomial distribution pi Ä k =

n&i

\ k + s (1&s) k

n&i&k

.

(11)

Accordingly, the frequency at equilibrium of sub-populations of type i can be shown to follow the binomial distribution u vi=

n i

s 1+s

i

1 1+s

\ +\ + \ +

n&i

.

(12)

Note that the variance in the proportion of old individuals per sub-population, i.e., the variance in local age structure, is then simply sn(1+s) 2, which is maximal, for a given n, when s is close to one. Probabilities of Identity In this section, our aim is to understand, first, whether genetic structure is different in an iteroparous species as compared to a semelparous species and, second, whether this different genetic structure is responsible for different selection pressures acting on the evolution of dispersal in an iteroparous species. Here, we address the first question; i.e., we compare the genetic structure (the probabilities of identity) in semelparous and iteroparous species, when the dispersal rate does not evolve. In order to perform meaningful comparisons, we assume no maternal effects on dispersal, and a dispersal rate equal at each age to the ES dispersal rate for a semelparous species living in the same habitat (Frank, 1986). We first consider the extreme case where there is no variance for age structure among sub-populations. These results are then compared to the case where the variance for age structure depends on the size of the populations, n, and on adult survival rate, s. For each case we consider the effect of adult survival on (i) the probabilities of identity among adults, (ii) the probabilities of identity among juveniles; and (iii) the average probabilities of identity among juveniles. No Variance in Age Structure. In Fig. 3A, identities among different classes of adults are shown as a function of the adult survival rate s, which actually corresponds to

150


FIG. 3. Probabilities of identity among different types of individuals versus adult survival rate, s. The curves give the results when there is some variance in age structure among populations. The symbols refers to the situation when there is no variance in age structure among populations. (A) Identity among adults. Thick solid line and closed circles: probability of identity between two adults of the same age (Q 22 =Q 11 ). Thin solid line and open triangles: probability of identity between two adults aged respectively 1 and 2 (Q 12 ). (B) Identity among juveniles. Thick solid line and closed circles: probability of identity between two juveniles born to parents aged 1 (Q$11 ). Dotted line and open circles: probability of identity between two juveniles born to parents aged 2 (Q$22 ). Thin solid line and open triangles: probability of identity between two juveniles born to parents aged respectively 1 and 2 (Q$12 ). (C) Average identity among juveniles. Thick solid line and closed circles: the average probability of identity between a juvenile born to a parent aged 1 and a random juvenile in the same sub-population before migration, Q$1 v . Dotted line and open circles: probability of identity between a juvenile born to a parent aged 2 and a random juvenile in the same sub-population before migration, Q$2 v . Thin solid line and open triangles: average probability of identity between two juveniles drawn at random in the same sub-population Q$v v.

151


the survival frequency given the age structure. These probabilities were computed as described in the Appendix. In the absence of variance in the age structure and for this simple life cycle, the probabilities of identity satisfy Q 11 =

h 2 h 2n 1 v (n 1 v &1) + Q 11 n n2 +

h 2n 2 v (n 2 v &1) h 2n 1 v n 2 v Q 22 +2 Q 12 2 n n2

(13)

h h(n 1 v &1) hn 2 v Q 12 = + Q 11 + Q 12 n n n

(14)

Q 22 =Q 11 ,

(15)

with h=(1&z vv)(1&cz v v) the proportion of philopatric juveniles in a sub-population. The probability of identity between adults belonging to the same age class Q 22 =Q 11 is lower than the probability of identity between adults belonging to different age classes Q 12 (Fig. 3A). This result may be explained because adults do not migrate. The probability that genes present in two adults aged 1 coalesce in the same parent the year before is lower than the probability that genes present in adults aged 1 and 2 coalesce the year before. Indeed, in the absence of maternal effects on dispersal, these probabilities of coalescence are respectively h 2n and hn. In the latter case, coalescence requires that the adult aged 1 was not an immigrant, while in the former, it requires that both adults aged 1 were not immigrants. When the number of old adults per sub-population increases, due to increasing adult survival s, both types of identity increase (Fig. 3A). This is not directly due to a change in the probabilities of coalescence since these probabilities are here independent of s. Rather, when old adults are more numerous, identities between young and old adults have a larger weight in the computation of other identities (see Eqs. (13) and (14)). As such identities are always larger than those between adults in the same age class, it results in a net increase in genetic identity. In conclusion, increasing adult survival limits juvenile recruitment. Site retention by adults limits gene flow between sub-populations since only juveniles disperse. This allows the building of high genetic identity among adults in the same sub-population. The probabilities of identity among different classes of juveniles can be derived from the identities among adults, as indicated in the Appendix. These identities are shown in Fig. 3B as a function of the adult survival rate s. The probability of identity among juveniles born to parents with the same age Q$11 or Q$22 is higher than that among juveniles issued from parents in different age classes Q$12 .

Indeed, in the first case, the two individuals may have the same parent, while coalescence in a single generation is impossible in the second case. Identity among juveniles born to old adults is generally higher than that for juveniles born to young parents. This happens because old individuals are always rarer than young adults. At most old adults represent 50 0 of the population when s=1. Therefore the probability that two juveniles born to old individuals have the same parent is higher than the same probability for two juveniles born to young adults. The probability that two juveniles born to old parents have the same parent decreases when s increases (and the number of potential old parents increases). This explains why identity among juveniles born to old parents decreases when s increases, while the identity between juveniles born to young parents increases. Fig. 3C shows the average probability of identity respectively between a juvenile born to a parent aged 1 and a random juvenile in the same sub-population before migration, Q$1 v , and between a juvenile born to a parent aged 2 and a random juvenile in the same sub-population before migration, Q$2 v . On the same figure is indicated the average probability of identity between two juveniles drawn at random in the same sub-population Q$v v. Juveniles born to young parents are less related to other juveniles in the same sub-population than are juveniles born to old parents. Differences between the identities for juveniles born to young and old individuals can be expressed as a function of the probability of identity among adults:

Q$2 v &Q$1 v =

n 1 v &n 2 v (Q 12 &Q 11 ). n

(16)

This quantity is always positive because young adults are more numerous than old adults and because the probability of identity among adults in different age classes is higher than the probability of identity among adults with the same age, as shown previously. Ultimately, the fact that juveniles born to old parents are more related to other juveniles in the same sub-population than juveniles born to young parents is due to the fact that the parents of the former have been reproducing for several years in that same sub-population without migrating. Equation (16) also reveals that the difference between Q$1 v and Q$2 v decreases when s increases because n 2 v gets closer to n 1 v . Eventually, if n 2 v =n 1 v , which may occur for s=1, the two probabilities of identity are equal. The relatedness of juveniles born to young parents to other juveniles in the same sub-population always

152 increases when s increases. This is also often the case for juveniles born to old parents. This is, in part, due to the increase in relatedness among the different classes of adults, as shown previously. As a result, the genetic identity among juveniles in the same sub-population Q$v v increases when the adult survival rate increases. Variance in Age Structure Among Local Sub-populations. Now the simplifying assumption concerning the absence of variance in age structure is relaxed. We use numerical methods to evaluate the matrix in Eq. (A12) and compute the identities for each sub-population type (see Appendix). Identities among the different classes of individuals are then averaged over the different population types. Figure 3 shows that the variation in age structure among populations does not affect qualitatively the results presented above. Note, however, that the variance in age structure tends to decrease the values of all the identity coefficients. Consider, for instance, a situation where there are exactly four young adults and one old adult per sub-population. The probability of identity between two juveniles born to an old adult in the same sub-population is then necessarily one. The average genetic identity between two such offspring is well below this value when the local age structure fluctuates. This can be easily understood as there will now exist some sub-populations with a greater number of old adults. The lower genetic identity between offspring born to old parents in these sub-populations will thus decrease the mean. More generally, when the local age structure fluctuates, two juveniles born to a given class of parents are more likely to be in a sub-population where parents of that class are abundant than in a sub-population where they are rare. This decreases the average probability of coalescence between these two juveniles, and consequently decreases all probabilities of identity.

Evolutionarily Stable Dispersal Rate In the following we use the conditions for evolutionary stability given in Eqs. (6), (7), (9), and (10) as well as the above results on identity coefficients to find the evolutionarily stable dispersal rates. Different cases are considered, depending, first, on the ability to evolve age-specific dispersal rates (i.e., maternal age effects) and, second, on whether we assume that there is some variation in age structure among populations. Results of individualbased simulations are also shown in Fig. 4: they are consistent with our theoretical expectations when the variance in local age structure is taken in account.


No Maternal Effect Fig. 4A shows that a higher survival rate increases the evolutionarily stable dispersal rate slightly. This result can be explained by our previous findings regarding the identity coefficients. A higher survival rate decreases the recruitment rate and, as a consequence, it decreases the level of gene flow between sub-populations since adults do not disperse. Lower gene flow increases the average level of relatedness within sub-populations and selects for higher dispersal rate, as seen in Eqs. (9) and (10). The variance in age structure does not affect this result qualitatively and the results, with or without variance in age structure, are almost indistinguishable. Note, however, that the ES dispersal rate is slightly higher when there is no variation in age structure. This result can be explained by the differences noted on the identity coefficients. In the absence of variance in age structure, identity coefficients tend to be higher, which selects for higher dispersal rates. Not surprisingly, a larger population size also selects for a lower ES dispersal rate as seen in Fig. 4. Indeed, a larger population size always decreases relatedness and, consequently, selects against dispersal, as previously shown by Comins et al. (1980), Frank (1986), and others.

Age-Specific Dispersal Rates Contrary to the previous case, the variance of age structure affects greatly the outcome of the evolution of age-specific dispersal rates. No Variance. First, in the absence of variance in age structure, Fig. 4B shows that old mothers always produce more dispersers than young mothers. This difference is maximized for low survival rates and decreases when survival rate increases. At the extreme case when s=1 there is no difference in the ES dispersal rates between young and old mothers. This effect can be fully explained by a kin competition argument. Indeed, agespecific dispersal rates evolve only because an offspring produced by a young mother may experience a different level of kin competition than an offspring produced by an old mother. These levels of kin competition are measured respectively by the average probability of identity between a juvenile born to a parent aged 1 and a random juvenile in the same sub-population before migration, Q$1 v , and between a juvenile born to a parent aged 2 and a random juvenile in the same sub-population before migration, Q$2 v . In the absence of variance in age structure,


153

FIG. 4. Evolutionarily stable dispersal rate versus adult survival rate, s. (A) With no age-specific dispersal strategy. The curves give the results when there is some variance in age structure among populations. The symbols refer to the situation when there is no variance in age structure among populations. (B) Age-specific dispersal rate. The curves give the results when there is some variance in age structure among populations (the bold curve is used for the ES dispersal rate of old females' offspring, and the dotted curve is used for ES dispersal rate of young females' offspring). The symbols refer to the situation when there is no variance in age structure among populations (the solid points are used for the ES dispersal rate of old females, and the open points are used for ES dispersal rate of young females). The ES dispersal rates are presented for different population sizes and the dispersal cost is c=0.1. In the upper right panel (n=10), we also show some estimates of average age-specific dispersal rates obtained in individual-based simulation and their standard errors (bars) for s=0.1, s=0.5, and s=0.9.

154


using Eq. (7), we can obtain an analytical expression for age-specific dispersal rates z 1 and z 2 and show that

z1 =

n (Q$1 v &c)(Q$22 &c 2 )&(Q$2 v &c)(Q$12 &c 2 ) n1 v (Q$11 &c 2 )(Q$22 &c 2 )&(Q$12 &c 2 ) 2 (17)

and

z 2 &z 1 =

n2 c(1&c)(Q$2 v &Q$1 v ) . n 1 v n 2 v (Q$11 &c 2 )(Q$22 &c 2 )&(Q$12 &c 2 ) 2 (18)

The probability of identity among offspring born to parents in the same age class was shown to be higher than the identity between juveniles born to parents with different ages. As a result, the denominator in Eq. (18) is positive and the difference between the dispersal rate of juveniles born to old and young parents has the same sign as Q$2 v &Q$1 v . Since Q$2 v Q$1 v , this difference is always positive. Offspring produced by old mothers experience higher risk of competing with related individuals. This selects for higher dispersal rates in old females' progeny. When s increases the difference in identities decreases as seen previously. This explains why the difference in dispersal rates decreases as well. Note that higher population sizes tend to increase the difference between age-specific dispersal rates (Fig. 4B). The ES dispersal rate of old females decreases only weakly with larger populations while the ES dispersal rate of young females is much more affected by this parameter. Variance in Local Age Structure. Second, when we relax the assumption of the absence of variation in age structure we get very different results. For low survival rates, again, we find that the ES dispersal rate is higher for old females than for young females. However, for high survival rates, we obtain exactly the opposite pattern: young females tend to disperse their offspring more than do old ones. This result is due to the fact that the age of the mother carries two conflicting pieces of information concerning the quality of the sub-populations. First, as above, offspring produced by old mothers experience higher levels of kin competition. This selects for a higher dispersal rate among the offspring of old mothers. Second, the age of the mother may give indirect information concerning the

probability of recruitment in the natal sub-population. Recruitment depends directly on the fraction of old adult females since these females will die and leave empty sites in the sub-populations. When there is some variation in the age structure among sub-populations the age of the mother is correlated with the average age of adults in the sub-population. Indeed an old mother is more likely to be found in a sub-population where old individuals are numerous than in a sub-population where they are rare, relatively to their average frequency. As a consequence, offspring produced by old females will benefit from being philopatric through an extra opportunity to be recruited. This selects for a higher dispersal rate in young mothers. This is the same mechanism as discussed in Ronce et al. (1998), except that maternal age now reflects imperfectly the probability of recruitment in the natal sub-population. When the variance in local age structure decreases, maternal age conveys less and less information about local recruitment probability. The evolution of age-specific dispersal rates results from a balance between these two opposing forces. This balance is very sensitive to the effect of the survival rate. Indeed, for a very low survival rate there is a strong selection for a higher dispersal rate in old mothers' progeny because (i) the difference between Q$1 v and Q$2 v is maximized and (ii) the variance of age structure is minimized (i.e., the variance of age structure increases with survival rate). At the other extreme, when the survival rate is high, there is selection for the reverse pattern because (i) the difference between Q$1 v and Q$2 v is minimized (it vanishes when s=1) and (ii) the variance of age structure is maximized. The reversal of this pattern occurs at intermediate levels of survival. Note that the survival rate at which reversal occurs decreases with lower population size. This is not surprising since lower population size increases the variance in age structure. In the extreme case where there is only one adult female per population the ES dispersal rate of old mothers is always lower than that of young mothers. We verified that numerical values for the age-specific dispersal rates then corresponded to analytical predictions derived independently by Ronce et al. (1998). In Fig. 4B, estimates of average dispersal rates obtained with the individual-based simulations are given for n=10 and for three values of the survival rate. Simulation results are in agreement with the predicted value, except when the predicted value for the agespecific ES dispersal rate is zero. Because of edge effects and recurrent mutations, the average dispersal rate in the simulations cannot indeed be equal to zero. In all simulations, the predicted dispersal rates appeared to be both convergent and evolutionarily stable.


DISCUSSION When each site in a fragmented population is occupied by a single adult, previous predictions (Hamilton and May 1977; Ronce et al., 1998) stated that adult survival rates had no effect on the evolution of unconditional dispersal strategies. Further, when dispersal is conditional on maternal age, dispersal rates are then predicted to decline with maternal age only if adult survival rates declined as well with age. Here we have allowed more than one adult to coexist in the same site and have tested the robustness of the previous predictions. It appears that none of them is entirely robust. First, adult survival does affect the evolution of unconditional dispersal. The ES dispersal rate increases when the survival rate increases. Second, when dispersal may change with maternal age, we find situations where the age-specific dispersal rates increase with maternal age as well as situations where they decrease with maternal age. The discrepancies between present predictions and previous results can be explained by interactions between kin selection phenomena and the age structure.

Kin Selection and Iteroparity For asexual haploid organisms, when only one adult lives in each local sub-population, then the local relatedness among its offspring is one whatever its age. On the contrary, when several adults reproduce in the same subpopulation, the relatedness among juveniles born locally will depend on the age structure in the local sub-population. We have shown that iteroparity and site retention by adults resulted in reduced dispersal per time unit between local groups and thus in a higher level of relatedness among juveniles born in the same local sub-population in comparison to the semelparous case. With the same assumptions as ours, Pen (2000) also showed analytically, in a case with no senescence, that relatedness within patch increased when the adult survival rate increased. This, in turn, selects for higher dispersal rates among juveniles, as an altruistic behavior to avoid the intense kin competition in the natal site. This explains why unconditional dispersal rates increase when the adult survival increases. In the simple case where adults survive for only 2 years, our numerical study shows that the increase in dispersal with survival rate is small. We would expect greater quantitative effects on average genetic identity and dispersal rate if individuals were allowed to survive longer. The fact that adult survival affected inbreeding coefficients and correlations between sites was already pointed out by Comins (1982) but the

155 consequences of this finding for dispersal evolution were not clearly investigated. Iteroparity not only has consequences on the average level of relatedness within local sub-populations, but it also generates differences in genetic identity among different classes of individuals within the same sub-population. We find that juveniles born to old mothers are more related to other juveniles in the same sub-population than are juveniles born to a young mother. Ultimately this result can also be explained by the absence of dispersal in adults and their repeated reproduction attempts in the same site. As a consequence, different levels of kin competition experienced by offspring born to old and young parents select for different dispersal rates depending on maternal age. This explains why, in the absence of variance in age structure between sites, offspring dispersal rates always increase with maternal age. A similar argument was used by Morris (1982). He predicted that, in an iteroparous species, mothers themselves should disperse when they age, because the intensity of competition with their offspring increases with the successive reproductive episodes in the same patch. Note that the previous argument does not require that old adults be senescent and that adult survival rates change with age. Even when adult survival rates are constant, we expect genetic identities among juveniles to be different depending on parental age, and thus that age-specific dispersal rates may potentially evolve. More generally, this simple model opens perspectives on the evolution of altruistic behaviors in age-structured populations. Genetic identities vary depending on which types of individuals are interacting. In particular, we find that adults in the same age class are less related than adults in different age classes. This might favor the evolution of age-specific altruism, so that adults may express a selfish behavior when interacting with other adults of the same age, but be more likely to help younger or older individuals.

Demographic Stochasticity and Life History Evolution The present results also suggest that taking into account demographic stochasticity may alter our predictions concerning life history evolution. Predictions for measures of genetic structure (probabilities of identity) are different when the age structure of local sub-populations is fixed versus when it is allowed to vary randomly due to the stochastic nature of survival events. In particular, we find that all average measures of genetic identity within sub-populations are smaller in the presence

156 of demographic stochasticity. However, differences remain small in many situations. Demographic stochasticity has much stronger effects on the evolution of age-specific dispersal rates. Indeed, strong qualitative and quantitative discrepancies emerged. In particular, when the variance of the age structure is large, we predict that the dispersal rates should decrease with age, contrary to the pattern observed in the absence of variance. Demographic stochasticity, by generating variance between the local sub-populations, thus creates new selection pressures on age-specific dispersal rates. More precisely, when survival rates and dispersal rates change with age, variance in local age structure represents variance in recruitment probability and reproductive values for juveniles. The habitat is now heterogeneous. In particular, recruitment probability is going to be higher in sub-populations containing numerous old adults if survival rates decline with age. Because of the variance of age structure, an old mother is more likely to be in such a sub-population. This favors the evolution of smaller dispersal rates in the progeny of old mothers. More generally, evolution of age-specific dispersal strategies could then be viewed as a strategy of habitat selection, where maternal age is an indirect cue for habitat quality, as measured by the reproductive value of juveniles in this habitat. Conflict between Habitat Selection and Kin Competition Few studies have investigated the combined effects of habitat selection phenomena and kin competition on the evolution of dispersal rates in heterogeneous environments. In particular, the concept of Ideal Free Distribution (Fretwell, 1972) and various models concerned with the evolution of habitat-specific dispersal strategies (McPeek and Holt, 1992) were derived without incorporating interactions among kin. Crespi and Taylor (1990), Ozaki (1995), Ezoe and Iwasa (1997), and Ronce et al. (1998) investigated cases where both kin competition and habitat heterogeneity were driving the evolution of dispersal. In the present model, kin competition and habitat selection represent conflicting forces acting on the evolution of age-specific dispersal rates. Indeed, based on relatedness alone, we would predict that higher dispersal rates should evolve in old mothers' progeny because of differences in age-specific genetic identities. On the contrary, habitat selection arguments predict that older mothers should disperse fewer of their offspring. Maternal age is thus associated with conflicting information, with respect to the probability of establishment and the intensity of competition among kin.


This explains the shift in our predictions when the adult survival rate s increases and the relative force of the two phenomena changes. According to our predictions, we would expect that, in stable habitats, semi-annual plant species that rarely reproduce a second year (low s) should then disperse a greater fraction of their seeds in their second reproductive attempt. The reverse prediction is made for species that regularly reproduce twice before dying (high s). We are also more likely to observe decreasing dispersal with maternal age when the size of local sub-populations is small as when it is large. The prediction of decreasing dispersal with maternal age derived by Ronce et al. (1998) when only one mother lives in a site is fairly robust as long as organisms live in small groups. More generally, our results suggest that we are not to expect a single pattern of variation for dispersal traits as a function of maternal age in natural populations. The relationship between dispersal and age has been little investigated empirically. Juvenile dispersal decreases with maternal age in the common lizard Lacerta vivipara (Ronce et al., 1998), in the perennial plant Cistus ladanifer (Acosta et al., 1997), and with grand-maternal age in the pea aphid Acyrtosiphon pisum (MacKay and Wellington, 1977). However, in the last case, dispersal was also found to sometime increase with grand-maternal age. We have provided a first theoretical approach to incorporate kin selection phenomena in an age-structured subdivided population and predict the evolution of agespecific dispersal rates. We found that predictions were indeed very sensitive to assumptions about the life cycle and demographic stochasticity. However, taking into account these factors requires manipulating an increasing number of different variables (the different types of genetic identities), which severely limits the range of situations that we can explore numerically, by using this method. A central assumption in the present model is that adults do not disperse. In the case of mobile adults, we do not expect the same trends. Both the relationship between maternal age and the intensity of kin competition and the relationship between maternal age and recruitment probability would then be different. Senescence is likely to have effects on both survival and fecundity. The effect of variation in fecundity with age was not investigated in our model. Ezoe and Iwasa (1997) found, when there was a single mother per site, that the dispersed fraction in the progeny of a given mother decreased when its fecundity decreased. However, we do not know whether this result can be generalized to situations where several adults coexist in the same site. More generally the present analysis assumes that adult fecundity is large. A limited

157


number of offspring per mother may result in another type of demographic stochasticity so that both the age structure and the size of local sub-populations fluctuate. The consequences of these fluctuations for the present predictions are unknown.

CONCLUSION Small population sizes have consequences on both the genetics and the demography of these populations. In particular, genetic drift may generate variance in allelic composition between local populations, while demographic stochasticity may generate variance in the age structure between populations. Here we have explored some of the evolutionary consequences of these two forms of stochasticity for the natal dispersal behavior and found that both greatly altered its evolution. How small population sizes affect the evolution of other life history traits in age-structured populations deserves further exploration.

if a{b,

Q$abk (t)=Q kab(t)

if a=b,

Q$abk (t)=Q$aak (t) =

n &1 k 1 + ak Q aa(t), n ak n ak

Calculation of Probability of Identity among Different Classes of Juveniles We assume that individuals are haploid and reproduce asexually. We consider an infinite number of sites, an island model of migration, and an infinite number of alleles. This means that the probability of identity in state is equal to the probability of identity by descent and that the probability of identity between genes issued from different sub-populations is null. We consider values of identity by descent in a low-mutation limit and, as in previous work (e.g., Taylor, 1988), the probabilities we compute are equilibrium values in this model. See Rousset and Billiard (2000) for full justification of such computations. We want to compute the probability that two genes drawn in two different juveniles born in the same subpopulation are identical. We note as Q$abi (t) the probability of identity between two juveniles born to parents respectively aged a and b, present in a sub-population k at time t, just after reproduction and before migration (see Fig. 1). This probability can be expressed as a function of the probability that the two parents aged a and b are themselves identical,

(A2)

where Q kab(t) is the probability of identity between two adults aged a and b just after recruitment in a sub-population of type k at time t. In Eq. (A2), the coalescence term 1n ak describes the probability that the two juveniles have the same parent aged a in the sub-population k. We now express the probabilities of identity among adults after recruitment at time t as a function of the same probabilities a generation before, at time t&1. Using the method proposed by Rousset (1999), we found that Q kab(t)=: : : m k Â i q ia Â j q ib Â l C ia Â j, b Â l i

j

l

+: : : m k Â i q ia Â j q ib Â l(1&C ia Â j, b Â l ) i

j

l

i jl

_Q (t&1),

APPENDIX

(A1)

(A3)

where m k Â i is the probability that the sub-population k (after adult survival and recruitment) was a sub-population i the time step before (before adult survival and recruitment). In the simple case where there are only two age classes, this probability is given by mk Â i =

p i Â k u vi n&k i = s (1&s) n&k&i. u vk i

\ +

(A4)

q ia Â j is the probability that the individual aged a is issued from an individual aged j at time (t&1) living in that same sub-population i before reproduction. These transitions between age classes may correspond to reproduction events, or to survival events. Individuals aged 1 at time t, after recruitment, are necessarily born the year before. The probability that they are issued from an individual aged j in that same sub-population i corresponds to the probability that they were philopatric offspring born to a mother aged j. Thus we have for a=1,

q ia Â j =q i1 Â j = =. ji h i .

n ji (1&z j ) n(1&z vi +(1&c) z vv) (A5)

Genes in individuals aged more than 1 at time t were necessarily in individuals of the previous age class at time

158


(t&1), which have survived. The transition probability is then for a{1 and j=a&1,

q ia Â j =q ia Â a&1 =1.

(A6)

All other transition probabilities are null. C ia Â j, b Â l is the probability that genes present in individuals a and b at time t coalesce in the same individual j in the sub-population i at time (t&1). This probability is expressed as C ia Â j, b Â l =' ia Â j, b Â l

1 . n ji

(A7)

The variable ' ia Â j, b Â l takes the value 1 when coalescence is possible and 0 otherwise. Coalescence in one time step can occur only between two adults aged 1 (when they were born to same parent the generation before) or between an older adult and an adult aged 1 (when the former gave birth to the latter the generation before): If j=l and (a=1 or b=1), ' ia Â j, b Â j =' i1 Â j, 1 Â j =' i1 Â b&1, b Â b&1 =1.

(A8)

The full proof for Eq. (A3) can be obtained upon request by contacting the authors. Equation (A3) is also written Q(t)=`+/ v Q(t&1),

(A9)

where Q(t) is the vector of identities among all pairs of classes of adults within all types of sub-populations at time t. We arrange the probabilities of identity so that the rth element of this vector corresponds to Q kab(t) when, for some (a, b, k), r=(k&1) a 2max +(a&1) a max +b. We define the vector ` the elements of which are ` r =: : : m k Â i q ib Â l C ia Â j, b Â l , i

j

(A10)

l

when r is as above. We also define the matrix / whose elements are / rc =m k Â i q ia Â j q ib Â l (1&C ia Â j, b Â l ),

(A11)

with r as above and c=(i&1) a 2max +( j&1) a max +l. At equilibrium, the probabilities of identity among adults can then be computed as Q=(I&/) &1 `.

the probabilities of identity among adults and Eqs. (A1) and (A2).

(A12)

We then compute the probabilities of identity among juveniles at equilibrium using the computed values for

ACKNOWLEDGMENTS We thank M. Kirkpatrick for helpful comments on the manuscript, J. Clobert for initially drawing our attention to questions regarding dispersal and age, and I. Pen for kindly providing Pascal code for the individual-based simulations. We also thank three anonymous referees for helping us to improve the clarity of the manuscript. O.R. and S.G. acknowledge a Ph.D. grant from the French Ministry of Research and Education. This is publication ISEM-2000-071 of the Institut des Sciences de l'Evolution, Montpellier.

REFERENCES Acosta, F. J., Delgado, J. A., Lopez, F., and Serrano, J. M. 1997. Functional features and ontogenic changes in reproductive allocation and partitioning strategies of plant modules, Plant Ecol. 132, 7176. Comins, H. N., Hamilton, W. D., and May, R. M. 1980. Evolutionary stable dispersal strategies, J. Theor. Biol. 82, 205230. Comins, H. N. 1982. Evolutionarily stable strategies for localized dispersal in two dimensions, J. Theor. Biol. 94, 579606. Crespi, B. J., and Taylor, P. D. 1990. Dispersal rates under variable patch density, Am. Nat. 135, 4862. Ezoe, H., and Iwasa, Y. 1997. Evolution of condition-dependent dispersal: A genetic algorithm search for the ESS reaction norm, Res. Popul. Ecol. 39, 127137. Frank, S. A. 1986. Dispersal polymorphism in subdivided populations, J. Theor. Biol. 122, 303309. Fretwell, S. D. 1972. Theory of habitat distribution, in ``Theory of Habitat Distribution'' (S. D. Fretwell, Ed.), pp. 79114 Princeton Univ. Press, Princeton, NJ. Gandon, S., and Michalakis, Y. 1999. Evolutionary stable dispersal rate in a metapopulation with extinctions and kin competition, J. Theor. Biol. 199, 275290. Gandon, S., and Rousset, F. 1999. Evolution of stepping stone dispersal rates, Proc. R. Soc. London B 266, 25072513. Hamilton, W. D., and May, R. M. 1977. Dispersal in stable habitats, Nature 269, 578581. Hastings, W. K. 1970. Monte Carlo sampling methods using Markov chains and their applications, Biometrika 57, 97109. MacKay, P. A., and Wellington, W. G. 1977. Maternal age as a source of variation in the ability of an aphid to produce dispersing forms, Res. Popul. Ecol. 18, 195209. Maynard-Smith, J. 1982. ``Evolution and the Theory of Games,'' Cambridge Univ. Press, Cambridge. McPeek, M. A., and Holt, R. D. 1992. The evolution of dispersal in spatially and temporally varying environments, Am. Nat. 140, 10101027. Morris, D. W. 1982. Age-specific dispersal strategies in iteroparous species: Who leaves when?, Evol. Theory 6, 5365. Motro, U. 1982. Optimal rates of dispersal. I. Haploid populations, Theor. Popul. Biol. 21, 394411. Motro, U. 1983. Optimal rates of dispersal. III. Parentoffspring conflict, Theor. Popul. Biol. 23, 159168. Ozaki, K. 1995. Intergall migration in aphids: A model and a test of ESS dispersal rate, Evol. Ecol. 9, 542549.

Natal Dispersal and Maternal Age Pen, I. 2000. Reproductive effort in viscous populations, Evolution 54, 293297. Ronce, O., Clobert, J. and Massot, M. 1998. Natal dispersal and senescence, Proc. Nat. Acad. Sci. USA 95, 600605. Rousset, F. 1999. Genetic differentiation in populations with different classes of individuals, Theor. Popul. Biol. 55, 297308. Rousset, F., and Billiard, S. 2000. ``A theoretical basis for measures of kin selection in subdivided populations: Finite populations and localized dispersal,'' J. Evol. Biol., in press.

159 Taylor, P. D. 1988. An inclusive fitness model for dispersal of offspring, J. Theor. Biol. 130, 363378. Taylor, P. D. 1990. Allele-frequency change in a class-structured population, Am. Nat. 135, 95106. Taylor, P. D., and Frank, S. A. 1996. How to make a kin selection model, J. Theor. Biol. 180, 2737. Williams, G. 1966. Natural selection, the cost of reproduction and a refinement of Lack's principle, Am. Nat. 100, 687690.