Selection, Subdivision and Extinction and Recolonization

Joshua L. Cherry1 National Center for Biotechnology Information, National Library of Medicine, National Institutes of Health, Bethesda, Maryland 20894 Manuscript received April 14, 2003 Accepted for publication November 2, 2003

ABSTRACT In a subdivided population, the interaction between natural selection and stochastic change in allele frequency is affected by the occurrence of local extinction and subsequent recolonization. The relative importance of selection can be diminished by this additional source of stochastic change in allele frequency. Results are presented for subdivided populations with extinction and recolonization where there is more than one founding allele after extinction, where these may tend to come from the same source deme, where the number of founding alleles is variable or the founders make unequal contributions, and where there is dominance for ﬁtness or local frequency dependence. The behavior of a selected allele in a subdivided population is in all these situations approximately the same as that of an allele with different selection parameters in an unstructured population with a different size. The magnitude of the quantity

Nese, which determines fixation probability in the case of genic selection, is always decreased by extinction and recolonization, so that deleterious alleles are more likely to fix and advantageous alleles less likely to do so. The importance of dominance or frequency dependence is also altered by extinction and recolonization. Computer simulations confirm that the theoretical predictions of both fixation probabilities and mean times to fixation are good approximations.

OPULATION subdivision has many population- between the Ne that applies to fixation probabilities of Pgenetic consequences. The amount of neutral varia- selected alleles and the Ne describing the behavior of tion maintained in a population, the distribution of neutral alleles can in some cases be resolved with the coalescence times, expected times to fixation or loss of notion of the effective selection coefficient se (Cherry alleles, and fixation probabilities of selected alleles can and Wakeley 2003). Nese, rather than Nes, determines all be altered by subdivision. An important determinant fixation probability. Ne is raised by subdivision, but se is of the magnitudes, and even the directions, of these lowered such that Nese is unaltered by subdivision. This effects is whether gene flow among subpopulations in- framework is consistent with the fact that although fixa- cludes local extinction and subsequent recolonization. tion probabilities are unaffected by subdivision, times In the absence of extinction and recolonization, sub- to fixation are increased. division increases the amount of standing neutral varia- If the assumption of simple genic selection is re- tion and lengthens coalescence times. By these measures laxed—if there is dominance for fitness or frequency- subdivision therefore increases effective population size, dependent selection—fixation probabilities are altered

Ne. Extinction and recolonization can diminish and by subdivision, even in the absence of extinction and even reverse these effects (Slatkin 1977; Maruyama recolonization (Slatkin 1981; Lande 1985; Spirito et and Kimura 1980), so that effective size can be either al. 1993; Cherry 2003a). Specifically, subdivision de- larger or smaller than actual size. creases the importance of dominance or frequency de- The simplest case involving natural selection is genic pendence. This effect can be described in terms of effec- selection in the absence of extinction and recoloniza- tive values of the additional parameters that describe tion. Under these conditions, so long as migration is the more complex selection regime (Cherry 2003a). symmetric, subdivision has no effect on the fixation Extinction and recolonization can alter fixation prob- probability of an allele under selection (Maruyama abilities, even with simple genic selection (Barton 1970, 1974). This would seem to suggest that for the 1993). The effect of this type of population structure purpose of understanding the behavior of selected al- is to decrease the importance of selection relative to leles Ne is unaffected by subdivision, since fixation prob- stochastic forces; extinction and recolonization is an ability is generally thought of as a function of Nes, where additional stochastic force that brings with it no addi- s is the selection coefficient. The apparent discrepancy tional directional change. The interaction between natural selection and extinction and recolonization has been a subject of much recent theoretical work (Cherry 2003b; Roze and Rousset 2003; Whitlock 2003). 1Address for correspondence: National Center for Biotechnology Infor- mation, National Library of Medicine, National Institutes of Health, 45 Cherry (2003b) derived results for genic selection Center Dr., Bethesda, MD 20894. E-mail: [email protected] in a finite island model with extinction and recoloniza-

Genetics 166: 1105–1114 ( February 2004) 1106 J. L. Cherry tion by a single founding allele. Here I extend these parameters of this equivalent panmictic population are, results in two ways. First, I allow for recolonization by by definition, the effective population size (Ne), the multiple founders, which may have a tendency to origi- effective selection coefficient (se), and, in the case of nate from the same deme. Second, I obtain results that dominance, the effective dominance parameter (he). apply when there is dominance for fitness or, what is Founders chosen independently: I first consider the formally equivalent, a form of frequency-dependent se- case of genic selection where the founders that recolo- lection. nize an extinct deme have no particular tendency to originate from a common source deme. This case corresponds to Slatkin’s (1977) “migrant pool” model. MODELS AND RESULTS The mean change in allele frequency in the popula- The model of population structure considered here tion as a whole is the mean of the within-deme mean is the finite island model. In this model D demes, each changes due to selection (because migration and recolo- consisting of N haploid or N/2 diploid individuals, ex- nization are symmetric they do not, on average, change change migrants among themselves and also serve as the allele frequency). For a deme whose allele frequency sources for recolonization of extinct demes. The ex- is x, the mean change due to selection is approximately pected fraction of migrant alleles entering a deme in a sx(1 Ϫ x), where s is the selection coefficient. Thus generation is m. The probability of extinction of any the expected value of the mean change in the entire sE[x(1 Ϫ x)], where the expectation isف given deme in any generation is ␭. Subsequent to extinc- population is tion, a deme is recolonized by k founding alleles. The taken across the quasi-equilibrium distribution of x. The subpopulation then immediately grows to full size. quasi-equilibrium value of E[x(1 Ϫ x)] can be obtained Every deme receives migrants and colonists from the from a recursion that gives this expected value in one population as a whole, in which the allele frequency is generation as a function of its value in the previous x. Suppose that x changes slowly compared to the allele generation and the previous value of E[x]. Let H ϭ frequency in any deme. From the point of view of any E[2x(1 Ϫ x)]. This quantity is the probability that two deme, the population as a whole looks, in the short copies of the gene sampled from the same deme, inde- term, much like a source population with constant allele pendently and with replacement, are in distinct allelic frequency x. The distribution of the within-deme allele states. Let Ht be the value of H in generation t. We seek frequencies will then attain a quasi-equilibrium, which an expression for Htϩ1 in terms or Ht and E[x]. corresponds to the equilibrium of a source-sink (conti- For two copies of the gene sampled at time t ϩ 1to nent-island) model. For x to change sufficiently slowly, be in different allelic states, it is necessary that the same two conditions must hold. First, the number of demes copy of the locus has not been sampled twice (probabil- must be large, so that the stochastic change in the overall ity 1 Ϫ 1/N). Assuming that this is the case there are allele frequency is small compared to the within-deme several possibilities to consider. There may have just stochastic change. Second, selection must be weak in been an extinction/recolonization event (probability the sense that the magnitudes of selective differences ␭), in which case the two copies of the gene may or are small compared to the reciprocal of the deme size may not be descended from distinct founders (probabil- N. This condition allows selection to have a significant ities 1 Ϫ 1/k and 1/k, respectively). The two can be effect on the behavior of an allele in the population as allelically distinct only if they are descended from dis- a whole because the size of the entire population is tinct founders, in which case they are different with much larger than N. This assumption has the additional probability 2x(1 Ϫ x). If there has not just been an ex- consequence that selection can be neglected in the deri- tinction/recolonization event (probability 1 Ϫ␭), zero, vation of the quasi-equilibrium distribution of within- one, or both of the sampled alleles may be migrants. If deme allele frequency. I make these two assumptions— neither is a migrant, the immediate ancestors of the that the number of demes is large and that selection is sampled pair are chosen independently and with re- weak compared to 1/N—in all that follows. placement from generation t. Thus the probability that The temporal trajectory of an allele’s frequency is they are allelically distinct is Ht. If exactly one is a mi- the outcome of the interaction between the directional grant, the probability is E[x](1 Ϫ x) ϩ (1 Ϫ E[x])x, effects of natural selection and the stochastic effects of where E[x] refers to the expectation in the previous genetic drift and extinction/recolonization. A diffusion generation. If both are migrants, they are chosen inde- approximation for these processes gives a complete sta- pendently from the population at large, so the probabil- tistical description of this trajectory. The diffusion is ity that they are in different allelic states is 2x(1 Ϫ x). specified completely by expressions for the mean and Therefore variance of the per-generation change in allele fre- ϭ Ϫ Htϩ1 (1 1/N) quency as functions of the allele frequency. For the ϫ Ϫ␭ Ϫ 2 ϩ Ϫ Ϫ ϩ Ϫ cases considered here, the diffusion turns out to be {(1 )[(1 m) Ht 2m(1 m)(E [x](1 x) (1 E [x])x) approximately the same as that describing a panmictic ϩ m22x(1 Ϫ x)] population whose size and selection parameters are different from those of the subdivided population. The ϩ␭(1 Ϫ 1/k)x(1 Ϫ x)}, (1) Selection, Subdivision and Recolonization 1107 where again E[x] refers to the expectation in the previ- ␭Ӷ1, E[xЈ] Ϸ x. The distribution of xЈ is a mixture of ous generation. At equilibrium, E[x] ϭ x. The equilib- two components, one corresponding to extinction and rium condition for H is therefore recolonization and another to the binomial sampling that occurs in the absence of extinction. Conditional H ϭ (1 Ϫ 1/N){(1 Ϫ␭)[(1 Ϫ m)2H ϩ (2m Ϫ m2 )2x(1 Ϫ x)] on no extinction, the second moment about E[xЈ]is Ϫ ف ϩ␭(1 Ϫ 1/k)2x(1 Ϫ x)}. (2) (1/N)x(1 x) (the familiar binomial sampling variance). The expected value of this quantity over the Solution for H gives Ϫ Ϫ distribution of x is (1/N)(1 FST )x(1 x). The case of (1 Ϫ 1/N)[(1 Ϫ␭)(2m Ϫ m2 ) ϩ␭(1 Ϫ 1/k)] extinction and recolonization is more complicated. Let H ϭ 2x(1 Ϫ x) 1 Ϫ (1 Ϫ 1/N)(1 Ϫ␭)(1 Ϫ m)2 y be the allele frequency after recolonization. The variance of y is about equal to the variance in the allele (N Ϫ 1)[(1 Ϫ␭)(2m Ϫ m2 ) ϩ␭(1 Ϫ 1/k)] ϭ 2x(1 Ϫ x). (3) frequency among the founders. The number of the k Ϫ Ϫ Ϫ␭ Ϫ 2 N (N 1)(1 )(1 m) founders that carry the A allele has a binomial distribu- Ϫ ϵ Ϫ Ϫ ϭ FST is defined by 1 FST E[x(1 x)]/x(1 x) tion with number of “trials” k and probability of “suc- H/2x(1 Ϫ x). Because H is proportional to x(1 Ϫ x), cess” x. The variance in the number of A alleles is Ϫ F will be independent of x, as it is in the absence of kx(1 x), and the variance in the fraction of A alleles ST Ϫ extinction and recolonization. From Equation 3 it fol- is (1/k)x(1 x). Thus, since the mean of y is x, the Ϫ ف 2 lows that second moment of y about zero, E[y ], is (1/k)x(1 x) ϩ x 2. The second moment about x, E[(y Ϫ x)2] ϭ Ϫ Ϫ␭ Ϫ 2 ϩ␭ Ϫ k)x(1 Ϫ x) ϩ x 2 Ϫ 2xx ϩ/1)فϪ ϭ (N 1)[(1 )(2m m ) (1 1/k)] E[y 2] Ϫ 2E[y]x ϩ x 2,is 1 FST N Ϫ (N Ϫ 1)(1 Ϫ␭)(1 Ϫ m)2 x 2. We know that at equilibrium E[x] ϭ x and E[x(1 Ϫ ϭ Ϫ Ϫ 2 ϭ Ϫ 2Nm ϩ N␭(1 Ϫ 1/k) x)] (1 FST )x(1 x), and therefore E[x ] x Ϸ , (4) (1 Ϫ F )x(1 Ϫ x). It follows from substitution and re- 2Nm ϩ N␭ϩ1 ST arrangement that the expected value of the second mo- ϩ Ϫ ف where the approximate equality holds for small m, large ment of y about x is [(1/k) FST ]x(1 x). Combin- N, and small ␭. For ␭ϭ0, the approximate expression ing the two components of the variance of xЈ gives reduces to 2Nm/(2Nm ϩ 1), a familiar approximation Ј Ϸ Ϫ␭ 1 Ϫ ϩ␭1 ϩ Ϫ for 1 Ϫ F for an island model with ordinary migration E[Var(x )] Ά(1 ) (1 FST ) ΄ FST ΅·x(1 x). ST N k (Wright 1940; Dobzhansky and Wright 1941). (7) The above shows that the mean change in x in a Ϫ Ϫ generation, M⌬x, is approximated by (1 FST)sx(1 This is the expected variance of the change in a single ⌬ x), with FST given by Equation 4. Because FST is indepen- deme. The variance of x, the change in the overall dent of x, the effective selection coefficient can be de- allele frequency, is given by fined by M⌬ ϭ s x(1 Ϫ x). s is equal to (1 Ϫ F )s, and x e e ST 1 therefore ⌬ ϭ ͚ ⌬ V x 2 Var( xi ) D i (N Ϫ 1)[(1 Ϫ␭)(2m Ϫ m2 ) ϩ␭(1 Ϫ 1/k)] s ϭ ΂ ΃s (5) e Ϫ Ϫ Ϫ␭ Ϫ 2 Ϸ 1 ⌬ N (N 1)(1 )(1 m) E[Var( xi )] D 2Nm ϩ N␭(1 Ϫ 1/k) Ϸ ΂ ΃s. (6) ϩ ␭ϩ Ϸ 1 Ϫ␭ 1 Ϫ ϩ␭1 ϩ Ϫ 2Nm N 1 Ά(1 ) (1 FST ) ΄ FST ΅·x(1 x), (8) D N k The effective population size can be defined in terms of the variance in the change in overall allele frequency where xi is the allele frequency in the ith deme. Thus Ϫ in a generation (⌬x). This overall variance is a combina- V⌬x is approximately proportional to x(1 x), as it is tion of within-deme variances. The variance of the in a panmictic population, and the variance effective ϭ Ϫ change within a deme is a function of the local allele size can be defined by V⌬x (1/Ne )x(1 x). From this ⌬ definition and Equation 8 it follows that frequency. To obtain the variance in x, V⌬x, I first derive an expression for the within-deme variance conditional Ϸ ND on allele frequency. I then take the mean across the Ne , (9) (1 Ϫ F ) ϩ N␭[(1/k) ϩ F ] quasi-equilibrium distribution of allele frequency, ST ST which leads to an approximate expression for the overall with FST given by Equation 4. Ne can be larger or smaller variance. than the actual population size ND, depending on the Let x be the allele frequency within a representative parameters N, m, ␭, and k. deme in one generation and xЈ be the frequency in the Founders that tend to come from the same deme: next. The variance of ⌬x, the change in allele frequency, So far it has been assumed that the individuals that is equal to the variance of xЈ, which is the second mo- recolonize an extinct deme have no particular tendency ment of xЈ about E[xЈ]. Because the mean change in to originate from a common deme. It is biologically allele frequency in a single generation is small for s, m, plausible that founders do tend to come from the same 1108 J. L. Cherry deme. Slatkin (1977) analyzed a model (the “propa- ance of their sum. The covariance of a pair is given by ⌼ ⌼ ϭ ⌼⌼ Ϫ ⌼ ⌼ ⌼ ⌼ ϭ gule pool” model) in which all of the founders always Cov( i, j) E[ i j] E[ i]E[ j]. Clearly E[ i]E[ j] 2 ⌼ ⌼ come from the same deme. Whitlock and Barton x . E[ i j] is the probability that both the ith and jth (1997) presented more general results that allowed for founders carry the A allele. Consider the case where i ϶ any degree of the tendency for a common deme of j. With probability 1 Ϫ φ, the corresponding founders origin. The methods used above can be adapted to give come from different demes and the conditional proba- results for alleles under selection with this pattern of bility that both carry A is x 2. With probability φ they recolonization. come from the same deme, and the conditional proba- 2 ϭ Ϫ Following Whitlock and Barton (1997), but using bility that they both carry the A allele is E[xi ] E[xi] φ Ϫ ϭ Ϫ Ϫ Ϫ ϭ Ϫ ϩ a haploid model, let be the probability that two distinct E[xi(1 xi)] x (1 FST )x(1 x) FSTx(1 x) founders come from the same deme. As above, we seek x 2. Therefore the unconditional probability that both Ϫ φ 2 ϩ φ Ϫ ϩ 2 ϭ φ Ϫ an equilibrium value of H, the probability that two genes carry A is (1 )x [FSTx(1 x) x ] FSTx(1 sampled from within a deme are allelically distinct. Now x) ϩ x 2 and there are two cases to consider when a sampled pair Cov(⌼ , ⌼ ) ϭ φF x(1 Ϫ x). (13) are immediate descendants of distinct founders. With i j ST Ϫ φ ⌼ Ϫ probability 1 the founders come from different The variance of i is simply the Bernoulli variance x(1 ⌼ demes and the conditional probability that they carry x). The variance of the sum of the i is the sum of their distinct alleles is 2x(1 Ϫ x). With probability φ they variances plus twice the sum of the covariances of dis- come from the same deme and the conditional probabil- tinct pairs: ity that they are distinct is H (of the previous genera- Var(͚⌼ ) ϭ k Var(⌼ ) ϩ k(k Ϫ 1)Cov(⌼ , ⌼ ) (14) tion). The equilibrium condition for H is therefore i i i j ϭ ϩ Ϫ φ Ϫ [k k(k 1) FST ]x(1 x). (15) H ϭ (1 Ϫ 1/N){(1 Ϫ␭)[(1 Ϫ m)2 H ϩ (2m Ϫ m2 )2x(1 Ϫ x)] The variance of the fraction of A alleles in the newly ϩ␭(1 Ϫ 1/k)[(1 Ϫ φ)2x(1 Ϫ x) ϩ φH]} formed population is given approximately by (10) Ϫ Ϸ 1 ϩ k 1φ Ϫ (compare to Equation 2). Solving for H, and using 1 Ϫ Var(y) ΄ FST ΅x(1 x). (16) ϭ Ϫ k k FST H/2x(1 x), we obtain

(1 Ϫ 1/N )[(1 Ϫ␭)(2m(1 Ϫ m) ϩ m2 ) ϩ␭(1 Ϫ 1/k)(1 Ϫ φ)] Thus the analog of Equation 7 for the more general 1 Ϫ F ϭ ST 1 Ϫ (1 Ϫ 1/N )(1 Ϫ␭)(1 Ϫ m)2 Ϫ (1 Ϫ 1/N )␭(1 Ϫ 1/k)φ case is Ϫ 2Nm ϩ N␭(1 Ϫ 1/k)(1 Ϫ φ) Ј Ϸ Ϫ␭ 1 Ϫ ϩ␭1 ϩ k 1φ ϩ Ϫ Ϸ . (11) E[Var(x )] Ά(1 ) (1 FST ) ΄ ΂ 1΃FST΅·x(1 x). 2Nm ϩ N␭[1 Ϫ φ(1 Ϫ 1/k)] ϩ 1 N k k (17) The approximate equality is equivalent to an expression This leads to for FST given by Whitlock and Barton (1997, Equation Ϫ 21). Ϸ 1 Ϫ␭ 1 Ϫ ϩ␭1 ϩ k 1φ ϩ Ϫ V⌬x Ά(1 ) (1 FST) ΄ ΂ 1΃FST΅·x(1 x) Again FST is independent of x,sose is well defined. It D N k k is given by (18) (1 Ϫ 1/N)[(1 Ϫ␭)(2m(1 Ϫ m) ϩ m2 ) ϩ␭(1 Ϫ 1/k)(1 Ϫ φ)] s ϭ ΂ ΃s and e 1 Ϫ (1 Ϫ 1/N)(1 Ϫ␭)(1 Ϫ m)2 Ϫ (1 Ϫ 1/N)␭(1 Ϫ 1/k)φ Ϸ ND 2Nm ϩ N␭(1 Ϫ 1/k)(1 Ϫ φ) Ne Ϸ ΂ ΃ s. (12) (1 Ϫ F ) ϩ N␭[(1/k) ϩ (((k Ϫ 1)/k)φ ϩ 1)F ] 2Nm ϩ N␭[1 Ϫ φ(1 Ϫ 1/k)] ϩ 1 ST ST (19) I now turn to the variance of the change in allele frequency in a deme, which leads to an expression for (compare to Equations 8 and 9). Equation 19 can be the effective population size. The component of the shown, with the aid of Equation 11, to be equivalent to variance due to ordinary drift is the same as in Equation a result obtained by Whitlock and Barton (1997). 7, but the component due to extinction and recoloniza- These results show that a selected allele in a subdi- tion must be modified. The number of A alleles among vided population with extinction and recolonization be- the k founders is the sum of k Bernoulli random vari- haves much like an allele with a different selection coef- ⌼ ables, i, that are 1 when the ith founder carries an A ficient in a panmictic population with a different size. allele (probability x) and 0 when it carries an a (proba- This follows from the fact that both the mean and the bility 1 Ϫ x). When founders are chosen without regard variance of the change in allele frequency are approxi- to the deme of origin, these Bernoulli random variables mately proportional to x(1 Ϫ x), as they are in a panmic- are independent, and the distribution is a binomial. In tic population (expressions for this mean and variance ⌼ the more general case the i are not independent and completely determine the diffusion approximation). we must consider their covariance to compute the vari- The parameters of the equivalent panmictic population, Selection, Subdivision and Recolonization 1109 ␷ ␷ ϭ ϩ Ϫ φ Ϫ se and Ne, are given by Equations 12 and 19. The value Cov( i, j) [1/kˆ (1 1/kˆ) FST]x(1 x). (22) of N s , which determines fixation probability, is altered e e The variance in the sum of the ␷ is given by by subdivision with extinction and recolonization. Spe- i ␷ ϭ ␷ ϩ Ϫ ␷ ␷ cifically, Var(͚( i)) N Var( i) N(N 1)Cov( i, j)

Ϫ ϭ Nx(1 Ϫ x) ϩ N(N Ϫ 1)[1/kˆ ϩ (1 Ϫ 1/kˆ)φF ]x(1 Ϫ x) Ϸ (1 FST) ST Nese Ϫ ϩ ␭ ϩ Ϫ φ ϩ NDs (1 FST) N [(1/k) (((k 1)/k) 1)FST] (23) (20) and the variance in the allele frequency after recoloniza- and the effect of extinction and recolonization is to tion is given by

reduce the magnitude of Nese. Ϫ ˆ Ϫ ϭ N 1 1 ϩ k 1φ ϩ 1 Ϫ Var(y) ΄ ΂ FST΃ ΅x(1 x). (24) More complicated patterns of colonization: In the N kˆ kˆ N foregoing it was assumed that the number of founders was fixed and that they contributed equally to the new For kˆ ϭ k, this equation differs slightly from Equation population. The number of founders might vary ran- 16. The reason is that Equation 16 took the variance in domly, and the contributions of founders might be un- allele frequency in the new population to be equal to equal. For the present purposes, these details can be the variance among the founders, whereas Equation 24 summed up by a single number ␣, the probability that includes additional variance due to multinomial sam- two distinct copies of the gene in a newly recolonized pling of the founders in the formation of the new popu- deme descend from the same founding allele. Suppose, lation. Which expression is more appropriate depends for example, that the number of founders is one with on the details of the colonization process, but the effect probability 1/4, two with probability 1/2, and three with of the difference on Ne is negligible. This is evident probability 1/4. Then ␣ϭ(1/4)(1) ϩ (1/2)(1/2) ϩ from comparison of Equation 19 with the expression (1/4)(1/3) ϭ 7/12. More generally, ␣ equals the ex- for Ne corresponding to Equation 24: pected value of 1/k when founders make equal contri- Ϸ ND Ne Ϫ ϩ Ϫ ␭ ϩ Ϫ φ ϩ ϩ␭. butions to the new population. Suppose, to take another (1 FST) (N 1) [(1/kˆ) (((kˆ 1)/kˆ) 1)FST] example, that there are always two founders, but that (25) one gives rise to, on average, a fraction a of the new Dominance or local frequency dependence: Suppose population while the other gives rise to the remaining that in a diploid population the fitnesses of the geno- 1 Ϫ a. Then ␣ϭa2 ϩ (1 Ϫ a)2 ϭ 1 Ϫ 2a(1 Ϫ a). types aa, Aa, and AA are given by 1, 1 ϩ 2hs, and 1 ϩ Because ␣ϭ1/k with fixed k and equal contributions, 2s, respectively. The average selective difference be- it makes sense to define the effective number of found- tween an A and an a allele, the marginal selection coef- ers, kˆ,by␣ϭ1/kˆ. ficient ˆs, depends on the allele frequency x and is given Expressions for F , and hence s , for the present gen- ST e by ˆs(x) ϭ 2hs(1 Ϫ x) ϩ (2s Ϫ 2hs)x in an unstructured eralization are straightforward modifications of results population with nonassortative mating. ˆs can be written for the special case presented earlier. In Equation 10, as k ϩ k x, with k ϭ 2hs and k ϭ 2s Ϫ 4hs. Thus the the number of founders k matters only in that it deter- 0 1 0 1 effect of dominance on fitness is formally equivalent to mines the probability of descent from the same founder. frequency-dependent selection where fitness is a linear Thus simply replacing k with kˆ gives the more general function of allele frequency. result: Consider now a subdivided population. The expected Ϫ Ϫ␭ Ϫ ϩ 2 ϩ␭ Ϫ ˆ Ϫ φ Ϫ ϭ (1 1/N)[(1 )(2m(1 m) m ) (1 1/k)(1 )] change in overall allele frequency x is the mean of the 1 FST 1 Ϫ (1 Ϫ 1/N )(1 Ϫ␭)(1 Ϫ m)2 Ϫ (1 Ϫ 1/N )␭(1 Ϫ 1/kˆ)φ expected changes due to selection across all demes:

2Nm ϩ N␭(1 Ϫ 1/kˆ)(1 Ϫ φ) ⌬ ϭ Ϫ Ϸ . (21) E[ x] (1/D)͚ˆs(xi)xi(1 xi) 2Nm ϩ N␭[1 Ϫ φ(1 Ϫ 1/kˆ)] ϩ 1 i Ϸ Ϫ ϩ 2 Ϫ ϭ Ϫ k0E[xi(1 xi)] k1E[xi (1 xi)]. (26) As before, se (1 FST)s. ␣ϭ ˆ Ϫ ϭ Ϫ That 1/k is sufficient to determine the variance We have already established that E[xi(1 xi)] (1 ⌬ Ϫ of x is more difficult to show. The number of A alleles FST)x(1 x) at quasi-equilibrium, with FST given by Equa- 2 Ϫ in a deme after recolonization is the sum of N Bernoulli tion 4. The problem is then to find E[xi (1 xi)]. In ␷ random variables, i, that are 1 if the ith individual the absence of extinction and recolonization, this ex- ␷ carries an A allele and 0 otherwise. The variance of i pectation followed from the moments of the beta distri- Ϫ ␷ ␷ ϶ is x(1 x). We seek the covariance of i and j for i bution that approximately characterizes within-deme ␷ ␷ ϭ ␷␷ Ϫ ␷ ␷ ␷ ϭ j. This is given by Cov( i, j) E[ i j] E[ i]E[ j]. E[ i] allele frequency under that condition (Whitlock 2002; ␷ ␷ ϭ 2 ␷ ␷ x,soE[ i]E[ j] x . E[ i j] is the probability that two Cherry 2003a). In the presence of extinction and distinct copies of the gene are both A’s. This is equal to recolonization this distribution is unknown. However, ϩ Ϫ Ϫ φ 2 ϩ φ Ϫ ϩ 2 ϭ ϭ (1/kˆ)x (1 1/kˆ)[(1 )x (FSTx(1 x) x )] just as we derived quasi-equilibrium conditions for H Ϫ ϩ Ϫ φ Ϫ ϩ 2 Ϫ (1/kˆ)x(1 x) (1 1/kˆ) FSTx(1 x) x . Thus 2E[xi(1 xi)] (Equations 2 and 10), we can write such 1110 J. L. Cherry

2 Ϫ Ϫ conditions for E[xi (1 xi)]. As in the case of E[xi(1 2. With probability 3/N, one pair of the triple repre-

xi)], it is convenient to think in terms of probabili- sents resampling of the same copy of the locus. There ties of sampling certain allele conﬁgurations. Let I ϭ will be two A’s and one a when the twice-sampled 2 Ϫ E[3xi (1 xi)]. I is then the probability that three copies copy is an A and the other is an a, which happens of the gene sampled from the same deme, indepen- with probability H/2. dently and with replacement, consist of two copies of the A allele and one of the a allele. Consideration of Putting this all together gives the ways in which such a sample can arise, and of their I Ϸ (1 Ϫ 3/N){(1 Ϫ␭)[(1 Ϫ 3m)I ϩ 3m(3(1 Ϫ F )x 2(1 Ϫ x) probabilities as functions of conditions in the previous ST ϩ Ϫ generation, yields the desired quasi-equilibrium condi- FSTx(1 x))] tion for I. ϩ␭ Ϫ Ϫ 2 Ϫ To limit the number of cases that must be considered, [3(1 1/k)(1 2/k)x (1 x) ␭ I immediately adopt approximations for small m and ϩ 3(1/k)(1 Ϫ 1/k)x(1 Ϫ x)]} and large N that neglect possibilities involving two or ϩ Ϫ Ϫ more unlikely events. For example, the probability that (3/N)(1 FST)x(1 x) (27) no member of a triple is a migrant is approximated by 1 Ϫ 3m, the probability that exactly one is a migrant is at quasi-equilibrium. Solution for I gives approximated by 3m, and the possibility that more than I Ϸ [(1 Ϫ 3/N ){3m[3(1 Ϫ F )x 2(1 Ϫ x) ϩ F x(1 Ϫ x)] one is a migrant is neglected. Even with such simpliﬁca- ST ST tions, there are many cases to consider, which I enumer- ϩ␭[3(1 Ϫ 1/k)(1 Ϫ 2/k)x 2(1 Ϫ x) ate hierarchically below for the migrant pool model. ϩ 3(1/k)(1 Ϫ 1/k)x(1 Ϫ x)]}

1. With probability 1 Ϫ 3/N, all three members of the ϩ Ϫ Ϫ Ϫ Ϫ Ϫ␭ Ϫ (3/N )(1 FST)x(1 x)]/[1 (1 3/N )(1 )(1 3m)]. sample are distinct (no copy of the locus has been (28) sampled twice). a. With probability 1 Ϫ␭, extinction has not oc- Neglecting second-order terms in the denominator, us- curred. ing 1 Ϫ 3/N Ϸ 1, and multiplying numerator and de- i. With probability 1 Ϫ 3m, none of the sampled nominator by N/3 gives alleles is a migrant. In this case the probability Ϸ Ϫ 2 Ϫ ϩ Ϫ of two A’s and one a is the value of I in the I [Nm[3(1 FST)x (1 x) FSTx(1 x)] previous generation. ϩ (N/3)␭[3(1 Ϫ 1/k)(1 Ϫ 2/k)x 2(1 Ϫ x) ii. With probability 3m, one member of the sam- ϩ Ϫ Ϫ ple is a migrant. There are two ways that there 3(1/k)(1 1/k)x(1 x)] can be two A’s and one a. The two nonmigrants ϩ Ϫ Ϫ ϩ ␭ ϩ (1 FST)x(1 x)]/[Nm N /3 1]. (29) can be allelically distinct and the migrant a copy of the A allele [probability Hx ϭ 2(1 Ϫ This daunting expression can be written in the form 2 Ϫ Ϫ ϩ 2 Ϫ FST)x (1 x)], or both nonmigrants can be A’s c0x(1 x) c1x (1 x), with c0 and c1 independent of and the migrant an a [probability E[x 2](1 Ϫ x and given by ϭ Ϫ Ϫ Ϫ ϭ Ϫ ϩ x) E[x x(1 x)](1 x) FSTx(1 x) Ϫ 2 Ϫ NmF ϩ N␭(1/k)(1 Ϫ 1/k) ϩ (1 Ϫ F ) (1 FST)x (1 x)]. The total probability is ϭ ST ST c0 (30) Ϫ ϩ Ϫ 2 Ϫ ϩ ␭ ϩ therefore FSTx(1 x) 3(1 FST)x (1 x). Nm N /3 1 b. With probability ␭ an extinction/recolonization Ϫ ϩ ␭ Ϫ Ϫ ϭ 3Nm(1 FST) N (1 1/k)(1 2/k) event has occurred. c1 . (31) Nm ϩ N␭/3 ϩ 1 i. With probability (1 Ϫ 1/k)(1 Ϫ 2/k) the three members of the sample are descendants of distinct founders, and the probability of two Thus the expected mean change in population-wide 2 Ϫ Ϫ ϭ ϩ A’s and one a is 3x (1 x). allele frequency, E[ˆs(x)x(1 x)] k0H/2 k1I/3, can Ϫ ϩ Ϫ ii. With probability 3(1/k)(1 1/k) two of the be written as (k0e k1ex)x(1 x), as it can in a panmic- sampled alleles are descendants of the same tic population with dominance or linear frequency de-

founder and the third is descended from an- pendence. The constants k0e and k1e are given by other founder. There will be two A’s and one ϭ Ϫ ϩ a when the first founder carries an A and the k0e (1 FST)k0 (c0/3)k1 second an a, which happens with probability ϩ ␭ Ϫ ϩ Ϫ ϭ Ϫ ϩ NmFST N (1/k)(1 1/k) (1 FST) Ϫ (1 FST)k0 k1 x(1 x). 3Nm ϩ N␭ϩ3 2 iii. With probability 1/k all three members of (32) the sample descend from the same founder and two A’s and one a are impossible. and Selection, Subdivision and Recolonization 1111 ϭ ␭ k1e (c1/3)k1 b. With probability an extinction/recolonization event has occurred. 3Nm(1 Ϫ F ) ϩ N␭(1 Ϫ 1/k)(1 Ϫ 2/k) ϭ ST i. With probability (1 Ϫ 1/k)(1 Ϫ 2/k) the three ϩ ␭ϩ k1. (33) 3Nm N 3 members of the sample are descendants of dis- k0e and k1e can be interpreted as the selection parameters tinct founders. Ϫ φ ϩ ␺ that, in a panmictic population of size Ne, give roughly A. With probability 1 3 2 the three found- the same behavior as do the selection parameters k0 and ers come from different demes, and the proba- 2 Ϫ k1 in the subdivided population. The selection coeffi- bility of two A’s and one a is 3x (1 x). cient and dominance parameter in the equivalent pan- B. With probability 3(φ Ϫ␺) two of the founders ϭ ϩ mictic population, se and he, are given by se k0e k1e/2 come from one deme and the third from an- ϭ ϩ and he k0e/(2k0e k1e). From Equation 4 it follows that other. By an argument similar to that for case Ϫ NmFST, which appears in Equation 32, equals (1/2)[(1 1.a.ii, the probability of two A’s and an a is ϩ ␭ Ϫ Ϫ ␭ Ϫ Ϫ ϩ Ϫ 2 Ϫ FST) N (1 FST) N (1 1/k)]. Some algebra then FSTx(1 x) 3(1 FST)x (1 x). gives C. With probability ␺ the three founders come from a single deme. The probability of two A’s ϭ Ϫ se (1 FST)s (34) and an a is I of the previous generation. and ii. With probability 3(1/k)(1 Ϫ 1/k) two of the Ϫ ϩ ␭ Ϫ Ϫ sampled alleles are descendants of the same ϭ 3Nm(1 FST) N (1 1/k)(1 2/k) Ϫ ϩ he (h 1/2) 1/2. founder and the third is descended from another (1 Ϫ F )(3Nm ϩ N␭ϩ3) ST founder. There will be two A’s and one a when (35) the first founder carries an A and the second an a. Ϫ φ Two noteworthy properties of se and he that were A. With probability 1 the founders do not found to hold in the absence of extinction and recoloni- come from the same deme, and the probability zation (Cherry 2003a) also hold in its presence. First, that the first is an A and the second an a is Ϫ se is independent of h and is directly proportional to s. x(1 x). As in the case with no extinction, the proportionality B. With probability φ the founders come from Ϫ constant is 1 FST, though the value of FST is altered by the same deme. The probability of an A and Ϫ Ϫ extinction and recolonization. Second, he is indepen- an a is (1 FST)x(1 x). 2 dent of s, and subdivision decreases the deviation of he iii. With probability 1/k all three members of the from 1/2 by a factor that is independent of h. sample descend from the same founder and two If founders tend to come from the same deme, analy- A’s and one a are impossible. sis of the case of dominance or frequency dependence The modified list of possibilities leads to a generalization is more complicated. There is an increase in the number of Equation 29. Like Equation 29, this generalization of cases that must be considered when there is an extinc- can be written as c x(1 Ϫ x) ϩ c x 2(1 Ϫ x). The con- tion/recolonization event. Furthermore, for k Ͼ 2an 0 1 stants are given by additional parameter must be specified to describe the φ ϭ ϩ ␭ Ϫ Ϫ φ Ϫ␺ ϩ ␭ Ϫ Ϫ φ pattern of recolonization; is insufficient to do so. c0 [NmFST N (1 1/k)(1 2/k)( )FST N (1/k)(1 1/k)(1 FST ) The need for an additional parameter can be seen ϩ Ϫ ϩ ␭ ϩ Ϫ ␭ Ϫ Ϫ ␺ as follows. For three distinct founders there are three (1 FST)]/[Nm N /3 1 N (1 1/k)(1 2/k) ] (36) possibilities with regard to demes of origin: (a) The and three founders come from three different demes; (b) ϭ Ϫ ϩ ␭ Ϫ Ϫ two come from one deme and the third from another; c1 [3Nm(1 FST) N (1 1/k)(1 2/k) and (c) all three come from the same deme. Suppose ϫ Ϫ φ ϩ ␺ ϩ φ Ϫ␺ Ϫ ϩ ␭ ϩ that φ ϭ 1/3. This value of φ can be realized in many [(1 3 2 ) 3( )(1 FST)]]/[Nm N /3 1 different ways. At one extreme, possibility c could occur Ϫ N␭(1 Ϫ 1/k)(1 Ϫ 2/k)␺]. with probability 1/3, possibility a with probability 2/3, (37) and possibility b with probability 0. At the other extreme, possibility b could occur with probability 1. Thus Thus the diffusion approximation for the subdivided φ does not uniquely determine the probabilities of the population with linear frequency dependence is again three outcomes. identical to that for a panmictic population with linear Let ␺ be the probability that all three distinct founders frequency dependence but different size and selection come from the same deme. Then the probabilities of parameters. outcomes a, b, and c are 1 Ϫ 3φ ϩ 2␺,3(φ Ϫ␺), and ␺, respectively. Specifying φ and ␺ uniquely determines COMPUTER SIMULATIONS the probabilities of all possible outcomes. To account for common demes of origin, item 1.b in To test the theoretical results obtained here, I have the list above must be modified to read as follows: run computer simulations of the population model and 1112 J. L. Cherry

Figure 1.—Predicted and observed fixation probabilities Figure 2.—Predicted and observed relative fixation proba- for various values of k and ␭. Fixation probabilities are given bilities when founders have some tendency to come from the relative to that of a neutral allele. In all cases the A allele was same deme. In all cases there are four founders (k ϭ 4), N ϭ initially present in a single copy, N ϭ 100, D ϭ 100, m ϭ 0.01, 100, D ϭ 100, m ϭ 0.001, s ϭ 10Ϫ3, and the A allele was initially Ϫ and s ϭ 10 3. Simulation results and predictions as functions present in a single copy. Simulation results and predictions of ␭ are shown for three values of k: k ϭ 1 (solid diamonds as functions of φ, the probability of a common deme of origin, and curve), k ϭ 2 (open circles and curve), and k ϭ 4 (solid are shown for three values of the extinction rate ␭: ␭ϭ0.01 triangles and curve). (solid diamonds and curve), ␭ϭ0.003 (open circles and curve), and ␭ϭ0.001 (solid triangles and curve). compared the results to analytic predictions. In these simulations the state of the population is represented question. p˜ would be the expected new allele frequency by D integers, each giving the number of copies of the if there were no selection. Incorporating the effect of A allele in a particular deme. Each generation new val- selection gives p ϭ (1 ϩ s)p˜/(1 ϩ sp˜). ues of these integers are chosen stochastically in accor- Figure 1 compares fixation probabilities estimated dance with the model. With probability ␭ an extinction/ by simulation to theoretical predictions for different recolonization event occurs in a deme. The allele fre- numbers of founders k with N ϭ 100, D ϭ 100, s ϭ quency among the founders is chosen probabilistically 10Ϫ3, m ϭ 0.01, and various values of ␭. In all cases according to the colonization pattern being simulated, the predictions are very close to the simulation results, and the number of copies of A is drawn from a binomial differing by no more than 2.6%. Predictions of mean distribution with this as its mean. With probability 1 Ϫ times to fixation are also close to observed values (within ␭ there is no extinction. In this case the new number 1.7%; results not shown). of A’s is drawn from a binomial whose mean depends Tendency for a common deme of origin: In these on the within-deme frequency in the previous genera- simulations founders have some tendency to come from tion, the pattern of selection, the overall allele fre- the same deme, characterized by the parameter φ. This quency x, and the migration rate. Theoretical predic- tendency modifies what happens after extinction. With tions of fixation probabilities and fixation times result probability φ, the founders come from the same deme. from substitution of Ne, se, and he into classical results A source deme is chosen at random. The allele fre- for panmictic populations (Kimura 1957; Kimura and quency in this source is then the probability parameter Ohta 1969). for a binomial random variable. With probability 1 Ϫ The effect of the number of founders: These simula- φ, the founders are chosen independently from the pop- tions examine the effects of altering the number of ulation at large and the probability parameter is x. founders. The case where there is no tendency for a Figure 2 compares predicted and observed fixation common deme of origin was simulated. In the case of probabilities over a range of φ values for different values extinction the number of A’s among the k founders was of ␭ with k ϭ 4, N ϭ 100, D ϭ 100, s ϭ 10Ϫ3, and m ϭ chosen from a binomial distribution with the probability 0.01. The predictions agree well with the simulation parameter equal to x. In the absence of extinction, the results (within 5.7%). Mean times to fixation are also number of A alleles in the next generation was chosen predicted well (within 2.2%; results not shown). from a binomial distribution whose probability parame- Dominance or local frequency dependence: These ϭ Ϫ ϩ ter p is determined as follows. Let p˜ (1 m)xi simulations involve frequency-dependent selection in a mx, where xi is the allele frequency in the deme in haploid population. They can also be interpreted in Selection, Subdivision and Recolonization 1113

Figure 3.—Predicted and observed relative fixation probabilities in the presence of local frequency dependence or Figure 4.—Predicted and observed mean times to fixation dominance. In all cases N ϭ 100, D ϭ 100, m ϭ 0.01, s ϭ in the presence of local frequency dependence or dominance. 10Ϫ4, and the A allele was initially present in a single copy. The plotted points represent the same simulation runs pre- Simulation results and predictions as functions of the domi- sented in Figure 3. nance parameter h are shown for three values of the extinction rate ␭: ␭ϭ0.01 (solid diamonds and curve), ␭ϭ0.003 (open ␭ϭ circles and curve), and 0.001 (solid triangles and curve). extended to cover dominance for fitness or frequency dependence. In all of the cases analyzed, the diffusion approxima- terms of dominance in a diploid population, and I use tion for the subdivided population was identical to that the parameterization involving s and h to describe the for a panmictic population with a different size and pattern of selection. In all of these simulations the num- different selection parameters. This fact allowed the ber of founders was one. Selection differed from that definition of effective values of population-genetic pa- described above only in that the selection coefficient s rameters: the effective population size Ne, the effective was replaced by the marginal selection coeffiecient ˆs, selection coefficient se, and, in the case of dominance, so that p ϭ (1 ϩ ˆs(p˜))p˜/(1 ϩ ˆs(p˜)p˜). the effective dominance parameter he. These effective Figure 3 shows predicted and observed relative fixa- parameters are equal to the actual parameters that char- tion probabilities for a range of values of the dominance ␭ ϭ acterize a hypothetical equivalent panmictic popula- parameter h for three different values of with k 1, tion. Substitution of effective for actual parameters in ϭ ϭ ϭ Ϫ4 ϭ N 100, D 100, s 10 , and m 0.01. The predic- classical results for panmictic populations allows predictions agree well with the observed values: With one tion of such quantities as fixation probabilities and ex- exception, all of the predictions are within 3.5% of the pected times to fixation. simulation results (the prediction for ␭ϭ0.001 and ϭϪ Computer simulations confirm that the theoretical h 4 differs from the simulation result by 6.4%). results give a good description of the trajectory of allele This figure illustrates that the dependence of fixation ␭ frequency under the conditions modeled. Probabilities probability on both h and is captured well by the of fixation are predicted to within a few percent, as are analytic results. Figure 4 shows that mean times to fixa- mean times to fixation. Figure 1 demonstrates that the tion are also predicted well by the theory (within 2.2%). theory captures the effect of multiple founders. Figure 2 shows that a tendency for founders to come from the same deme is properly accounted for. Figures 3 and 4 DISCUSSION illustrate that predictions are good for cases of domi- The results presented here describe the fates of alleles nance or frequency-dependent selection. under selection in a subdivided population with local The effect of extinction and recolonization varies with extinction and subsequent recolonization by an arbi- the (effective) number of founders and their tendency trary number of founding alleles. In the simplest case for a common deme of origin. FST can be either raised treated, selection is genic and frequency independent or lowered by extinction and recolonization. This is and founding alleles have no tendency to come from evident from Equation 21. It can also be seen from the same subpopulation. These results were extended consideration of two extreme cases. If recolonization to cases where founding alleles come from the same involves a singe haploid founder, the allele frequency deme with any specified probability. They were also in the deme immediately goes to zero or one, which 1114 J. L. Cherry corresponds to maximum local differentiation from the Cherry, J. L., 2003b Selection in a subdivided population with local extinction and recolonization. Genetics 164: 789–795. population average. This leads to higher FST. If, on the Cherry, J. L., and J. Wakeley, 2003 A diffusion approximation for other hand, there is a very large number of founders, selection and drift in a subdivided population. Genetics 163: which have no tendency to come from the same deme, 421–428. Dobzhansky, T., and S. Wright, 1941 Genetics of natural popula- then an extinction/recolonization event moves the local tions. V. Relations between mutation rate and accumulation of allele frequency to approximately the population mean. lethals in populations of Drosophila pseudoobscura. Genetics 26: 23–51. This destroys local differentiation and lowers FST. Kimura, M., 1957 Some problems of stochastic processes in genetics. The product Nese, which determines fixation probabil- Ann. Math. Stat. 28: 882–901. ity, is also affected by extinction and recolonization. Kimura, M., and T. Ohta, 1969 The average number of generations until fixation of a mutant gene in a finite population. Genetics Despite the fact that FST can be either raised or lowered 61: 763–771. by extinction and recolonization, the direction of the Lande, R., 1985 The fixation of chromosomal rearrangements in a effect on Nese does not depend on the pattern of recolo- subdivided population with local extinction and colonization. Heredity 54 (3): 323–332. nization: |Nese| is always lowered by extinction and recol- Maruyama, T., 1970 On the fixation probability of mutant genes onization, as is clear from Equation 20. The change in in a subdivided population. Genet. Res. 15: 221–225. FST, whatever its direction, has the same effect on the Maruyama, T., 1974 A simple proof that certain quantities are inde- mean change in allele frequency due to selection and pendent of the geographical structure of population. Theor. the component of the variance due to ordinary genetic Popul. Biol. 5: 148–154. Maruyama, T., and M. Kimura, 1980 Genetic variability and effec- drift. Extinction/recolonization events are always a tive population size when local extinction and recolonization source of additional variance, but no additional direc- of subpopulations are frequent. Proc. Natl. Acad. Sci. USA 77: tional change. Thus the magnitude of the ratio of the 6710–6714. Roze, D., and F. Rousset, 2003 Selection and drift in subdivided mean to the variance, which is given by |Nese|, is de- populations: a straightforward method for deriving diffusion ap- creased by extinction and recolonization, and in its pres- proximations and applications involving dominance, selfing and ence selection is made effectively weaker relative to sto- local extinctions. Genetics 165: 2153–2166. Slatkin, M., 1977 Gene flow and genetic drift in a species subject chastic change. to frequent local extinctions. Theor. Popul. Biol. 12: 253–262. These results extend the situations in which a subdi- Slatkin, M., 1981 Fixation probabilities and fixation times in a vided population with selection can be related to a subdivided population. Evolution 35: 477–488. Spirito, F., M. Rizzoni and C. Rossi, 1993 The establishment of roughly equivalent panmictic population. This equiva- underdominant chromosomal rearrangements in multi-deme sys- lence allows the application of a wealth of established tems with local extinction and colonization. Theor. Popul. Biol. results for panmictic populations (e.g., Kimura 1957; 44: 80–94. Kimura and Ohta 1969) to the subdivided populations. Whitlock, M. C., 2002 Selection, load and inbreeding depression in a large metapopulation. Genetics 160: 1191–1202. Whitlock, M. C., 2003 Fixation probability and time in subdivided populations. Genetics 164: 767–779. LITERATURE CITED Whitlock, M. C., and N. H. Barton, 1997 The effective size of a subdivided population. Genetics 146: 427–441. Barton, N. H., 1993 The probability of fixation of a favoured allele Wright, S., 1940 Breeding structure of populations in relation to in a subdivided population. Genet. Res. 62: 149–157. speciation. Am. Nat. 74: 232–248. Cherry, J. L., 2003a Selection in a subdivided population with dominance or local frequency dependence. Genetics 163: 1511–1518. Communicating editor: M. A. Asmussen