The Reinforcement of Mating Preferences on an Island

Corresponding author: Mark Kirkpatrick, Department of Zoology, University of Texas, Austin, TX 78712., kirkp{at}mail.utexas.edu (E-mail)

Communicating editor: M. A. ASMUSSEN

	ABSTRACT

TOP ABSTRACT THE GENERAL MODEL HYBRID INCOMPATIBILITY FROM... A FOUR-LOCUS MODEL DISCUSSION APPENDIX 1 APPENDIX 1 APPENDIX 1 LITERATURE CITED

We develop a haploid model for the reinforcement of female mating preferences on an island that receives migrants from a continent. We find that preferences will evolve to favor island males under a broad range of conditions: when the average male display trait on the island and continent differ, when the preference acts on that difference, and when there is standing genetic variance for the preference. A difference between the mean display trait on the continent and on the island is sufficient to drive reinforcement of preferences. Additional postzygotic isolation, caused, for example, by either epistatic incompatibility or ecological selection against hybrids, will amplify reinforcement but is not necessary. Under some conditions, the degree of preference reinforcement is a simple function of quantities that can be estimated entirely from phenotypic data. We go on to study how postzygotic isolation caused by epistatic incompatibilities affects reinforcement of the preference. With only one pair of epistatic loci, reinforcement is enhanced by tighter linkage between the preference genes and the genes causing hybrid incompatibility. Reinforcement of the preference is also affected by the number of epistatically interacting genes involved in incompatibility, independent of the overall intensity of selection against hybrids.

WHILE mating preferences are clearly fundamental as isolating mechanisms in many groups of animals, it is unclear whether they are the cause or the effect of speciation. A common view is that mating preferences are an important cause of speciation. The hypothesis here is that mating preferences diverge in isolated populations as a by-product of genetic change caused by either adaptation or genetic drift. This divergence causes prezygotic isolation that results in speciation (MAYR 1963 ; RICE and HOSTERT 1993 ). The converse possibility, that speciation causes preference evolution, was suggested by DOBZHANSKY 1940 . Under his idea of reinforcement, an important side-effect of divergence in allopatry is partial postzygotic isolation. Crosses between individuals from different populations produce offspring with reduced fitness. When two populations come into contact, there is a selective premium for individuals from each population to avoid mating with individuals from the other. Consequently, mating preferences evolve to reinforce the postzygotic isolation, accelerating the process of speciation.

Although intuitively appealing, the reinforcement hypothesis has a checkered history of support from evolutionists (reviewed by BUTLIN 1987 , BUTLIN 1989 ; HOWARD 1993 ). On the empirical side, weaknesses have been identified in many early studies that were offered in support of the idea, making their interpretation ambiguous. Evidence of reinforcement has, however, been found in several detailed studies of geographical variation in mating behavior within species, for example, in frogs (GERHARDT 1994 ), flies (NOOR 1995 ), and birds (SAETRE et al. 1997 ). Strong support for the widespread operation of reinforcement comes from the work of COYNE and ORR 1989B , COYNE and ORR 1997 (see also NOOR 1997 ). Their review of studies on 171 species of Drosophila shows that sympatric pairs of species have far stronger behavioral isolation than allopatric pairs that have been isolated for comparable amounts of time.

Theoretical objections to the reinforcement hypothesis have also been raised. Reinforcement involves two competing forces that act on a mating preference. The first is indirect selection. Even if preferences do not directly affect survival or the number of gametes produced, preference alleles favoring heterotypic matings are indirectly selected against because they are associated with low-fitness hybrid genotypes. This is the engine that drives reinforcement. The countervailing force is gene flow: hybridization causes the preference genes in the two populations to become homogenized. The most obvious theoretical problem for reinforcement is that the force of gene flow might overwhelm indirect selection and prevent reinforcement from succeeding (MOORE 1957 ; MAYR 1963 ). Recombination breaks down the genetic associations between preference genes and the loci selected against in hybrids (FELSENSTEIN 1981 ; BARTON and HEWITT 1985 ), which weakens the force of indirect selection favoring reinforcement.

Several models have established that reinforcement of mating preferences can succeed, however, at least in some circumstances. SVED 1981 considered the situation of complete hybrid inviability or sterility and derived analytic expressions for the reinforcement of a specific type of mating preference. Three simulation studies since then have found that reinforcement of mating preferences succeeds in some but not all cases. SPENCER et al. 1986 developed a model that includes demography as well as evolution and tracked the fate of a single population when hybrids are completely infertile or inviable. LIOU and PRICE 1994 extended that model by allowing reciprocal introgression between a pair of species that have only partial postzygotic isolation. SERVEDIO and KIRKPATRICK 1997 compared situations in which there is reciprocal introgression with situations involving one-way gene flow. Because these simulation models make quite restrictive assumptions about genetics and behavior, it is not known how general their qualitative conclusions might be or how the results can be applied quantitatively to natural populations.

This article develops an analytic model for investigating the reinforcement of mating preferences on an island that receives immigrants from a continent. It considers what happens when postzygotic isolation is weak and so applies to the early phases of divergence. We choose to focus on this situation for two reasons. First, islands are sites of rapid and profuse speciation (MAYR 1963 ), and it is of interest to see how reinforcement might play a role in these radiations. Second, it is simpler to study a single island population than two or more populations in which the picture is complicated by the effects of reciprocal gene flow and spatial structure.

Our main aim is to determine if and how mating preferences that discriminate against the continental immigrants will evolve on an island. Major questions are: Can reinforcement of mating preferences occur, and if so how big of an impact will it have? Does the way in which hybrids are selected against determine if reinforcement succeeds? How is reinforcement affected by genetic details such as the number of loci, their linkage, and the distribution of their effects? When can observable phenotypic characters be used to make predictions about the outcome of reinforcement?

The article begins by describing a general haploid model. Initially, we make no specific assumptions about what causes hybrids to have reduced fertility and/or viability. The results therefore apply to cases where ecological factors select against hybrids that have phenotypes intermediate between the parental forms (e.g., GRANT and GRANT 1993 ; SCHLUTER 1995 , SCHLUTER 1996 ), for example, and to those where epistatic interactions cause developmental problems in hybrids (reviewed by COYNE 1992 ; WU and PALOPOLI 1994 ; TURELLI and ORR 1995 ). We find that evolution will typically reinforce preferences on the island so that females there discriminate against males from the continent. We then develop a specific model in which hybrid unfitness is caused by epistatic interactions. Results there show how the genetic basis of hybrid unfitness can affect the outcome of reinforcement.

There are four features of the model we emphasize at the start. First, we assume that mating preference genes are free of direct selection; that is, preference genes do not affect the survival or the number of gametes produced by an individual. This allows us to look at the effects of reinforcement on preference evolution in isolation from other evolutionary forces. In our model, reinforcement results from genetic associations (linkage disequilibrium) between preference alleles and other genes that are directly selected, for example those that cause hybrids to have low fitness, as envisioned by Dobzhansky. Second, the model is quite general in several respects regarding genetics (e.g., there can be any number of genes and any linkage relations between the genes that affect hybrid fitness, the preference, and the display) and behavior (any form of mating preference is allowed). The model is therefore free of several assumptions that restrict the generality of earlier models. Third, the model treats the average value of the display trait as a known quantity, rather than accounting for its evolution explicitly. This approach makes the results general to all forms of natural and sexual selection that might act on a display. Fourth, the migration rate of continental individuals onto the island is also viewed as a known quantity. Again, this buys generality: no restrictive assumption is made about how the values of the mating preference and display trait affect rates of hybridization. But another consequence is that our model does not describe how evolution might shut off introgression and lead to complete reproductive isolation, as Dobzhansky suggested. Thus ours is a model for the reinforcement of mating preferences, but not for the reinforcement of prezygotic isolation.

	THE GENERAL MODEL

TOP ABSTRACT THE GENERAL MODEL HYBRID INCOMPATIBILITY FROM... A FOUR-LOCUS MODEL DISCUSSION APPENDIX 1 APPENDIX 1 APPENDIX 1 LITERATURE CITED

Our model considers the evolution of three kinds of characters: a female mating preference, a male display trait that the preference acts on, and incompatibility traits that reduce the fitness of hybrids. To make the analysis tractable, we assume that the genome is haploid. There can be any number of autosomal loci underlying the traits. The model therefore covers the special cases where only a single locus affects the female preference or male display trait and situations where these characters vary quantitatively as the result of contributions from many loci. Two alleles, denoted 0 and 1, segregate at each locus. Any linkage map is allowed; no assumption is made about recombination rates. The population size on the island is large enough that the effects of drift can be ignored.

To analyze the model, we use notation and results developed by BARTON and TURELLI 1991 , who introduced a general framework for studying evolution in multilocus systems. Their approach is extended here to allow for migration and a series of selection events that occur over the course of each generation.

In our model, the mating preference can be any trait that affects the probabilities that a female will mate with different kinds of males (KIRKPATRICK and BARTON 1997 ). This general view of a preference subsumes many earlier models of mating preferences as special cases. For example, a preference might control a mating preference function (as in LANDE 1981 ) or the tendency of a female to mate with one of two types of males (as in O'DONALD 1980 ). But the present model is more general and allows a female's preference, P, to be any measurable aspect of her phenotype: the period of day when she searches for mates, for example, or the average number of males she inspects before making a decision.

We assume that genetic variation in the preference is contributed by one or more loci with additive effects, and we denote this set of loci as P. These loci do not affect survival or immediate fecundity (that is, the number of eggs a female produces). The preference phenotype of a particular female can always be written in the form

(1)

The summation is over each of the loci that contribute to variation in the preference (the symbol indicates there is one term in the sum for each locus in set P), is the mean of the preference on the island at the start of a generation, ^P_i is the difference in the effects on the preference of the two alleles at locus i, and e_P is a random environmental effect. The variable X_i takes on value 1 if the individual carries allele 1 at locus i and value 0 if it carries allele 0. The frequency of allele 1 among zygotes on the island is p_i, and the frequency of the allele 0 is q_i = 1 - p_i. The quantity ^P_i (X_i - p_i) in Equation 1 measures how much locus i causes that individual's preference to deviate from the average preference. (Two quantities on the right side of Equation 1, and p_i, change across generations but those effects cancel in such a way that genotypic values remain constant.) The notation used throughout the article is summarized in Appendix 1.

Females choose mates on the basis of a trait that males display. The model allows this trait to be expressed either in males alone or in both sexes. Natural selection acts on the trait, but no specific assumption is made about the form of the fitness function. Likewise, the model allows for any form of sexual selection that might be caused by the female preference. The value for a male's display trait phenotype, T, is affected by a set of loci denoted T. We assume these loci have additive effects on T and that the genes are expressed the same way in both sexes if the trait is present in females. The value of a display trait phenotype can therefore be written in the same form as Equation 1, but with replacing , T replacing P, ^T_i replacing ^P_i, and e_T replacing e_P on the right-hand side. It is possible to generalize the model to allow for epistatic interactions among the display trait genes (see KIRKPATRICK and BARTON 1997 ).

When a preference and a display trait are framed in this general way, KIRKPATRICK and BARTON 1997 found that a key parameter governing evolution of the preference is the phenotypic correlation between the preference in females and the display trait in males among mated pairs. We denote this correlation by . It is a phenotypic measure and so is free of any assumptions about the genetics underlying the preference and display trait. This correlation is, however, responsible for creating the genetic associations between preference alleles and the genes that affect the display trait and hybrid fitness. The value for can be calculated given a specific set of assumptions about how mating occurs. Alternatively, it could also be measured directly in nature. Consider a species of frog where pairs of individuals can be caught in copula. For each pair, one would measure the male's display trait (e.g., the fundamental frequency of the call) and also the female's preference (e.g., the percentage of times she approaches a speaker playing a low-pitched rather than a high-pitched call). The correlation between these two numbers gives an estimate of . The value of may change as a population evolves; in that case, the value relevant to the equations below is the one that prevails in the current generation.

The last factor relevant to the model is hybrid incompatibility. Because many things reduce the viability and/or fertility of hybrids in nature, for now we do not commit to any specific assumption about the causes of incompatibility. The model therefore allows for many possible modes of selection against hybrids. (Later in the article we focus on hybrid incompatibility caused by epistasis.) The set of all the loci involved in determining hybrid incompatibility is denoted H.

Regarding migration, a proportion m of the individuals on the island arrives from the continent in each generation. No assumption about the sex ratio of the migrants is made. Without migration, the model reduces to a "null model" of sexual selection: because there is no other force acting on the preference, it is free to equilibrate at any value (KIRKPATRICK and RYAN 1991 ). With gene flow from the continent, however, migration and indirect selection produce forces that act on the preference, and our goal is to find out how these forces interact. The model does not make assumptions about the history of the two populations. It therefore applies to cases of secondary contact (in which the island and continent populations initially diverge in isolation) and also to cases where an island is colonized from a continent and continuously receives gene flow thereafter.

The order in which the events of selection and migration occur in the life cycle varies according to the ecology of the species. Dispersal in some species occurs before natural selection acts on the male display trait, for example, but afterward in other species. Accordingly, we make no restrictions about the ordering of migration and natural selection in the life cycle and let the model determine when and why the ordering matters.

Evolution of a preference gene:
This section introduces the model for the evolution of a preference locus, explaining the major components and notation. Each term is then considered in detail in the subsequent section.

Evolution of the preference can be described exactly using the notation of BARTON and TURELLI 1991 . The change in the frequency of an allele at preference locus i in a single generation is

(2)

The right-hand side is a sum of four terms, corresponding to the four events that change preference allele frequencies over the course of a generation. The first term represents the effect of migration, which causes evolution whenever there are differences in the allele frequencies of individuals on the island and the continent. The next three terms on the right of Equation 2 are summations that reflect the impact on the preference of natural selection acting on the male display trait, of sexual selection acting on that trait, and of selection against hybrid incompatibility. The symbol indicates that the summations are taken over all subsets of the set listed to the right of the symbol (including that set itself).

To better understand this notation, consider the first summation. It is taken over all sets of male display trait loci in the genome, including the full set T. For example, if genetic variation in the male trait is attributable entirely to the two loci j and k, then that summation is

(3)

The sums in Equation 2 and Equation 3 involve the ã and C, which are, respectively, the selection coefficients and genetic disequilibria defined by BARTON and TURELLI 1991 . The ã are selection coefficients that represent the force of selection in favor of allele 1 at the combination of loci listed in the subscripts, averaged over males and females. Thus ã_j is strength of selection favoring allele 1 at locus j, while ã_jk measures selection that brings together allele 1 at locus j with allele 1 at locus k. Selection coefficients can be calculated for any given assumptions about how selection acts; details are given in Appendix 1 and in BARTON and TURELLI 1991 .

The C measure the degree of disequilibrium or association in the population between alleles 1 at the loci listed in the subscript. In the case of two loci, C_ij is equal to the standard measure of linkage disequilibrium (often denoted D). Generalizing to situations involving any number of loci, the disequilibria are defined by the relation

(4)

where E[·] stands for the expectation taken over all genotypes in the population. The value of C in the population changes both within and between generations as the result of selection, migration, and recombination. Calculations for the C that appear in this model are given in Appendix 1.

The superscript n in the ãⁿ and Cⁿ of Equation 2 and Equation 3 indicates that those quantities are evaluated at the point in the life cycle when natural selection acts on the male display trait. Likewise, terms carrying an s superscript are evaluated at the time that sexual selection (mate choice) happens, while the superscript h indicates quantities evaluated when selection acts against hybrids.

An important implication of Equation 2 is that the associations between alleles at two, three, and larger numbers of loci can contribute to the evolution of the preference. The Barton-Turelli framework, in contrast to many other theoretical approaches, can account for these associations exactly. In practice, however, expressions for the C can quickly become unmanageable. The approach is made practical by a clever method of approximation that they devised. Provided the forces of selection and migration acting on genotype frequencies are weak and recombination rates are not too small, then a population will rapidly evolve to a state called "quasi-linkage equilibrium" (QLE; KIMURA 1965 ; NAGYLAKI 1976 ). At this point, allele frequencies will change relatively slowly and the disequilibria between sets of loci (the C) will be small compared to their maximum possible values. These facts allow one to find simple approximations for the ã and C that appear in Equation 2.

To make the model tractable, we analyze it assuming the population is in QLE. Biologically, this means that migration is weak, the forces of selection on alleles and sets of alleles are small, differences between females in their mating preferences are not large, and recombination rates are not very small (see Appendix 1 and C). The results will be most accurate for the early stages of divergence, when postzygotic isolation is still weak.

Forces acting on the preference:
With an overview of the model's notation in hand, we now consider each of the components of Equation 2 in detail. The subsequent section then combines the results to find the overall dynamics of the preference.

Migration: The first term on the right side of Equation 2 represents the effect of migration on preference locus i:

(5)

The migration rate m is defined as the proportion of individuals on the island that have immigrated that generation from the continent, averaged over the two sexes, and d^m_i is the difference between the frequencies of allele 1 on the continent and on the island:

(6)

The prime (') indicates values on the continent, and so p_i' is the frequency of the preference allele there. A superscript m denotes quantities that are evaluated at that point in the life cycle when migration occurs. Migration is a direct evolutionary force, as it causes an allele's frequency to change independent of that gene's statistical associations with other genes. In contrast, all the other forces acting on the preference in this model are indirect: the preference evolves as a result of its genetic associations with other traits. This fact is evident from the three summations that appear in Equation 2, all of which involve the C.

Natural selection on the male display trait: The second factor affecting evolution of the preference is natural selection acting on the male trait. Equation 2 expresses the effects of natural selection in terms of selection coefficients (the ã) and the genetic disequilibria (the C).

The selection coefficients are calculated in Appendix 1 assuming that the fitness function is relatively smooth over the trait's range of phenotypic variation (specifically, that its third and higher derivatives are small near the trait mean). In that case, selection coefficients involving sets of three or more male trait loci are negligible, and we need to consider only the effects of selection on single loci and on pairs of loci. The selection coefficient for a single male trait locus j is

(7)

while the selection coefficient for a pair of male trait loci j and k is

(8)

Here ß_n and _n are, respectively, the directional and stabilizing selection gradients acting on the display trait phenotype during natural selection, averaged over males and females. If the display trait is expressed only in males, then these gradients are simply half of the values measured for the males (because the values for females are 0). Methods with which to estimate these selection gradients in natural populations are discussed by LANDE and ARNOLD 1983 . The values of ß_n and _n will change as the display evolves; the relevant values are those that obtain in the current generation.

These and following expressions for the selection coefficients and genetic disequilibria have been greatly simplified by dropping terms of order ã² or smaller. Thus these approximations are valid when the selection coefficients and migration rate are much smaller than 1. Explicit bookkeeping of the terms that have been dropped is done in Appendix 1 and C.

The genetic disequilibria (the C) are calculated in Appendix 1, which shows that at QLE the disequilibrium between a preference locus i and a trait locus j is

(9)

while the disequilibrium between a preference locus i and two male trait loci j and k is

(10)

In these expressions,

(11)

_P and _T are, respectively, the phenotypic standard deviations for the preference and trait on the island, and is the correlation between the preference phenotype of females and the display trait phenotype of males in mated pairs. The indicator variable _n takes the value 0 if natural selection on the display trait occurs before migration and the value 1 if not. R_U is a function of the recombination rates among the loci listed in its subscript:

(12)

Here r_U is the multilocus recombination rate among the loci in set U, that is, the probability that there is at least one recombination event among the loci in that set. (For example, r_U is the standard recombination rate between loci i and j.) For our QLE approximations to hold, r_U must be substantially larger than 0 (see BARTON and TURELLI 1991 ). The function R_U takes on larger values as linkage between the loci in set U becomes tighter.

Equation 9 shows that two factors contribute to the covariance between a preference gene and a display trait gene: migration, which builds up genetic correlations across the whole genome, and mate choice, which generates associations specifically between preference and display trait loci. The magnitude of the covariance at the time that natural selection acts is affected by the order of selection and migration, because migration generates linkage disequilibrium within the generation. Recombination rates affect the contribution from migration, but not from mate choice: the linkage disequilibrium at QLE caused by sexual selection is independent of the recombination rates (LANDE 1981 ; KIRKPATRICK 1982 ; BARTON and TURELLI 1991 ). Equation 10 implies that only migration generates genetic associations between sets of three loci: under our assumptions of weak additive effects for the display trait and preference loci, mate choice does not contribute to these three-way disequilibria.

The overall impact of natural selection on the display trait is found by summing over sets of trait loci (the first summation on the right side of Equation 2), using the results from Equation 7Equation 8Equation 9Equation 10. The change at preference locus i is

(13)

where

(14a)

(14b)

(14c)

Equation 13 has been simplified by dropping terms of order ã³ and smaller.

Sexual selection on the male display trait: The third factor influencing evolution of the preference is sexual selection on the male display trait. The calculation for its effect follows that for natural selection. A difference is that migration always comes before mate choice, as mating is the last event in each generation before the start of the next. Thus the summation in Equation 2 corresponding to sexual selection is given by Equation 13 with three changes: the directional sexual selection gradient ß_s replaces its natural selection counterpart ß_n, the stabilizing sexual selection gradient _s replaces _n, and _n is set equal to 1.

Selection against hybrid incompatibility: The final factor causing evolution of a preference gene is selection on the loci that cause hybrid incompatibility. The last summation on the right side of Equation 2 accounts for all forms of selection acting on F₁ hybrid offspring, backcrosses, and all other types of recombinants.

Because for now we are not committing to the details of how selection acts against hybrids, the selection coefficients ã^h_U remain undetermined. The disequilibria C^h_iU can, however, be calculated. Using the results from Appendix 1, we find

(15)

where _h = 0 if selection on hybrid incompatibility occurs before migration and 1 otherwise. This shows that the associations between preference loci and incompatibility loci are proportional to the migration rate, m, and the differences in allele frequencies on the island and continent at these loci, d_iU.

The effect on preference locus i of selection against hybrid incompatibility is therefore

(16)

where

(17)

is a measure of the force of selection on all the incompatibility loci with respect to preference locus i. As with Equation 13, terms of order ã³ and smaller have been dropped from Equation 16. (Despite the minus sign, we will see that I_i is typically positive and that selection against hybrids does indeed favor reinforcement.)

This is the last of the terms that enters into Equation 2 for the evolution of a preference gene. Next we will assemble these results to find the overall dynamics of the preference.

Evolution of the mean preference:
A general expression for the rate of evolution of a single preference allele is obtained by substituting the results from the last section into Equation 2. The result can be simplified by substituting d_i for d_i^m (which appears in Equation 5), a step that is justified because the change in preference allele frequencies within a generation caused by selection is small enough (of order ã²) that it can be neglected in our approximations.

To find the rate of change in the mean preference at QLE, we sum across all the preference loci,

(18)

where h²_P is the heritability of the preference and h²_T the heritability of the male trait. The direct effect of migration appears as the first term of Equation 18. It reduces the difference between the average preferences on the island and continent, which is ( - '). The next four terms, involving the selection gradients ß and , reflect the impact of natural and sexual selection on the display trait. This is our most general result for the evolutionary dynamics of the preference.

Reinforcement at equilibrium: The outcome of reinforcement depends on a balance struck between migration, which makes preferences on the island similar to those on the continent, and indirect selection on the preference genes, caused by selection on hybrid incompatibility and the display trait. How large is the difference between preferences on the island and continent when an equilibrium is reached? To answer that question, we will introduce an additional assumption, define a new way to measure the preference, and calculate the equilibrium value of a term in Equation 18.

The new assumption is that the preference loci are unlinked to the loci affecting the display trait and hybrid incompatibility. The reason for this assumption is that it greatly simplifies Equation 18. The terms S_3i and I_i no longer depend on i and can be written simply as S₃ and I. Further, the recombination function R_ij becomes equal to 1.

To describe the preference equilibrium, it is helpful to focus on the difference between the mean preferences on the island and the continent. In the absence of reinforcement of the mating preference, migration will homogenize the preference genes in the two populations and the difference will vanish. Thus any difference between the preferences on the island and continent at equilibrium is the result of reinforcement. Further, if we divide this difference by the phenotypic standard deviation of the preference on the island, we have a dimensionless index of reinforcement. This allows one to make comparisons across different species and different kinds of preferences. Doing the same for the male display trait gives us these two new measures:

(19)

The last ingredient needed is an expression for the overall selection gradients acting on the male display trait at equilibrium. When the display trait is at equilibrium, the force of directional selection is offset by gene flow. When these two forces are in balance, it is straightforward to show that

(20)

where h²_T is the heritability of the display on the island, that is, its additive genetic variance divided by its phenotypic variance. (This result is only an approximation because it neglects the force of indirect selection on . But that force is O(ã²), however, and therefore can be neglected for our present needs.)

We finally arrive at an expression for the degree of reinforcement in the mating preference at equilibrium. Set the left side of Equation 18, use Equation 19 and Equation 20, and assume that the migration rate is not zero. Then we find

(21)

This is our most general result for the divergence of the island and continent mating preferences at equilibrium.

An immediate message that Equation 21 conveys is that divergence of the island preference happens under a broad range of conditions, despite the presence of gene flow and regardless of the mechanism of postzygotic isolation. This conclusion follows because only a very unusual combination of parameters will make _P exactly equal to zero. Complete postzygotic isolation, which has sometimes been thought to be necessary for reinforcement to evolve (e.g., BUTLIN 1989 ; RICE and HOSTERT 1993 ), is not required. The outcome depends in part on quantities that one might anticipate, such as the heritability of the preference (h²_P) and the amount of hybrid incompatibility (I). The equilibrium also depends, however, on the intensity of stabilizing selection on the male trait (the 's) and on whether migration in each generation happens before or after natural selection on the display (_n).

Several of the terms in Equation 21 can be estimated from phenotypic data. One is the difference in the average value of the display trait on the island and continent, D_T. The result also involves the S terms, however, which depend on the genetic details of the display trait (that is, recombination rates, allele frequencies, and allelic effects). Unfortunately, those quantities cannot be estimated for any species with the data now available, and there do not seem to be any obvious generalizations that can be made about their magnitude or even their sign.

But things simplify a lot under some circumstances. When migration and stabilizing selection are sufficiently weak, and when migration occurs before natural selection acts on the display trait, then

(22)

This simple result tells us how much reinforcement causes preferences on the island to diverge from those on the continent. That difference is proportional to _T, the difference at equilibrium in the display trait means on the island and continent. The constant of proportionality depends on only three quantities: the phenotypic correlation between the male display and the female preference among mated pairs , the heritability of the preference h²_P, and the intensity of hybrid incompatibility I. The factor of ¹/₂ results from our assumption that only one sex (females) choose their mates. Many other variables, such as the type of selection acting against hybrids, number of genes affecting the display trait and preference, and the migration rate, apparently do not affect the outcome.

Figure 1 illustrates how divergence of the island preference changes with _T. Although reinforcement causes the island preferences to differ from those on the continent at equilibrium, the difference is not necessarily large. With = 0.25, h²_P = 0.4, and I = 0.2, for example, a difference of three phenotypic standard deviations between the display trait means on the island and continent will cause the mean preferences to differ by only 18% of a phenotypic standard deviation.

View larger version (16K): In this window In a new window Download PPT slide   Figure 1. The difference between the average mating preference on the island and continent as a function of the difference between the average display traits in those two populations, from Equation 22. Both differences are measured in units of phenotypic standard deviations. I is the intensity of selection against hybrid incompatibility, as defined by Equation 17. Other parameter values are h2P = 0.4 and = 0.25.

Equation 21 and Equation 22 are our main results. Before going further, we recapitulate the major simplifying assumptions and then draw several implications. The fundamental approximation that makes the analysis possible is that selection coefficients (the ã) are small. This situation holds when fitness effects of individual loci and sets of loci are small. It also implies that the impact of migration is weak, meaning that the migration rate and/or genetic differences between the island and continent are small. Quantitatively, each of the terms following the 1 in the denominator of Equation 21 must be small. In biological terms, the upshot is that our results are most accurate when prezygotic and postzygotic isolation between the island and continent are weak. Regarding recombination, the basic model allows any pattern of linkage between the loci (as long as the recombination rates are not very small). But to find an expression for the equilibrium of the average preference, we assumed that preference loci are unlinked to other genes in the model. If there is linkage, one can solve for the equilibrium if values for the recombination rates and the allelic effects are given.

Two interesting conclusions flow from these results. First, a difference between the average display on the continent and island, no matter what causes that difference, is sufficient to drive reinforcement of the preference. But if the island and continent displays are the same (_T = 0), then females cannot distinguish the two types of males, and reinforcement is doomed. Additional postzygotic incompatibility is not needed: Equation 21 and Equation 22 show that the average preference on the continent and island can differ (that is, _P 0) even when there is no hybrid incompatibility (that is, I = 0). There is an intuitive explanation for this result. If the combined forces of natural and sexual selection keep males on the island distinct from those on the continent, then females that mate with immigrant males will have offspring that are poorly adapted to the local ecological conditions, that will have difficulty finding a mate, or both.

A second conclusion applies when migration and stabilizing selection on the display trait are weak. Then the amount of reinforcement in the preference is directly proportional to the difference between the average male trait on the island and continent (Equation 22). Further, when postzygotic isolation is negligible (I 0), the constant of proportionality depends only on things that can in principle be estimated from phenotypic data: the phenotypic correlation between the display in males and the preference in females among mated pairs, , and the heritability of the preference, h²_P.

Up to now, no assumption has been made about what mechanism causes hybrid incompatibility, that is, postzygotic isolation. But several interesting questions are still unanswered, such as: How much is reinforcement strengthened by selection against hybrids? How does the genetic architecture of postzygotic isolation affect reinforcement? We now introduce a specific model for postzygotic isolation to address those issues.

	HYBRID INCOMPATIBILITY FROM EPISTASIS

TOP ABSTRACT THE GENERAL MODEL HYBRID INCOMPATIBILITY FROM... A FOUR-LOCUS MODEL DISCUSSION APPENDIX 1 APPENDIX 1 APPENDIX 1 LITERATURE CITED

DOBZHANSKY 1937 and MULLER 1942 hypothesized that postzygotic isolation between species could be caused by epistatic interactions between sets of loci. They pointed out that populations isolated from the same ancestral stock could undergo substitutions at different loci. If the new alleles at these loci cause problems in development or gametogenesis when they are combined in the same individual, these genes will contribute to postzygotic isolation when the populations come back into contact. There is now abundant evidence that hybrid incompatibility in some groups of animals is indeed caused by epistatic interactions (COYNE 1992 ; WU and PALOPOLI 1994 ; TURELLI and ORR 1995 ). Further, some cases involve interactions between sets of more than two loci (MULLER 1942 ; WU and PALOPOLI 1994 ). Motivated by those data, we now look at a specific model in which hybrid incompatibility is caused by epistasis acting on sets of interacting loci. We commit to specific assumptions about how epistasis operates, and so the results of this section are not general to all forms of epistatic incompatibility.

We allow there to be any number of epistatically interacting sets of loci, and within each set there may be any number of loci. For convenience, we designate the allele that is most common on the island as allele 1. When all of the alleles carried by an individual in one of these epistatic sets are of the same type, either 0 or 1, then development and meiosis proceed normally. Within a set consisting of four loci, for example, these high fitness genotypes are {0, 0, 0, 0} and {1, 1, 1, 1}. But if an individual carries a mixture of alleles of both kinds, then fitness is reduced. We assume that fitness is reduced by the same amount for all mixtures of alleles in a given set (e.g., genotypes {0, 0, 1, 1} and {0, 1, 0, 0} have the same fitness), but different sets can have different fitness effects. We number the epistatic sets to refer to them. The number of loci in set H_i is n_i, and the loss of fitness caused by a mixture of alleles in that set is s_i. As in the rest of the article, the set of all loci in the genome involved in hybrid incompatibility is referred to as H.

Having laid out assumptions for how incompatibility works, we can now calculate its impact on the mating preference. We saw earlier that the rate of evolution of the preference depends on I_i (Equation 17 and Equation 18). That quantity measures the impact on preference locus i of selection on all the loci involved in hybrid incompatibility. The selection coefficients that appear in I_i are calculated in Appendix 1 for our model of epistasis. For any subset U of loci in epistatic set H_j, the selection coefficient is

(23)

where

(24)

and |U| denotes the number of loci in set U. (The products, which are over all loci in H_j that are not also in U, are defined to equal 1 when H_j = U.) Substituting into Equation 17, the combined effect of selection on all the epistatic loci is

(25)

The effects of the incompatibility genes simplify when the continent and island are near fixation for opposite alleles at these loci, which requires that migration is weak relative to epistatic selection (m s_i). In that case, d_U is ~(-1)^|U|, and k(j,U) is approximately equal to 1 if U is not equal to H_j, to 2 if U equals H_j and |U| is even, and to 0 otherwise. Then

(26)

Two useful observations can be made here. The timing of migration relative to epistatic selection does not affect reinforcement: _h has disappeared. This is because when migration introduces only pure continental genotypes to the island, maladapted recombinant genotypes are produced and selected against only after recombination. Second, we can assess the effects of recombination on reinforcement. Consider what happens when hybrid incompatibility depends on pairs of epistatically interacting loci. Focusing on a single preference locus i and a single epistatic pair j and k, with the help of Equation 12, Equation 26 becomes

(27)

This quantity is always positive and is proportional to the selection coefficient s against epistatic recombinants. Thus stronger selection against hybrids increases reinforcement, as expected. Further, one can show that reinforcement is always strengthened whenever linkage between the preference locus and the epistatic loci is tightened. Intuitively, the reason is that tighter linkage keeps the island-specific preference allele more highly correlated with high fitness genotypes (FELSENSTEIN 1981 ). The effect of the recombination rate between the epistatic loci depends on the linkage map. If the preference locus does not lie between the two epistatic loci, one can show that decreased recombination between the epistatic loci decreases reinforcement. Intuitively, lower recombination means less opportunity for selection against recombinants. On the other hand, if the preference locus lies between a pair of epistatic loci on a chromosome, decreasing any of the recombination rates increases reinforcement.

How does the number of loci in each epistatic set affect reinforcement of the preference? That question can be answered when the loci within each epistatic set are unlinked to each other and to the preference loci. For any set U of unlinked loci, the recombination function R_U then equals 1/(2^|U-1| - 1). Using that fact and Equation 26 gives the overall impact of epistatic incompatibility,

(28)

where

(29)

This expression for I can be entered directly into the earlier results for the equilibrium of the mean preference (Equation 21 and Equation 22), which also assume the preference loci are unlinked to the incompatibility loci.

The results show how the degree of preference reinforcement is amplified by selection against hybrid recombinants. Equation 28 shows that the overall effect is determined by a simple sum of the selection coefficients acting on each epistatic set, weighted by the factor f(), which is determined by the number of loci in each set. This factor increases from a value of f = , when the set consists of a pair of loci, to a value of f = 2, when there are three loci, to a value of 2.8 with five loci. The rise is slow, approximately logarithmic, as shown in Figure 2. This trend reflects the outcome of two conflicting factors. As the number of loci increases, there is a larger number of recombinant genotypes for selection to act against, which enhances reinforcement. On the other hand, the high fitness genotypes are broken apart rapidly when there are many freely recombining loci. That makes the high fitness genotypes more rare, decreasing the effectiveness of the indirect selection that favors reinforcement. The net effect is that although reinforcement is enhanced when the number of epistatically interacting factors increases, the increase is not dramatic. In sum, preference reinforcement is affected by the number of loci that interact to cause hybrid incompatibility, as well as the strength of selection against hybrid recombinants.

View larger version (12K): In this window In a new window Download PPT slide   Figure 2. The relative impact on reinforcement of the number of epistatically interacting incompatibility loci under free recombination, from Equation 29.

	A FOUR-LOCUS MODEL

TOP ABSTRACT THE GENERAL MODEL HYBRID INCOMPATIBILITY FROM... A FOUR-LOCUS MODEL DISCUSSION APPENDIX 1 APPENDIX 1 APPENDIX 1 LITERATURE CITED

The genetically simplest case to which these results apply is one where the preference and trait are each controlled by one locus and a single pair of loci interact to cause epistatic incompatibility. This section uses the results of the last to study evolution of the preference in this simple situation. We have four motivations. First, we show how the results developed above apply to small numbers of loci as well as to more genetically complex situations. Second, we develop some new biological results. For example, we will see that there are cases in which reinforcement is doomed to failure. Third, we check the accuracy of the analytic results using exact simulations. Fourth, simulations are used to see if the analytic results for the haploid model generalize to diploidy.

Suppose that females carrying preference allele P₀ prefer to mate with males carrying display trait allele T₀, while P₁ females prefer T₁ males. Given a choice between one of each, a P_i female will mate with her preferred male 1 + _i times more often than the other male. Thus _i = 0 implies random choice, while a large value of _i implies a strong preference. (More details about this mate choice model can be found in KIRKPATRICK 1982 , in which the preference strength is parameterized slightly differently.) On the island, T₁ males have a viability advantage of s_T over the T₀ males, and we assume the display trait is not expressed in females. A pair of loci denoted A and B cause epistatic incompatibility; the recombinant genotypes A₀ B₁ and A₁ B₀ have fitness (1 - s_E) relative to A₀ B₀ and A₁ B₁ individuals. For simplicity, we assume all four loci are unlinked.

The continental population is fixed at the display trait and incompatibility loci (that is, for alleles A₀, B₀, and T₀). The preference can be polymorphic, however, and the frequency of allele P₁ on the continent is denoted p^'_P. On the island, we assume that the overall impact of natural and sexual selection favors the T₁ allele and is sufficiently strong to keep the display trait locus polymorphic in the face of migration. At the epistatic loci, we imagine that selection is sufficiently strong that the alleles A₁ and B₁ are near fixation on the island (m s_E), but migration from the continent continually introduces the alternative alleles. Last, for consistency with the analytic results we assume that all the parameters (m, s_T, s_E, ₀, and ₁) are much smaller than 1.

Given those assumptions, the first job is to relate the parameters just described to those of the general model. The simple expression for the preference equilibrium given by Equation 22 applies here. Without any loss of generality, we can measure the means of the preference and display trait in terms of allele frequencies, and so we have - ' = p_P - p^'_P and - ' = p_T. Because allele frequencies are free of nonheritable environmental effects, the preference heritability is h²_P = 1 and the phenotypic variance is equal to the genetic variance for both characters: ²_P = pq_P and ²_T = pq_T. A calculation shows that for weak preferences the correlation among mated pairs is (₀ + ₁). Last, epistatic selection involves only a pair of loci, and from Equation 27 we find that the intensity of hybrid incompatibility is I = .

Reinforcement at equilibrium:
How much will the preference on the island diverge from the continent? The answer, found using Equation 22 and the results above, is

(30)

To complete the picture, we need an expression for the display trait equilibrium, _T. One can show at the migration-selection equilibrium that its value is 1 - 2m/(s_T + ₁p_P - ₀q_P) (to first-order accuracy in the ã). Substituting that into Equation 30 leads to a cubic equation in _P that can be easily solved using, for example, Mathematica (WOLFRAM 1996 ). The solution is long, so we do not show it here, but we illustrate its most interesting features with examples shown in Figure 3 and Table 1.

View larger version (16K): In this window In a new window Download PPT slide   Figure 3. Difference between the frequencies of preference allele 1 on the island and continent as a function of its frequency on the continent for the four-locus model. and , results from the analytic approximation based on Equation 30; and , exact results from simulations. and , sT = 0.15, sE = 0.1, 0 = 1 = 0.1, m = 0.01. and , sT = 0.07, sE = 0.05, 0 = 1 = 0.05, m = 0.005.

As seen in Figure 3, there can be no divergence of the island preference (_P - p^'_P = 0) if there is no genetic variation for the preference among migrants from the continent (p^'_P = 0 or 1). When the migrants are polymorphic for the preference, however, genetic variation for the preference is also maintained on the island. That allows disequilibrium to build up between the preference locus and the other loci, causing reinforcement. (One can show that reinforcement also occurs if genetic variation for the preference is maintained on the island by mutation.) In short, we conclude that for reinforcement to succeed in the four-locus model with weak selection and migration, there must be standing genetic variation in the preference.

Simulations:
We ran exact simulations of the four-locus model for three reasons: to check the accuracy of the analytic approximations, to determine the stability of the equilibria, and to see how results from the haploid model might apply to diploids.

The simulations show that the analytic approximations are quite accurate. We measured the relative error of the approximations as (_P - _P)/(_P - p^'_P), where _P is the analytic approximation based on Equation 30 and _P is the exact result from simulation. This is a sensitive test of our analytic results for the effects of reinforcement: it measures the error in the approximation for the equilibrium (which is _P - _P) relative to the amount of reinforcement in the preference (which is _P - p^'_P). For the examples shown in Figure 3, the error is always <8.5%. Further, it decreases as the parameters become smaller. Table 1 shows that the relative error is proportional to the size of the parameters and becomes very small (at most a few percent) when all parameter values are much smaller than 1. These two properties strongly support our analytic results for the amount of reinforcement in the preference at equilibrium.

The simulations suggest that the equilibria for the four-locus model are globally stable. For any given set of parameters, populations evolve to the same equilibrium regardless of the initial conditions. (That conclusion can be supported analytically: we derived an approximation for the leading eigenvalue using Equation 18 and found that the equilibrium is always locally stable.) Thus it seems that history does not affect the outcome: it does not matter whether the island is isolated for a period before it starts to receive migrants from the continent.

Last, we built a diploid simulation model to see if the analytic results, which assume haploidy, might apply there. Diploidy raises the issue of dominance. Rather than explore all the possible effects of dominance, we made assumptions intended to make the diploid simulations as comparable to the haploid model as possible (and that avoid introducing additional parameters). Appropriate parameterizations for selection on the male display locus are provided by GOMULKIEWICZ and HASTINGS 1990 . We took the relative viabilities of the T₀ T₀, T₀ T₁ and T₁ T₁ genotypes, respectively, as 1/(1 + s_T), 1, and 1 + s_T. Matings occurred at the frequencies shown in Table 2. Regarding dominance and epistatic incompatibility, several studies of hybrid fitness imply that hybrid genotypes that are further from either of the pure parental genotypes have lower fitness (COYNE 1985 ; WU and DAVIS 1993 ; DAVIS et al. 1994 ; HOLLOCHER and WU 1996 ). We therefore assigned nonrecombinants (e.g., A₀ A₀ B₀ B₀) fitness of 1 + s_E, genotypes with one incompatible allele (e.g., A₀ A₀ B₀ B₁) fitness of 1, and genotypes with two pairs of incompatible alleles (e.g., A₀ A₁ B₀ B₁ and A₀ A₀ B₁ B₁) fitness of 1/(1 + s_E).

We quantified the discrepancy between the analytic haploid approximations and the exact result from simulations of the diploid model with the same measure that we used for the haploid simulations. Table 1 shows that the discrepancy for diploids is roughly an order of magnitude larger than it is for haploids. That is perhaps not surprising because the analytic results are an approximation for a haploid system and not a diploid one. Nevertheless, the discrepancy with the diploid simulations declines rapidly as the parameters become smaller. We conclude that our analytic results may give a reasonable quantitative guide for diploids when there is no dominance and when parameter values are very small.

	DISCUSSION

TOP ABSTRACT THE GENERAL MODEL HYBRID INCOMPATIBILITY FROM... A FOUR-LOCUS MODEL DISCUSSION APPENDIX 1 APPENDIX 1 APPENDIX 1 LITERATURE CITED

These results show that reinforcement of a mating preference is expected quite generally when one population receives migrants from another. We expect to see females prefer males from their own population when only three conditions are met: there is standing genetic variation for a female mating preference, mate choice produces a phenotypic correlation in mated pairs between the preference in females and some trait in males, and the average value of that trait differs between the populations. These criteria make sense: reinforcement requires that the preference can evolve, that the preference affects mate choice, and that males from different populations can be distinguished. Hybrid incompatibility will amplify reinforcement but, perhaps surprisingly, is not necessary.

Postzygotic isolation is weak in some groups of animals, such as birds (GRANT and GRANT 1997 ). Our results show that reinforcement of preferences can contribute to speciation in these cases, before strong hybrid unfitness develops. While the model here considers only a single preference and display trait, clearly the same process could develop in multiple preference-trait pairs, thus amplifying the overall contribution of reinforcement to prezygotic isolation. The strong postzygotic isolation emphasized as necessary to reinforcement in earlier theoretical studies (e.g., SVED 1981 ; SPENCER et al. 1986 ; LIOU and PRICE 1994 ) is apparently not necessary for the process to begin. Our results do not, however, show if or how reinforcement of preferences might finish the process of speciation by reducing genetic introgression to nil.

Hybrid incompatibility in some groups of animals is caused by epistatic interactions between loci that have diverged in the two populations (DOBZHANSKY 1937 ; MULLER 1942 ; COYNE 1992 ). When postzygotic isolation is caused by a pair of epistatically interacting loci, the model shows that reinforcement is enhanced by tighter linkage between those loci and the preference genes. Reinforcement is also affected by the number of epistatically interacting loci. When several sets of epistatic loci contribute to postzygotic isolation, their effects on reinforcement are additive. But because hybrid incompatibility of this sort is not necessary for reinforcement, there is apparently no paradox in the experimental results that suggest reinforcement that has occurred between some species pairs that show no postzygotic isolation in the laboratory (COYNE and ORR 1989B ).

Earlier models of reinforcement differ in one or more critical ways from ours. First, rather than studying a mating preference acting on a display trait, most models focus on reinforcement via assortative mating. For example, reinforcement has been modeled via the spread of genes that decrease migration between populations (MAYNARD SMITH 1966 ; BALKAU and FELDMAN 1973 ), that cause a change in flowering time (CROSBY 1970 ; DICKINSON and ANTONOVICS 1973 ;