Sébastien Gourbiere, Laboratoire de Théorie des Systèmes, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan Cedex, France Tel.: +33(0)4 68 66 1763; fax: +33(0)4 68 66 1760; e-mail: email@example.com
I use explicit genetic models to investigate the importance of natural and sexual selection during sympatric speciation and to sort out how genetic architecture influences these processes. Assortative mating alone can lead to speciation, but rare phenotypes’ disadvantage in finding mates and intermediate phenotypes’ advantage due to stabilizing selection strongly impede speciation. Any increase in the number of loci also decreases the likelihood of speciation. Sympatric speciation is then harder to achieve than previously demonstrated by many theoretical studies which assume no mating disadvantage for rare phenotypes and consider a small number of loci. However, when a high level of assortative mating evolves, sexual selection might allow populations to split into dimorphic distributions with peaks corresponding to nearly extreme phenotypes. Competition then works against speciation by favouring intermediate phenotypes and preventing further divergence. The interplay between natural and sexual selection during speciation is then more complex than previously explained.
Recent developments in adaptive dynamics and quantitative genetics theory allow for the simultaneous description of ecological interactions based on quantitative traits and of the genetics underlying the traits involved in these ecological interactions (see Dieckmann & Doebeli, 1999; Doebeli & Dieckmann, 2000, 2003 for the development of Adaptive Dynamics, and Doebeli, 1996; Kondrashov & Shpak, 1998 and Drossel & McKane, 2000 for the development of Quantitative Genetics). These studies use earlier ecological models of selection devoted to the study of ecological character displacement (e.g. Bulmer, 1974, Roughgarden, 1976, Slatkin, 1980, Taper & Case, 1985) and add a description of the genetics of reproductive isolation. They lead to compelling theoretical studies of speciation triggered by different types of ecological interactions. The majority of these studies demonstrated that sympatric speciation can easily occur because of competitive interactions (Doebeli, 1996; Dieckmann & Doebeli, 1999; Doebeli & Dieckmann, 2000, 2003; Drossel & McKane, 2000), but predation and mutualism could also lead to such ecological speciation processes (Doebeli & Dieckmann, 2000). Indeed, frequency-dependent selection due to all these different kinds of ecological interactions provides the disruptive selection which itself leads to the evolution of reproductive isolation to avoid production of unfit hybrids. These studies then strongly support the idea that sympatric speciation is likely to occur because of disruptive natural selection and subsequent evolution of assortative mating.
Surprisingly, little attention has been given to the relative importance of natural selection and sexual selection during the speciation process. Obviously, these models are not meant to be models of sexual selection as they do not include loci for a male specific trait and loci underlying female preference for that trait (as has been done with similar genetic frameworks by Higashi et al., 1999, and Kondrashov & Kondrashov, 1999), but they do implicitly account for sexual selection. Indeed, they consider evolution of quantitative traits in the female part of the population and males are assumed to have exactly the same phenotypic and genotypic distribution for these traits. ‘Female preference’ and ‘male trait’ involved in this choice are thus determined by the same set of loci. As pointed out by Kirkpatrick & Ravigné (2002), such assortative mating in and of itself is equivalent to sexual selection. In this paper, sexual selection is thus considered in a broad sense to include more or less any kind of assortative mating. Thus, given that both natural and sexual selection are included in these models, we can actually determine how each of them contributes to speciation.
Drossel & McKane (2000) made the only attempt to investigate the relative importance of sexual selection and natural selection in competitive speciation. They worked out the impact of assortative mating alone on the phenotypic distribution, before including competitive interactions. They demonstrated that neither selective assortative mating (hereafter ‘SAM’), i.e. when rare phenotypes are less likely to mate, nor nonselective assortative mating (hereafter ‘NSAM’), i.e. when all individuals are equally likely to mate, can lead to speciation. Accordingly, they conclude that formation of new species, after natural selection was subsequently included into the model, was because of natural selection. However, using another kind of genetic model, Kondrashov & Shpak (1998) clearly showed the possibility of speciation by means of NSAM and they argued that increasing the number of loci makes speciation easier. These two studies produced conflicting results: why did Drossel & McKane never obtain assortative mating speciation, even though they are implicitly dealing with an infinite number of loci? Drossel & McKane (2000) have proposed that such discrepancies may be due to the lack of genetic flexibility in Kondrashov & Shpak's model. This could either be due to the artificial constraint used by Drossel & McKane (2000) by assuming that the genetic variance of the offspring distribution does not vary through time and with respect to the parental genotypes (a common assumption of the infinitesimal model, Bulmer, 1980) or it could be due to the assumption of equal allele frequencies across the loci that Kondrashov & Shpak (1998) made in order to use the hypergeometric framework. So, in the absence of any clear explanation for these conflicting results, there is no coherent picture of how assortative mating can split a population into a bimodal distribution and (with the notable exception of Drossel & McKane, but using a nonexplicit genetic background) no attempt has been made to work out the influence of the mating process itself on competitive speciation models.
These results leave us with the question: how do assortative mating and competition contribute to speciation when the quantitative trait involved in both natural and sexual selection is determined by a finite number of loci? Clearly, to sort out the relative importance of assortative mating and natural selection in explicit genetic models of competitive speciation, where the number of loci and the allelic effect can vary, would also address the related question: does the genetic architecture change the relative importance of natural and sexual selection in competitive speciation?
Here, I investigate these questions using a hypergeometric model and an individual based model (IBM) of competitive speciation. First, I look at these models to sort out the effect of different levels of NSAM and SAM, before I include natural selection. Interestingly, as the number of loci (and the allelic effect) can vary, comparisons can be made between results obtained using explicit genetic models (Doebeli, 1996; Kondrashov & Shpak, 1998, and this study) and Drossel and McKane's quantitative genetic model. Secondly, I check the sensitivity of the results obtained with the hypergeometric framework to the assumption of equal allele frequencies at all the loci using the IBM. The questions are then ultimately addressed within a framework relaxing assumptions made to set up the analytical frameworks.
In the first part of this paper I consider no competition between individuals having different phenotypes. I show that assortative mating can drive speciation when considering a finite number of loci. However, as expected, rare (extreme) phenotypes’ disadvantage in finding mates and intermediate phenotypes’ advantage because of stabilizing selection strongly impede speciation requiring higher levels of assortativeness and particular initial conditions to proceed. Interestingly, increasing the number of loci also decreases the likelihood of speciation. This probably explains why Drossel and McKane did not obtain speciation by means of assortative mating alone, since they implicitly assumed an infinite number of loci.
In the second part of this study I include competition between individuals having different phenotypes. As suggested by earlier investigations, this generates frequency dependent competition which provides disruptive selection that contributes to speciation. However, the relationship between the strength of competition (i.e. the range of neighbouring phenotypes an individual competes with) and the disruptiveness of selection is highly nonlinear. This is mostly because both local and global competition (that is, competition between individuals having only very similar phenotypes and competition between all the individuals) lead to weak frequency dependence. It means that the usual criterion for speciation (which is that competition occurs only between individuals whose phenotypic differences are smaller than an upper limit) is not valid. There must also be a lower limit for the range of phenotypes an individual competes with, since competition that is not localised does not allow for speciation, because the phenotypic distribution then simply fits the carrying capacity.
From these two sets of results I conclude that sympatric speciation is harder to achieve than is claimed in recent papers. This is because they usually (1) do not take into account a mating disadvantage of rare phenotypes and (2) consider few loci (to save simulation time). I also suggest that the interplay between natural and sexual selection during speciation is slightly more complex than previously assumed. In the early stage, as is usually thought, natural selection must allow for the evolution of assortative mating. But as soon as sufficient assortment has evolved, the population is expected to split into two monomorphic sets of individuals with extreme phenotypes. Such a split obviously provides a selective advantage to intermediate phenotypes so that, in a second stage, natural selection actually acts against speciation, by preventing further divergence. I also argue that natural selection and sexual selection eventually lead to a non-linear evolution of reproductive isolation, a dynamical behaviour which has been briefly reported by Doebeli (1996).
The hypergeometric model for competitive speciation I investigated is closely related to Doebeli's model (1996). This framework has already been described elsewhere (Barton, 1992; Doebeli, 1996; Shpak & Kondrashov, 1999) and its use in modelling competitive speciation has been very clearly presented by Doebeli (1996).
The model deals with the simplest scenario of competitive speciation where mate choice is based on a quantitative character which also determines the competitive ability of individuals. Organisms are haploid, have sexual reproduction and generations are discrete and nonoverlapping. The quantitative trait is assumed to be defined by a set of diallelic loci (with allele 0 and 1). Loci have equivalent and additive phenotypic effects and recombine freely. I will refer to nl and a as the number of loci and the allelic effect, respectively. The phenotypes then lie between 0 and nla.
The mating process is described by the set of mating probabilities defined as a function of the difference between individual phenotypes. The function used is a normal distribution with a mean of 0 and a variance denoted . The probability for an individual of phenotype i to mate with an individual of phenotype j is then proportional to (eqn 12 in Doebeli, 1996):
Clearly, the lower σm, the stronger the degree of assortative mating. More precisely, the actual mating probabilities are derived from eqn 1 in two different ways to consider either selective or nonselective mating. To obtain the nonselective mating probabilities, the m(i,j) are first multiplied by the genotype frequencies. For any i, they are then divided by their sum over j to ensure that (for any i) the sum of mating probabilities equals 1. Pairing then occurs according to these set of normalized probabilities so that no genotype gets less mating. To model selective mating because of rare phenotype disadvantage, the m(i,j) are first multiplied by the genotype frequencies and then divided by their sum over i and j. This leads to weaker mating probabilities for rare extreme phenotypes.
The density-dependent fitness function used by Doebeli corresponds to the model of Bellows (1981) describing the population dynamics of species as a function of three parameters; the intrinsic growth rate of the population, a competition parameter and a parameter influencing the carrying capacity. Unfortunately, the carrying capacity is then defined as a function of these three parameters (see eqn 15 in Doebeli, 1996). This made the sensitivity analysis with respect to the three parameters impractical. Therefore I used the Lotka and Volterra equation (e.g. Kot, 2001, p. 51):
This function takes only two independent parameters, r and K, which I refer to as the intrinsic growth rate of the population and the carrying capacity, respectively. It is a discrete-time counterpart of the continuous-time Lotka–Volterra model which has been used in other investigations of competitive speciation (Dieckmann & Doebeli, 1999; Doebeli & Dieckmann, 2000, 2003).
For the purpose of this study, individuals have to compete with respect to the quantitative character which also determines their mating probabilities. First, the carrying capacities of different phenotypes are given by a normal distribution with a mean corresponding to the middle of the phenotypic range (i.e. nla/2) and a variance denoted . The carrying capacity of an individual with phenotype i is then:
Clearly, the lower σK the lower are the carrying capacities of extreme phenotypes relative to those of intermediate phenotypes. Secondly, the competition between individuals is assumed to decrease with the difference in their phenotypic values. As for the mating probabilities, the competition coefficients follow a normal distribution with a standard deviation denoted as σ. Thus, the competition coefficient between an individual having phenotype i and an individual having phenotype j is:
Obviously, the lower σ, the narrower is the range of phenotypes an individual with a given phenotype competes with. If σ tends toward 0, only individuals with the same phenotype compete with one another. If σ tends towards infinity, all individuals compete with one another whatever their phenotypes.
Accordingly, the fitness of an individual with phenotype i is:
where Ni,t denotes the effective density that an individual of phenotype i competes with. This density is given by:
Nj,t being the density of individuals with phenotype j at time t. Equation 6 can also be set as:
where Nt denotes the population size and pj,t the frequency of individual having phenotype j before viability selection.
The above equations are combined to allow the model to describe the evolution of the phenotypic distribution during the viability selection stage of the life cycle. The frequencies are then changed according to:
where qi,t denotes the frequency of individuals with phenotype i after selection, and where Wt is the mean fitness of the population.
Equations determining the changes in the phenotypic distribution during the reproductive stage are more complex because they describe both the mating process and the phenotypic distribution of offspring for any pair of parental phenotypes. The final distribution is given by (Doebeli, 1996, eqn 13):
where pij(k) describes the offspring phenotype distribution from parents with phenotypes i and j and di are normalizing constants obtained when modelling selective or NSAM as explained above. The values of pij(k) are defined according to some binomial and hypergeometric probabilities described by eqns 8–10 in Doebeli (1996).
The dynamics of the hypergeometric model including both natural and sexual selection is then given by iterating eqns 8, 9 together with the equation for the density of the population:
As in any hypergeometric model, the competitive speciation model described above (and more specifically, eqns 8–10 in Doebeli, 1996) is built on assumptions that allele frequencies are equal across the loci and that all genotypes in the same phenotypic class are equally frequent. The conditions under which this assumption would hold have been investigated by Shpak & Kondrashov (1999) and Barton & Shpak (2000). Such symmetrical models are likely to be useful under disruptive natural selection, as allele frequencies then stay or tend to become equal across the loci. However, this assumption becomes more problematic under stabilizing selection. Indeed, allele frequencies tend to become asymmetric because natural selection leads to the fixation of optimal haplotype with either allele fixed at different loci. Furthermore, it is not clear whether allele frequencies tend to be symmetric when assortative mating alone is considered, although this is strongly suggested by simulations run by Kondrashov & Kondrashov (1999). Thus, I set up an IBM encapsulating all the assumptions of the hypergeometric model described above, but relaxing the assumption of equal allele frequencies across the loci. This implies that offsprings’ genotypes are no longer defined according to the distribution given by eqns 8–10 in Doebeli (1996). Instead, any offspring phenotype is determined by random choice between the two parental alleles at each locus as expected under Mendelian inheritance and free recombination.
All the simulations run with the IBM start with asymmetric initial allele frequencies. For any individual, the genotype is determined according to a nonuniform pattern of mean allele frequencies across the loci. More specifically, every 1-allele is randomly assigned to one of two equally large sets of loci, one set with a probability of 1/3 and the other with a probability of 2/3. Within a set, all loci are equivalent and the exact position of each 1-allele random.
I investigated the hypergeometric model and the IBM by first considering assortative mating alone (i.e. considering only sexual selection and neglecting natural selection), then adding natural selection due to the carrying capacity distribution and finally including frequency-dependent competition due to interactions between individuals having different phenotypes. The first step is to look at the possibility of speciation by the mean of NSAM and SAM. I then assume that there is no viability selection, i.e. that qi,t = pi,t. The second step is to look at the possibility that stabilizing selection generated by the carrying capacities distribution prevents speciation by assortative mating. I then let σ tend toward 0, so that any individual competes only with individuals having exactly the same phenotype. Indeed, under these conditions, eqn 6 becomes Ni,t = Ni,t. The last step was to check how frequency-dependent competition contributed to speciation. The assumption that σ tends toward 0 is then relaxed to allow for nonzero σ values.
In this study, speciation is considered to occur when the phenotypic distribution becomes bimodal and when the mating probability between individuals belonging to each of the two modes falls off below a threshold value fixed to 10−5.
Below, I first report the results obtained using the hypergeometric model. I then check whether the conclusions are still valid when the assumption of equal allele frequencies across the loci is relaxed using the IBM. Thus, where unspecified, the results described will be those corresponding to the hypergeometric model.
I first considered the genetic architecture used by Doebeli, i.e. a set of nl = 20 loci with the same allelic effect a = 1. As shown in Fig. 1a, the phenotypic distribution always evolves to an equilibrium within 200 generations. There are only two types of equilibrium distributions. As long as individuals mate randomly or with a weak assortativeness (i.e. σm > 2.4), the asymptotic distribution is an approximately Gaussian distribution (lower panel in Fig. 1b) with a nearly constant phenotypic variance (slightly >5 in Fig. 1b). When assortativeness is strong enough (i.e. σm ≤ 2.4), the phenotypic distribution becomes bimodal (upper panel in Fig. 1b). The two modes correspond to the extreme phenotypes, whose frequencies increase to 0.5. The reproductive isolation between modes quickly increases to 1 and speciation is completed. Both species are then fully monomorphic. Allele 0 is fixed at all loci in the first species, whereas allele 1 is fixed at all loci in the other species. The phenotypic variance reaches its maximal value; (anl/2)2. These findings do not rely on the use of a hypergeometric model as a very similar nonlinear relationship emerges when using the IBM (Fig. 1b). However, there are two interesting differences between results obtained with these two frameworks. First, when speciation occurs (for lower values of σm), assortative mating hardly leads to fixation when using the IBM, although it does using a hypergeometric framework. Secondly, when speciation does not occur, loci become fixed for either loci allele running the IBM whereas (by assumption) all the loci stay polymorphic in the hypergeometric model. None of these differences alter the main result about NSAM speciation, as the threshold value for speciation to proceed is always around 2.4.
Such splits because of NSAM confirm and expand the previous results of Kondrashov & Shpak (1998). Speciation by means of NSAM is a basic feature of genetically explicit models, depending neither on the type of mating function used nor on the key assumption of equal allele frequencies across the loci made to set up the hypergeometric framework. These results all together apparently contrast with those obtained by Drossel & McKane (2000) who used the same mating function, but a nonexplicit genetic framework. This suggests that the possibility of NSAM speciation depends on the genetic architecture considered.
How does genetic architecture influence NSAM speciation?
To look at the importance of the genetic architecture for NSAM speciation, I examine the sensitivity of the threshold value to the number of loci nl and the allelic effect a.
Increasing nl or a both lead to a wider range of degree of assortative mating allowing speciation by pure assortment (Fig. 2a). When speciation occurs, the phenotypic distribution always splits into a purely dimorphic population with only the two extreme phenotypes. The range of degree of assortative mating allowing speciation increases almost linearly with both nl and a. An increase in a favours speciation much more than an increase in nl. For instance, starting with nl = 5 and a = 1, a ten-fold increase in the a or in nl increases the threshold value by more than 10- and four-fold, respectively.
These results are in good agreement with those of Kondrashov & Shpak (1998), although the threshold value does not seem to increase with the square root of nl, and contrast with the result obtained by Drossel & McKane (2000). Indeed, I show that an increase in nl (or a) increases the likelihood of NSAM speciation, whereas (implicitly) considering a very large number of loci Drossel & McKane (2000) reached the conclusion that such NSAM cannot lead to speciation. This contrast actually disappears when considering an increase in nl although scaling the allelic effect with 1/n. Indeed, such a scaling has to be done to keep the phenotypic range as a constant. Otherwise, for a given value of σm, an increase in nl or a (which leads to an increase of the total phenotypic range; anl) would artificially accentuate the actual level of assortativeness (by decreasing the ratio between the range of potential mating partners and the total phenotypic range). An increase in nl (and the corresponding decrease in a) then does not make speciation by the mean of NSAM easier (as suggested by Kondrashov & Shpak and by the results of Fig. 2a), but on the contrary more difficult (Fig. 2b). Again, the results obtained using the IBM are consistent with those obtained with the hypergeometric framework, although speciation then requires stronger degree of assortative mating (Fig. 2b). This unifies the results obtained from explicit genetic frameworks (Kondrashov & Shpak, 1998, and this study) and quantitative genetic models (Drossel & McKane, 2000). That the relationships displayed in Fig. 2b have a nonzero asymptote (Fig. 2b) is also consistent with the analytical results produced by Drossel & McKane (2000) to describe how the broadening effect of NSAM increases when sigma σm is decreased. Indeed, they established that the phenotypic variance increases with time but reaches an asymptotic value as long as σm is larger than a (nonzero) threshold value. Otherwise the variance of phenotypic distribution increases in an unlimited way, although this never allows for speciation (i.e. for the evolution of a bimodal distribution). However, it is clear that, by using an explicit genetic framework, speciation is allowed because such an increase in the phenotypic variance leads to the fixation of either allele. That speciation is always allowed when nl is increased in my explicit genetic model (potentially mimicking the framework used by Drossel & McKane, 2000), is then also consistent with their theory.
Thus, when the quantitative trait involved in mate choice is determined by a finite number of loci, NSAM can lead to speciation. However, mating can also select against individuals having rare phenotypes are less likely to find a suitable mate. We know from Drossel & McKane (2000) that when mating is selective, not only does speciation not proceed, but the phenotypic variance of the population no longer depends on degree of assortative mating. Unfortunately, as Kondrashov & Shpak (1998) only considered NSAM, we still do not know if the conclusion of Drossel & McKane (2000) holds for a genetic system involving a finite number of loci.
Does selection against rare phenotypes prevent assortative mating speciation?
To look at the importance of rare phenotypes’ disadvantage, I checked the threshold value in the same genetic conditions as in Fig. 2a, b. Figure 3a, b are then strictly analogous to Fig. 2, but for SAM.
As expected, rare phenotypes’ disadvantage often prevents assortative mating speciation to occur. Indeed, the threshold value is very much lower than when NSAM is considered. The threshold value still increases when the phenotypic range is widened by an increase in a, but surprisingly it no longer depends on nl. To estimate the actual mating probabilities under SAM, the mating probabilities (given by eqn 1) are normalized to ensure that their sum over all i and j phenotypes add up to one. But, this sum does not significantly increase when nl increases, because it reaches its maximum as soon as nl = 5. By contrast, to estimate the actual mating probabilities under NSAM, mating probabilities are normalized to ensure that their sum over j phenotypes adds up to one. The sum over j does significantly increase when nl is increased from 5 to 50. The actual mating probabilities then depend on nl. This explains why the threshold value of degree of assortative mating increases when the phenotypic range is widened by an increase in nl under NSAM (Fig. 2a), whereas it does not under SAM (Fig. 3a). However, scaling the allelic effect to keep the phenotypic range as a constant, I found that (as for NSAM) to consider more loci with fewer effect decreases the likelihood of speciation to proceed. Here, it is important to note that, using the IBM, speciation still arises for some level of degree of assortative mating, but only if the initial genetic variance is large enough. The threshold value of degree of assortative mating and the minimal value of the initial genetic variance allowing for speciation are reported on Fig. 3b. As for NSAM, a comparison with the results of Drossel & McKane (2000) can be made about the asymptote of the relationship displayed in Fig. 3b. According to Drossel & McKane (2000), when nl is very large, the phenotypic variance is expected to take on finite values not depending on σm. The phenotypic variance is then expected not to increase infinitely. That speciation appears to happen in more and more restrictive conditions with apparently no possibility of split when nl is very large (i.e. a zero asymptote) is thus consistent with the Drossel & McKane's theory.
To sum up, both NSAM and SAM lead to speciation when the quantitative trait involved in the mating choice is determined by a finite number of loci. For a fixed phenotypic range, the likelihood of such a speciation decreases with nl, but never reaches zero if mating is non selective. As stated by Kirkpatrick & Ravigné (2002), if mating is selective, speciation also depends on initial conditions: the initial genetic variance needs to be large. So far, I have not considered that the quantitative trait is under natural selection. To generalize these previous results, I assessed whether stabilizing selection (because of the differences in the carrying capacities of phenotypes) can prevent NSAM or SAM speciation from happening.
Does stabilizing selection prevent NSAM or SAM speciation?
The strength of stabilizing selection is determined by the parameter σk in my model. Low σk values lead to sharp distributions of the phenotypic carrying capacity corresponding to strong stabilizing selection. On the contrary, high σk values lead to very weak stabilizing selection.
Both NSAM and SAM can still produce speciation when stabilizing selection is included in the hypergeometric model (see Fig. 4a for NSAM and the upper series of Fig. 4b for SAM). As expected, the threshold values obtained are lower than in the absence of selection because stabilizing selection is opposite to speciation in that it favours intermediate phenotypes. The way the threshold value varies with the level of stabilizing selection is different under NSAM and SAM. Interestingly, including stabilizing selection does not prevent speciation by the mean of SAM as long as σk is larger than 10 and there is almost no change in the threshold values obtained with different σk values. This explains why there is only one series corresponding to the hypergeometric model in Fig. 4b. However, any further decrease in σk value strongly impedes evolutionary diversification so that there is virtually no more speciation whatever the genetic architecture considered (corresponding threshold values are not displayed as they are always lower than 0.05). On the contrary, when mating is nonselective, strong stabilizing selection gradually lowers the range of degree of assortative mating, allowing speciation (Fig. 4a). However, for both NSAM and SAM, stabilizing selection does not change the shape of the relationship between the threshold value and nl: the likelihood of speciation by NSAM or SAM still decreases exponentially with nl.
The figure can be very different when using the IBM. It is well-known that stabilizing selection leads to the fixation of 0 and 1 alleles at different loci as it allows producing only the optimum phenotype. Accordingly, there is a conflict between the effects of assortative mating and stabilizing selection. Stabilizing selection tends to produce a population of individuals all having the same genome consisting of an optimal mixture of 0 and 1 alleles; assortative mating tends to produce two sets of individuals whose genomes consist only of either 0 or 1 alleles.
When mating is nonselective, stabilizing selection always prevents speciation. The population starts splitting because of assortative mating, but after a few generations individuals with intermediate phenotype increase in frequency because of natural stabilizing selection. Most of the loci get fixed, although an equilibrium between the two processes allows maintaining some level of polymorphism (data not shown). Accordingly, speciation by the mean of NSAM when stabilizing selection is included mostly relies on the artificial assumption of equal allele frequencies across the loci imposed by the hypergeometric framework. Such an assumption does not allow for fixation of 0 and 1 alleles at different loci (as expected under stabilizing selection). Lowering the effect of stabilizing selection, hypergeometric models then clearly favour evolutionary diversification.
When mating is selective, speciation can still proceed although it requires specific conditions (Fig. 4b). It obviously requires stabilizing selection not to be strong and a high level of assortment. An additional and important requirement is a large enough initial genetic variance (see required values displayed in Fig. 4b). Indeed, a large genetic variance means that individuals with extreme phenotypes are initially present, although they are still less abundant than individuals with intermediate phenotypes. Because of assortative mating, those individuals with extreme phenotypes can increase in frequency while intermediate phenotypes eventually start becoming the less abundant ones. As mating is selective, intermediate phenotypes then experience difficulty in finding mates and the corresponding selective disadvantage can overcome the advantage those individual phenotypes have because of stabilizing selection. Clearly, the required level of degree of assortative mating for speciation is higher than in the absence of natural selection so that assortative mating is no longer able to generate speciation when nl is too high. Such a speciation never happens when mating is nonselective, because intermediate phenotypes do not suffer disadvantages in finding mates. Stabilizing selection then prevents the split of the population. If the initial genetic variance is weak, SAM no longer allows for speciation. On the contrary, SAM backs the effect of stabilizing selection as extreme phenotypes, being rare, also experience difficulty in finding mating partners. Alleles 0 and 1 then become fixed and the population is only made up of the optimal phenotype.
These results add to those of Kondrashov & Shpak (1998), Kirkpatrick & Ravigné (2002) and the previous results of this study demonstrating speciation by means of assortative mating in absence of stabilizing selection. Here, I show that SAM can still lead to speciation when stabilizing selection acts on the mating trait, although it requires the initial genetic variance to be large. Interestingly, selective mating can also oppose speciation if the genetic variance is initially weak.
Now, as long as only stabilizing selection is included, competition occurs between individual with the same phenotype. To understand the interplay between assortative mating and frequency dependent competition, I included interactions between different phenotypes.
Does frequency-dependent competition act for or against NSAM speciation?
I investigated the threshold value of nonselective degree of assortative mating allowing for different strengths of competition between phenotypes, i.e. different values of σ.
Similar relationships emerge while using the hypergeometric model (Fig. 5a) and the IBM (Fig. 5b), although quantitative differences exist between them. The common figure is a nonlinear relationship with a σ value allowing for speciation in the largest set of conditions.
This finding partially contrasts with a canonical result of the recent theory on competitive speciation: evolutionary diversification happens if the width of the competition coefficient distribution is lower than the width of the carrying capacity distribution. That is, competitive speciation happens if σ < σk. This criterion has been demonstrated analytically for asexual organisms (Dieckmann & Doebeli, 1999; Day, 2000; Doebeli & Dieckmann, 2000) and numerical investigations have shown this to be a good approximation for sexual organisms (Dieckmann & Doebeli, 1999; Doebeli & Dieckmann, 2000). The rationale is that under global competition (i.e. large σ values), all phenotypes suffer the same amount of competition as they all compet one another. As extreme phenotypes have lower carrying capacities, they never become the more abundant and the frequency distribution stays a unimodal one. If the range of competitors is decreased, extreme phenotypes compete with fewer individuals than intermediate phenotypes do. In this case, they get a selective advantage which eventually overcompensates for the disadvantage because of their lower carrying capacity. To decrease the range of competitors (lowering σ) then widens the range of degree of assortative mating that leads to speciation.
However, if σ is strongly decreased, the competition becomes local, i.e. it occurs only between individuals having very close phenotypes. The actual level of competition then strongly decreases and the fitness differences between intermediate and extreme phenotypes are then mostly due to their carrying capacities. Accordingly, when σ tends toward 0, the phenotypic distribution simply fits the carrying capacity distribution.
Thus, explanation for the nonlinear relationships displayed in Fig. 5a, b is simple. Frequency-dependent selection (required for competitive speciation) decreases when σ takes on high values (because the impact of competition on extreme phenotypes is increased), but also when σ is too weak (because competition is relaxed on intermediate phenotypes). Consequently, the usual criteria for speciation (i.e. σ < σk) cannot be a sufficient condition. There must be a lower limit for the range of phenotype an individual competes with (i.e. σ), as too local competition does not allow for speciation.
The two main purposes of this study were to sort out (1) how sexual selection and frequency-dependent natural selection contribute to influence sympatric speciation and (2) how this depends on the genetic architecture of the quantitative trait under both sexual and natural selection.
I investigated a hypergeometric model of competitive speciation closely related to Doebeli's (1996) model which initiated the recent Dieckmann & Doebeli (1999) and Doebeli & Dieckmann (2000, 2003) adaptive dynamic models. This model is interesting for two reasons. It uses the same hypergeometric framework that Kondrashov & Shpak (1998) used to demonstrate the possibility of speciation by the mean of assortative mating alone and results of this model can also be compared with those obtained using the quantitative genetic theory (Drossel & McKane, 2000). All these links allow addressing the two above questions. However, as any hypergeometric model, this model is built on the assumption that allele frequencies are equal at all the loci. As this assumption may be misleading in generating unstable solutions (Shpak & Kondrashov, 1999; Barton & Shpak, 2000), I also used an IBM to back the conclusions drawn from the hypergeometric framework. This strategy conforms to the need to unify existing speciation models (Kirkpatrick & Ravigné, 2002).
Speciation under sexual selection and stabilizing selection
Looking at the evolution of the phenotypic distribution under nonselective mating alone, I confirm that a strong enough level of assortative mating allows an initially unimodal distribution to split into two purely monomorphic species (Kondrashov & Shpak, 1998). This kind of assortative mating speciation (with complete loss of polymorphism) appears to be a general feature of hypergeometric models, and does not depend on the kind of assortative mating function used (threshold and interval-based mode functions in Kondrashov & Shpak (1998) or the classical exponential function in this study). Furthermore, speciation by NSAM is also widely obtained using the IBM, which means that it is a basic feature of any NSAM model involving a finite set of loci. I further extend Kondrashov & Shpak's (1998) results by considering rare phenotypes’ disadvantage in finding a mate. Such SAM still allows for speciation, although it requires higher levels of assortative mating and large enough initial genetic variance. These results also extend a study by Kirkpatrick & Ravigné (2002), who showed that SAM can easily lead to speciation, but working with the most favourable genetic architecture for speciation to proceed, i.e. a two-locus model (see results on the effect of the number of loci on the likelihood of speciation discussed below).
That speciation by assortative mating is more likely when the quantitative trait is determined by a high number of loci (Kondrashov & Shpak, 1998) apparently contrasts with the result obtained using the quantitative genetic theory (Drossel & McKane, 2000). Indeed, implicitly assuming a very large number of loci by using quantitative genetic models, these authors never observed speciation. However, this is only an apparent paradox, which is solved when the allelic effect is scaled to keep the phenotypic range as a constant. Speciation by means of assortative mating is then less likely when considering more loci with smaller effects.
Thus, by considering selective and nonselective mating under different genetic architectures, this study bridges the gaps between results previously obtained by Kondrashov & Shpak (1998), Drossel & McKane (2000) and Kirkpatrick & Ravigné, (2002) and provides us with a coherent picture of how assortative mating can lead to speciation when neither stabilizing selection nor competitive interactions are considered.
Speciation can still proceed when stabilizing selection is included giving intermediate phenotypes a selective advantage. However, this requires the initial genetic variance to take on high enough values. If the phenotypic distribution is already broad, assortative mating allows individuals with extreme phenotypes to increase in frequency. As they become rare, intermediate phenotypes then experience difficulty in finding mates and the corresponding selective disadvantage can overcome the advantage of intermediate phenotypes because of stabilizing selection. Such a possibility does not appear if mating is nonselective, so that stabilizing selection generally prevents nonselective mating speciation. Given the level of genetic variance required for SAM speciation to occur, it is very likely that this process actually acts only in a second stage of speciation, after natural selection has already broaden the phenotypic distribution. As suggested by the final phenotypic distributions I obtained, it could then be very efficient in splitting incipient species apart. On the contrary, as long as the initial genetic variance is low, a selective disadvantage of rare phenotypes in finding mates reinforces the effect of stabilizing selection. In this case, alleles 0 and 1 quickly get fixed so that the population contains only the optimal phenotype. Thus, if no other process allows an initial increase of the genetic variance, selective mating acts against speciation preventing extreme phenotypes to increase in frequency.
An important conclusion from these results is that recent papers supporting sympatric speciation have probably overestimated the likelihood of sympatric speciation, because they do not account for mating disadvantage of rare phenotypes which, in the initial stage of speciation, correspond to extreme phenotypes (Doebeli, 1996; Dieckmann & Doebeli, 1999; Doebeli & Dieckmann, 2000, 2003). Another reason why these models have probably overestimated the likelihood of sympatric speciation is that they generally deal with a small number of loci, which strongly enlarges the range of parameters allowing speciation. These results provide two possible answers to the question (Bridle & Jiggins, 2000): why is sympatric speciation theoretically so easy?
Importance of frequency-dependent selection for speciation by reinforcement and pleiotropy
Following Drossel & McKane (2000), I did not explicitly account for the evolution of assortative mating but looked at the relative importance of assortative mating, stabilizing selection and competition. The results of this study apply directly when a trait determining the habitat specialization pleiotropically affects the mate choice. But, these results also give new insights into sympatric speciation by reinforcement as explicitly modelled in recent papers (Dieckmann & Doebeli, 1999; Doebeli & Dieckmann, 2000, 2003).
Pleiotropy is an easy scenario for speciation as we do not have to consider the evolution of assortative mating which appears as a by-product of habitat specialization. The usual picture of this type of speciation is host shift exemplified by the apple maggot flies, treehoppers, phytophageous moths and mimetic butterflies (Jiggins et al., 2004). This generally implies a few morphs corresponding to a new host. However, there is no reason why finer adaptive differentiation could not lead to some prezygotic isolation, as encapsulated in the multiple phenotypes model presented in this paper. New phenotype then increases in frequency, exactly in the same way as when an emerging well-adapted morph increases in frequency by using a still unexploited (unique) host. If mating is not selective, it then contributes to strongly increase the genetic differentiation between individuals. On the contrary, evolutionary diversification can be strongly impeded if individuals with a new phenotype or morph suffer a selective disadvantage in looking for still rare mates. This is especially important if habitat specialization pleiotropically leads to strong level of assortative mating, for instance when only individuals with exactly the same niche mate together. However, if natural selection has previously allowed for the evolution of some large enough genetic variance, SAM is also expected to strongly contribute to the genetic differentiation of the population into two monomorphic sets of individuals having extreme phenotypes. Interestingly, it means that sexual selection can induce maladaptive differentiation by generating ecologically unfit extreme phenotypes. Natural selection then acts to prevent the split, which means to unexpectedly reduce evolutionary diversification.
The trickier scenario for sympatric speciation is reinforcement, as we need to explain why assortative mating would evolve in the first place. The usual understanding of competitive speciation is that prezygotic isolation evolves to reinforce post-zygotic isolation due to density and frequency-dependent competition. The cost of hybridization is because of the production of offspring with intermediate phenotype, who experience the strongest level of competition. This is thought to generate a gradual increase of prezygotic isolation until the natural selection gradient acting against intermediate phenotypes vanishes. Here, I show that if strong enough assortative mating evolves, it is able to split the population into two extreme phenotypic clusters. This happens suddenly around a threshold value of assortativeness. Interestingly, the evolution of such a level of assortative mating then generates the conditions for the invasion of mutant playing a strategy corresponding to lower level of assortative mating. Indeed, any mutant with a lower level of assortative mating would get a selective advantage by producing offspring with intermediate phenotypes, who can then fit the empty space in the phenotypic range. Evolution of prezygotic isolation is then expected to go back to a lower level of assortative mating. However, as a consequence of this backward evolution, the frequencies of intermediate phenotypes will strongly increase and conditions for the evolution of a stronger level of assortative mating will be recovered. Thus, the highly nonlinear relationship between the level of assortative mating and the phenotypic variance raises up the possibility for nonlinear evolution of prezygotic isolation with the possibility of speciation-despeciation cycles. This might explain the nonlinear evolution of assortative mating briefly reported by Doebeli (1996, p. 903).
Another point of this paper is the relationship between the level of differentiation and the strength of competition between phenotypes. This relationship is also nonlinear with an intermediate level of competition providing the best condition for species differentiation. Indeed, both low and high levels of competition between phenotypes do not generate the frequency dependent selection required for speciation to happen. Although a simple finding, that low σ values can make speciation harder or impossible has not been investigated yet. This contrasts with the classical result that competition lead to speciation when σ < σK (e.g. Dieckmann & Doebeli, 1999; Day, 2000; Doebeli & Dieckmann, 2000, 2003). I show that σ must also constraint by a lower limit as when σ approaches 0 in value, only individuals with exactly the same phenotypes compete one another and natural selection then make the phenotypic distribution fitting the carrying capacity distribution, which is assumed to be unimodal. More generally, it means that conditions for a branching process to happen (as determined in the analytical Adaptive Dynamics context) do not necessarily correspond to conditions for speciation, as suggested in a recent review on Adaptive Dynamics (Waxman & Gavrilets, 2004).
To conclude, competitive speciation is harder to achieve and slightly more complex than claimed in recent key papers about sympatric speciation. It is harder to achieve because recent models did not account for selective disadvantage of rare phenotypes and dealt with a small number of loci, two conditions which strongly favour speciation. It is slightly more complex than previously thought as natural selection favours divergence in the earliest stage of speciation, but latter acts against divergence to prevent ecological maladaptation generated by sexual selection.
I am deeply grateful to James Mallet for all his comments throughout the achievement of this work and for his extremely helpful suggestions on early versions of this manuscript. I would also like to thank John Welch and John Maynard-Smith for stimulating discussions. It is a pleasure to thank Adam Eyre-Walker who kindly invited me to the Centre for the Study of Evolution (CSE) at the University of Falmer, during the years 2002 and 2003. Financial support was provided by a Marie Curie post-doctoral fellowship (HPMF-CT-2001-01230). This contribution is dedicated to the loved memory of my father whose honesty, ability to observe and knowledge will stay forever as invaluable landmarks for my professional and family life.