COMPETITION FOR A CONTINUOUS RESOURCE: POLYGENIC MODELS OF DIVERGENCE
Jitka Polechová1,2
1Institute of Evolutionary Biology, University of Edinburgh, Edinburgh, Scotland. 2Department of Ecology and Evolutionary Biology, University of Tennessee, 37996 Knoxville, Tennessee.
Present & corresponding address: 2 Tel. ++1 865−974−0049 Fax. ++1 865−974−3067 email:
[email protected]
Running title: Competition for a continuous resource
Abstract I show that disruptive selection readily arises under competition for a single resource and frequency−dependent selection when a trait range is limited so that the optimal variance cannot be reached under linkage equilibrium. However, with competition for a single resource (and density dependence), disruptive selection arising from frequency−dependent competition on a limited range is likely to be too weak to drive divergence. I illustrate the importance of cost−free assortment for the evolution of population divergence under moderate disruptive selection. Further, I show that even infinitesimal disruptive selection is sufficient to stabilize the symmetric (a.k.a. hypergeometric) model and that the core assumption of the symmetric model, that frequencies of equivalent genotypes are stable, indeed holds under a wide range frequency−dependent competition and disruptive selection. Therefore, the symmetric model can be a valid approximation of a quantitative trait under frequency−dependent and disruptive selection − for example in assessing ecologically driven speciation.
Keywords disruptive selection, frequency−dependent competition, speciation, hypergeometric model, stability
1
Introduction Frequency dependent selection has long been known to play an important role in sympatric speciation, since it allows coexistence of nascent species − disruptive selection will tend to eliminate variation unless variance is stabilized by frequency−dependent competition (Maynard Smith, 1966; Udovic, 1980; Felsenstein, 1981). Competition among conspecifics is as well long considered to promote sympatric and parapatric speciation (Darwin 1859, Lack 1954; see Schluter 1999, Chap 6 for a review on character displacement within species.) Recently, however, the frequency dependent competition among individuals sharing a single resource has been argued to be an important force, that may drive sympatric speciation (Doebeli, 1996a; Dieckmann and Doebeli, 1999). It has been argued that in these examples, it is the limitation of the trait range that leads to disruptive selection, and that the divergence is mainly facilitated by the absence of a cost to assortative mating, rather than by the modest disruptive selection (Gavrilets, 2004: Ch. 9, p. 309, 328−330; Polechová and Barton, 2005). In this paper, I emphasise that combination of the strength of frequency−dependent selection and limitation of the trait range which together drive disruptive selection can be easily deduced from the estimated equilibrium distribution on an unlimited range, and only weak disruptive selection arises unless the limitation of the trait range is strong. I use the symmetric model (Kondrashov, 1984; Barton, 1992; Doebeli, 1996a,b) to model a quantitative trait that is determined by 2 − 25 loci, and test when the assumptions of the model are valid. I concentrate on evolution in single populations (i.e., in sympatry) and model the evolution of a quantitative trait in order to understand how the disruptive selection that arises from competition for a one−dimensional resource could drive the evolution of reproductive isolation. Mostly, I model divergence in a single quantitative trait, and only briefly illustrate the evolution of association with an independent assortment trait. I begin with a Gaussian resource distribution, and competition between similar phenotypes, as in Roughgarden’s (1972) model. This results in no disruptive selection at equilibrium on an infinite range; disruptive selection arises only when the phenotypic range is limited (as e.g. in Doebeli, 1996a; Dieckmann and Doebeli, 1999; Bürger and 2
Gimelfarb, 2004).
briefly illustrate the evolution of association with an independent assortment trait. I begin with a Gaussian resource distribution, and competition between similar phenotypes, as in Roughgarden’s (1972) model. This results in no disruptive selection at equilibrium on an infinite range; disruptive selection arises only when the phenotypic range is limited (as e.g. in Doebeli, 1996a; Dieckmann and Doebeli, 1999; Bürger and Gimelfarb, 2004). Various approximations can be made to model a quantitative trait: at one extreme, modelling a single locus or an asexual genome, and at the other, assuming a large or even infinite number of unlinked loci of small effect (Bulmer, 1980; Zhivotovsky and Gavrilets, 1992; Nagylaki, 1993). Bürger and Gimelfarb (2004) recently carried out extensive analysis of 2−8 loci with additive effect under frequency and density dependent selection, revealing numerical and some analytical results, especially regarding maintenance of polymorphism. General multilocus methodology in principle allows analysis of an infinite number of loci (Barton and Turelli 1991), and gives approximations under various assumptions: quasi linkage−equilibrium (Barton and Turelli, 1991; Kirkpatrick et al. 2002), epistatic fitness (Barton and Turelli, 1991), or for any fitness with a smaller number of loci. However, direct computations are hardly feasible for more than a few loci, since n loci imply at least 22 n genotypes in the diploid model. Here I use the symmetric (a.k.a. hypergeometric) model (Kondrashov, 1984; Barton, 1992; Doebeli, 1996a,b), which assumes that all genes are biallelic, and have equal and additive effect. I denote the two alleles as + and −. The key assumption is that equivalent genotypes (coding the same phenotype, e.g.: +−− , −+−, −−+) are equally frequent. This allows numerical analysis of the evolution of a large number of loci (up to about a hundred loci). Multimodal and asymmetric distributions (Kondrashov, 1983a,b) can be described with the model. However, the equilibrium may or may not be stable towards fluctuations in frequencies of equivalent genotypes. If this assumption fails, then symmetric solutions are not relevant. Symmetric solutions have high variance under no selection and so they are unstable under stabilizing selection that favours low variance − any particular genotype (++− vs. +−+, say) close to the optimum can fix. It has been suggested (Kondrashov, 1984) that in general, the symmetric model is stable under disruptive selection but not under stabilizing selection. It is therefore essential to determine under what conditions the symmetric model is asymptotically unstable 3 (Note that the existence of locally towards fluctuations that violate its assumptions.
stable symmetric solutions does not mean that asymmetric solutions with widely
It has been suggested (Kondrashov, 1984) that in general, the symmetric model is stable under disruptive selection but not under stabilizing selection. It is therefore essential to determine under what conditions the symmetric model is asymptotically unstable towards fluctuations that violate its assumptions. (Note that the existence of locally stable symmetric solutions does not mean that asymmetric solutions with widely different frequencies of genotypes cannot also be stable.) The variance is not generally maintained: some of the alternative optima would fix unless high variance is stabilised − e.g. by frequency−dependent selection. Although the importance of frequency dependent selection and disruptive selection for stabilizing either of type of instability (shifts in the mean and equivalent allele frequencies, respectively) has been demonstrated (Barton and Shpak, 2000), there has been no detailed analysis that shows how to predict what strength of selection would be sufficient for both unimodal and bimodal equilibria to be stable towards fluctuations in the mean as well as towards genotypic fluctuations. The symmetric model allows fast computations and efficient testing of the accuracy of analytical predictions based on approximations that rely on assumptions about the variance of phenotypes, for example, the infinitesimal model (Bulmer, 1980). It has been used as a computationally tractable approximation for a quantitative trait in several speciation studies (Kondrashov, 1983b; Doebeli, 1996a; Kondrashov and Kondrashov, 1999) but the stability of the solution towards fluctuations in genotype frequencies, as assumed by this symmetric model, has never been tested in these studies. I show that when modelling additive traits, the core assumption of the symmetric model (that equal genotype frequencies for a given phenotype are stable) is fulfilled for a wide range of conditions which we can specify. That extends the justification for the use of this model.
4
Methods Population growth: I model a population with survival dependent on a single quantitative trait, z. I first assume that population competes for a single resource and later extend the model to bimodal resource distribution. Following Roughgarden (1972), logistic growth of the population is regulated by its effective density Ψ relative to carrying capacity (K). Intrinsic growth rate, r(z, Ψ) representing Malthusian fitness W(z, Ψ)) is defined as follows: ΨHz, ΨL y i z, j1 - z W Hz, ΨL = r Hz, ΨL = r0 j KHzL { k where Ψ = HΨ * GL HzL
(1)
Ψ Hz, t + ∆tL = Ψ Hz, tL H1 + ∆t r Hz, ΨLL Here, Ψ is the effective density, determined by a convolution Ψ = Ψ * G between Ψ, the population density and G, Gaussian competition function with variance Σ2c . That is, population density Ψ is weighed by intensity of competition G between similar 1 phenotypes of intensity .Carrying capacity K has optima at z0 ± L and variance Σ2k Σ2 c
around the optimum. K is unimodal in the basic model where L = 0 − the selection would be stabilizing on an infinite range. Later, I generate disruptive selection directly, using a bimodal resource distribution L > 0 (and L > Σk ) or a U−shaped disruptive S z2
2 − 1 + k. (I formally set Σ2 selection. For the latter, the fitness is given by: r(z) > e k
1 equal to , choose S* < 0 and modify the growth rate (Eq. 1) by setting the fitness of S*
intermediate phenotypes fixed to k (small)). In numerical solutions, I use the
Ψ Hz,ΨL approximation r(z) = r0 Exp[- ] to avoid negative fitnesses; ro being the KHz,Σk L
maximum growth rate. I scale the allelic effect Α according to the number of loci, Α = Zm . The phenotype then lies in the phenotypic range < -Zm , Zm > − trait range is n
independent on the number of loci. The exact definitions are below. The diploid phenotype, z, is determined as a sum of the phenotypic values of two gametes, i and j.
5
-Hi+ j-n-z0 ± LL n m E K(z) = !!!!!!!!!! ExpA n2 2 Σ2k 2 2 Π Σk Z 2n -Hi+ j-yL2 Zm2 n ExpA E ΨHyL ΨHz, Σc , ΨL = !!!!!!!!!!! â 2 Σ2 n2 Zm
2
Z2
m
By definition, any f @± LD º f @+LD + f @-LD. 2Π
y=0
c
(2)
The life cycle is: random union of gametes ® selection ® meiosis. (The population densities are shown after random union, unless otherwise stated). In the main model, selection is on diploids; the model with an independent trait for assortment is haploid for simplicity. There is no mutation in the model: genetic variability arises from the initial distribution of phenotypes with symmetric genotypic frequencies. Generations are discrete, the generation time is ∆t = 1 and maximum growth rate is r0 = 1. Also, without a loss of generality, I set z0 = 0. Therefore, the phenotypic values are both positive and negative, as they represent departures from the optimum set to zero. Throughout, I assume that the boundary to phenotypic range Zm is symmetrical. Note that the regulation of population growth is the same as the one used in Dieckmann and Doebeli (1999). The determination of the trait is similar but symmetry between loci is not assumed by Dieckmann and Doebeli (1999) − their analysis of sexually reproducing populations is based on individual based simulations. I use the hypergeometric (= symmetric) model, as its symmetry assumption allows easier exploration of a wider range of key parameters (e.g. trait range, strength of frequency−dependent selection) without the complications of random drift that are inevitable with individual based simulations. Analytical predictions for the infinite range (Roughgarden 1972) can be used to predict the occurrence of disruptive selection on the limited range and hence the stability of symmetric allele frequencies in the numerical solutions.
Symmetric model, stability analysis: In the symmetric model (Kondrashov, 1984; Doebeli, 1996a; Barton and Shpak, 2000), a phenotype is determined by additive biallelic loci (labeled + and −) which have equivalent effect (that is, through the number of plus or minus alleles). Crucially, it assumes that the solution is symmetric, in the sense that equivalent genotypes (coding the same phenotype, e.g., +− and −+) are equally frequent. The validity of the symmetry 6 assumption, as well as the stability of the solutions towards other fluctuations, is tested.
In the symmetric model (Kondrashov, 1984; Doebeli, 1996a; Barton and Shpak, 2000), a phenotype is determined by additive biallelic loci (labeled + and −) which have equivalent effect (that is, through the number of plus or minus alleles). Crucially, it assumes that the solution is symmetric, in the sense that equivalent genotypes (coding the same phenotype, e.g., +− and −+) are equally frequent. The validity of the symmetry assumption, as well as the stability of the solutions towards other fluctuations, is tested. The stability analysis of the symmetric model was developed by Barton and Shpak (2000). I therefore provide only general description of their analysis. We can distinguish three kinds of instabilities in the symmetric model. First, concordant fluctuations that result in a change of phenotypic distribution (e.g. ++, −−¯) but keep equivalent genotypes equally frequent. Here, means an increase of the frequency of specified genotype; ¯ a decrease. The concordant fluctuations are denoted by p = 0 and are described by n nonzero eigenvalues (and corresponding eigenvectors). Second, there can be discordant changes in allele frequencies that keep the phenotype constant but equivalent phenotypes diverge in frequencies, so that the assumption of the symmetric model is violated (e.g. ++− , +−+ ¯. These are denoted as class p = 1 and described by n − 1 eigenvalues). Third, class p = 2 describes discordant changes in associations between loci which keep both the phenotype and allele frequencies constant. Here, the class p = 2 corresponds to an increase in pairwise linkage disequilibrium between one pair of loci, and a decrease in the disequilibrium between another pair (e.g.++−−, −−++, −++−¯, +−−+¯). Classes with p > 2 corresponds to discordant changes in associations of higher orders (defined as in Turelli and Barton, 1990). The fluctuations n+2 in associations are described by p = 2 to p = , and the number of eigenvalues 2
decreases by 2 with each class. We will see that fluctuations in associations never result in any instability in the numerical calculations: the higher order terms decay very quickly with no linkage, so selection would need to be extraordinarily strong to compensate. In the symmetric model, testing the stability of the phenotypic distribution is feasible for about 30 loci, since the number of variables traced is of order n3 . Stability towards fluctuations in genotype frequencies is a key assumption of the symmetric model − if the distribution is unstable towards discordant changes (i.e. Λ1,. > 1), the symmetric model fails. Here, Λ1,. is any eigenvalue describing stability towards discordant fluctuations in genotype frequencies; the first index describes the class of fluctuations, p; the second index describes the order of eigenvalues within a class. The first 7
eigenvalue is by definition the largest in its class. The solution stays stable if every
fluctuations in genotype frequencies is a key assumption of the symmetric model − if the distribution is unstable towards discordant changes (i.e. Λ1,. > 1), the symmetric model fails. Here, Λ1,. is any eigenvalue describing stability towards discordant fluctuations in genotype frequencies; the first index describes the class of fluctuations, p; the second index describes the order of eigenvalues within a class. The first eigenvalue is by definition the largest in its class. The solution stays stable if every eigenvalue Λ < 1. Frequency dependence directly induces stability towards shifts of the mean phenotype (p = 0), and can affect other fluctuations (especially, for p = 1) via disruptive selection on a limited range. Therefore, the differential that determines the stability matrix needs to include a frequency−dependent component, proportional to the derivative of fitness as a function of genotype frequencies (see Barton and Shpak, 2000, Eq. 4 and 5). Code in Mathematica (Wolfram Research) is available on request.
Results
Frequency−dependent competition for a single resource Single trait
First, I focus on evolution in a population competing for a single resource. The fitness of a phenotype z is defined following Roughgarden (1972). There is a continuous solution to this model, where r(z) = 0 for all phenotypes, and effective density ΨHzL = HΨ * GL HzL matches the carrying capacity KHzL (* denotes convolution). The solution has mean equal to the optimum (for symmetric competition function, G) and 2 the variance of the phenotypic distribution is Σ` = Σ2 − Σ2 (where Σ2 is the variance z
k
c
k
of carrying capacity K and Σ2c is the variance of the competition function G). Phenotypic variance Σ2z is maintained by frequency−dependent competition as soon as it is stronger than stabilizing selection (Σ2c < Σ2k ); and it can be shown that the solution is exactly Gaussian for Gaussian G and K. However, in asexual populations, the continuous solution can be unstable towards perturbations of the growth rate (specifically, G and K) if mutation is very low, leading to impossibility of stable coexistence of divergent phenotypes (see Gyllenberg and Meszéna, 2005). For sexually reproducing populations, recombination prevents persistence of discrete clusters, resulting in a continuous distribution. Slatkin (1979) shows that a Gaussian solution is favoured by selection 2 under different underlying genetics, and if the8distribution is Gaussian with variance Σ`z
= Σ2k − Σ2c , there is no selection at equilibrium.
solution can be unstable towards perturbations of the growth rate (specifically, G and K) if mutation is very low, leading to impossibility of stable coexistence of divergent phenotypes (see Gyllenberg and Meszéna, 2005). For sexually reproducing populations, recombination prevents persistence of discrete clusters, resulting in a continuous distribution. Slatkin (1979) shows that a Gaussian solution is favoured by selection 2 under different underlying genetics, and if the distribution is Gaussian with variance Σ`z
= Σ2k − Σ2c , there is no selection at equilibrium. Clearly, competition for a single resource in the Roughgarden (1972) model allows maintenance of phenotypic variance but does not drive divergence. Although recent analysis (Dieckmann and Doebeli, 1999) seemed to show that the divergence in both clonal and sexually reproducing populations is driven by competition for a continuous resource, divergence in sexual populations (Fig 3 in Dieckmann and Doebeli, 1999; Σ2c = 0.16, Σ2k = 1, Zm = 0.5) occurred because the trait range was limited so that the 2 optimal variance (Σ` = Σ2 − Σ2 = 0.84) could not be reached, even with two marginal z
k
c
phenotypes at Zm = ± 0.5. Divergence in asexual populations (Dieckmann and Doebeli, 1999) has a different cause: transient selection, which can arise in a process of matching the continuous Gaussian solution. Transient disruptive selection is manifested to the greatest extent if mutational variance is very small (Σ2m d 0.001), when the clonal population evolves via branching, in direction of a fitness gradient (as in Dieckmann and Doebeli, 1999). A single phenotype first evolves towards the highest carrying capacity at z = z0 , and then diverges as the growth rate at z0 ± Ε is greater than zero. Divergence continues further, under low mutation after the N phenotypes match the respective unstable equilibrium: selection diminishes rapidly as more phenotypes evolve, so that the residual selection tends to zero (Barton and Polechová, 2005). 2 If the phenotypic distribution moves away from the optimum (z` = 0, Σ`z = Σ2k − Σ2c ), the
mean fitness of the population decreases. Figure 1 shows that the mean fitness of the population changes slowly as variance increases if the mean phenotype is at the optimum; if the population mean is away from the optimum, average fitness increases with increasing variance. Figure 1
With frequency−dependent selection, restricting the phenotypic range to be narrower than it is favoured by the selection will always generate disruptive selection for extreme phenotypes, even though it might be weak and the fitness might be convex only near the 9
margin of the trait. The disruptive selection arises because marginal phenotypes
With frequency−dependent selection, restricting the phenotypic range to be narrower than it is favoured by the selection will always generate disruptive selection for extreme phenotypes, even though it might be weak and the fitness might be convex only near the margin of the trait. The disruptive selection arises because marginal phenotypes experience a lower impact of competition as the density of similar phenotypes is disproportionally lower near the margins (there are no phenotypes outside the trait range), giving them a net advantage. I show that there is a simple condition that determines when disruptive selection arises on the whole phenotypic range and stabilizes the fluctuations in genotype frequencies in the symmetric model. I assess how severe the range limitation and frequency dependence should be to possibly facilitate the evolution of a bimodal distribution. First, I show that weak disruptive selection arises readily with moderate range restrictions, and is sufficient to stabilize the unimodal solution under the symmetric model. (Note that the symmetric solution under stabilizing selection is unstable towards fluctuations in genotypic frequencies, as they can fix for any combination with same phenotypic value; yet, the symmetric model assumes that frequencies of all genotypes are equal.) I begin with a simple numerical example: 10 loci, Σ2c = 0.5 and phenotypic range , Σ2 º 1, r = 1, ∆t = 1. The variance favoured by selection is Σ` = Σ2 − Σ2 = 0.5. k
0
z
k
c
(Half of the range therefore corresponds to about 3Σ`z : that means a rather weak restriction, as only 2% of the hypothetical equilibrium distribution on the infinite range are outside of the trait range.) Although disruptive selection is clearly generated, the solution stays unimodal (Fig. 2A, B) and the variance after random mating, Σ2z = 0.21 does not largely differ from the variance at linkage equilibrium Σ2LE = 21 n Α2 = 0.2 (the variance after one generation of selection is Σ2z = 0.23). Pairwise linkage disequilibrium D1,2 stays very low − around 0.0018. Due to frequency−dependent competition, high variance is maintained and the mean phenotype is stable, Λ0,1 = 0.83 (Λ0,1 is the largest eigenvalue describing stability towards shifts of the mean phenotype. Details of the stability analysis are described in the methods.) The solution is stable towards discordant fluctuations in allelic 10
frequencies as well (described by Λ1,1 ), although rather weakly: 1 - Λ1,1 = 0.004. The equilibrium solution is always stable towards fluctuations in associations between loci
Due to frequency−dependent competition, high variance is maintained and the mean phenotype is stable, Λ0,1 = 0.83 (Λ0,1 is the largest eigenvalue describing stability towards shifts of the mean phenotype. Details of the stability analysis are described in the methods.) The solution is stable towards discordant fluctuations in allelic frequencies as well (described by Λ1,1 ), although rather weakly: 1 - Λ1,1 = 0.004. The equilibrium solution is always stable towards fluctuations in associations between loci (p³2). The largest eigenvalue in the corresponding class of fluctuations, p ³ 2 is Λ2,1 = 0.50 (Λ2,1 is the largest eigenvalue describing stability of pairwise linkage disequilibria) − that is because there is no linkage and mating is random, so that linkage disequilibrium decays rapidly. Figure 2
Next I estimate when the limitation of the range with frequency−dependent competition stabilizes the solution towards discordant fluctuations in genotypic frequencies. I assess the effect of the number of loci, width of the range and the strength of competition Σ2c . As the number of loci increases to infinity, each locus becomes effectively neutral (the Zm allelic effect is Α = ) and the solution stays asymptotically stable. This is illustrated n
at Figure 3, which shows how the selection on the variance changes from stabilizing to disruptive as the number of loci decreases. We can determine when the solution gets unstable (selection is stabilizing) as the number of loci decreases. If the variance at linkage equilibrium, Σ2LE = 21 n Α2 , is 2 greater than the equilibrium variance on the infinite range, Σ` = Σ2 − Σ2 , the overall z
k
c
selection is still stabilizing and the distribution will be unstable towards fluctuations in frequencies of equivalent genotypes (Fig 3, two to four loci). Conversely, the solution is Zm selections arises on the limited range: Zm < "########################## 2 n H1 - Σ2c L (Α = , Σ2k º 1). I have n
stable towards the discordant fluctuations in genotype frequencies if disruptive
tested numerically (for 6 and 10 loci, variable Σ2c and Zm ) that this prediction for the stability of the symmetric model holds (see Appendix). Figure 3
It follows that for a strictly limited range: |Z| = 1 (Z = ), Σ2k = 1 (e.g. Bürger and Gimelfarb, 2004; Dieckmann and Doebeli, 1999), frequency dependent 2
Zm competition will generate disruptive selection if Σ2c < 1 − − therefore, weak 2n
competition between similar phenotypes is sufficient (Σ2c < 0.9375) to generate 11 !!!! disruptive selection − beginning with 2 loci. Similarly, Σ2c £ 0.5 and Zm £ n will
It follows that for a strictly limited range: |Z| = 1 (Z = ), Σ2k = 1 (e.g. Bürger and Gimelfarb, 2004; Dieckmann and Doebeli, 1999), frequency dependent 2
Zm competition will generate disruptive selection if Σ2c < 1 − − therefore, weak 2n
competition between similar phenotypes is sufficient (Σ2c < 0.9375) to generate !!!! disruptive selection − beginning with 2 loci. Similarly, Σ2c £ 0.5 and Zm £ n will result in disruptive selection. The net effect of frequency−dependent selection on a strongly limited range will therefore favour high polymorphism in allelic frequencies. I note that the distributions are stable towards all other fluctuations within the symmetric model: shift of the mean and fluctuations in associations (disequilibria) between loci. Allele frequencies are stable at frequency 1/2. The disruptive selection generated by reduction of the trait range is too weak to overcome recombination; the solution stays unimodal for most of the solutions. Strong frequency−dependent competition of variance Σ2c = 0.16 still leads to a stable unimodal distribution on the phenotypic range , even though selection is disruptive: marginal phenotypes have about twice the fitness of intermediate phenotypes. This disruptive selection leads to only a slight increase of variance: under linkage equilibrium, the phenotypic variance would be Σ2LE = 0.0125, variance under selection after random mating is Σ2z = 0.0132. Linkage disequilibrium stays low, about 0.0015 for a pairwise association. Note that more than about a two fold increase between the variance after recombination and the variance after selection for one generation would lead to continuing divergence (Bulmer, 1980). That is because under no selection, the variance of offspring equals half of the variance of the parents (infinitesimal model, Bulmer, 1980). Then if selection favours lesser than a two−fold increase of variance, density stays unimodal. (See Polechová and Barton, 2005 for analysis of evolution of the trait under stabilizing and frequency dependent competition using the infinitesimal model). With competition for a single resource, strong disruptive selection can be generated only under very specific conditions . This occurs if the stabilizing selection is very weak (fitness is nearly uniform with in the absence of frequency−dependent selection) and 1 frequency dependent selection is very strong (approx. Σ2c ~ ) − so that strong 2 n ÈZÈ
12
disruptive selection can arise on the limited range.
With competition for a single resource, strong disruptive selection can be generated only under very specific conditions . This occurs if the stabilizing selection is very weak (fitness is nearly uniform with in the absence of frequency−dependent selection) and 1 frequency dependent selection is very strong (approx. Σ2c ~ ) − so that strong 2 n ÈZÈ
disruptive selection can arise on the limited range. I confirm that for weak frequency dependent selection, Σ2c > Σ2k , the equilibrium density of phenotypes is always unstable towards fluctuations in genotype frequencies. (If frequency dependent competition is to maintain polymorphism, it has to be stronger than stabilizing selection: Σ2c < Σ2k , Roughgarden (1972), Slatkin (1979), Bürger and Gimelfarb (2004)). Weaker frequency dependent selection cannot maintain the variance: without mutation, no solution with positive variance can exist (Roughgarden, 1972; Slatkin, 1979), and the solution is always a spike at the optimum.
Evolution with assortative mating based on a different trait It would be interesting to know whether such frequency−dependent competition between similar phenotypes generates disruptive selection that it is likely to be strong enough to drive evolution of differentiated populations with assortative mating. With random mating, the increase of the variance favoured by selection is rather minute, and the phenotypic distribution stays unimodal. I briefly assess the possibility of evolution of association (disequilibria) between an "ecological" trait, z and a trait x allowing assortment. I note that the symmetric model with an independent trait for assortment has been analysed before, by Kondrashov (1983b, 1999). In order to analyse two traits, I use a haploid model. The second trait, x, allows for assortment between similar phenotypes: the strength of assortment is based on phenotypic proximity in the assortment trait x, described by the variance Σ2a of the Gaussian preference function A(x−x’, Σ2a ). The probability of matching between the
Hx-x’L phenotypes x and x’ is therefore ExpA- E, decreasing with the distance between x 2 Σ2 2
a
and x’ and increasing with the variance
Σ2a
of the preference function A. Numerical
analysis shows that with the competition for a single resource, divergence in the ecological trait z is possible only with strong restrictions of the trait range, strong competition (low Σ2c ) and strong assortment (low Σ2a ) with virtually no cost of assortative mating. 13
2 Σa
and x’ and increasing with the variance Σ2a of the preference function A. Numerical analysis shows that with the competition for a single resource, divergence in the ecological trait z is possible only with strong restrictions of the trait range, strong competition (low Σ2c ) and strong assortment (low Σ2a ) with virtually no cost of assortative mating. First, the assortment is normalized to ensure that all phenotypes (with respect to the assortment trait x) have equal mating success. This cost−free assortment, however, itself leads to divergence (Moore, 1979; Gavrilets, 2004 Chap. 9 and 10; Polechová and Barton, 2005). I introduce a cost of assortment, which increases with smaller effective density of matching partners Ψa , where Ψa HxL = H Ψ * AL HxL, relative to the total population density N. Specifically, the previous preference function, normalized for Ψa HxL N
c equal mating success for all phenotypes, is divided by PΑ (x) = 1 + ; c ranges
from zero to infinity. In this illustration, I have chosen the parameters as in Dieckmann and Doebeli (1999). With no cost of assortment c = 0, range = , Σ2c = 0.16 and variance in assortment preference fixed at Σ2a = 0.025. (Σ2a actually evolves in Dieckmann and Doebeli (1999), and it ranges between 0 and 0.05 according to the proportion of differences in the "marker" trait: Σ2a equals zero if there are no differences and 0.05 if all alleles in the two marker traits are different.) Then, the distribution is bimodal at 2000 generations and consists of two distinct clusters after 10000 generations and the covariance between the traits converges to 0.18. The divergence slows considerably with even minute cost of assortment, c = 0.001. After 5000 generations, the distribution is only slightly bimodal (Figure 4): the covariance between the traits is 0.02. However, covariance then increases fast, and converges to c = 0.205 after 15000 generations. A small cost of assortment c = 0.01 restricts the divergence: the distribution stays unimodal, and even if there is initially small covariance between the traits, it quickly decreases towards zero: covariance is 10-5 at T=3000 (at T=20000, cov = 10-17 ). Speciation in a single population is rather unlikely under the specified conditions: severe restriction of the trait range, very strong frequency−dependent selection and strong and virtually cost−free assortment are all necessary to drive divergence via evolution of association between the ecological and assortment trait. Under strong (direct) disruptive selection, the divergence would be more feasible.
14
Speciation in a single population is rather unlikely under the specified conditions: severe restriction of the trait range, very strong frequency−dependent selection and strong and virtually cost−free assortment are all necessary to drive divergence via evolution of association between the ecological and assortment trait. Under strong (direct) disruptive selection, the divergence would be more feasible. Figure 4
Direct disruptive selection Strong disruptive selection can clearly be generated if imposed directly. I assess the interaction between the frequency−dependent selection and disruptive selection, and the stability of the symmetric model under these conditions. I model two forms of z2
2 , S < 0 or a Gaussian disruptive selection: U−shaped disruptive selection, K(z) ~ e-S Hz ±LL - 2 Σ2 2
bimodal resource, K(z) ~ e
k
(see Methods). Without prezygotic reproductive
isolation (e.g., assortative mating), however, progeny with central phenotypes always form initially, even if inviable or infertile. I start with U−shaped disruptive selection, which approximately takes the form of S z2
2 −1+k (see Methods for details). Figure 5 shows the effect of strong WHzL > e
U−shaped disruptive selection, where the fitness of central phenotypes is 0.1 relative to the fitness of marginal phenotypes (S U 0.2). Because fitness is high where the trait range is truncated, frequency−dependent selection strengthens the overall disruptive selection. Selection over a generation time ∆t can substantially increase the variance, and lead to bimodal solution. The variance under selection (after recombination) is Σ2z = 0.7, and increases about 1.8 times after one generation of selection (under linkage equilibrium, the variance would be Σ2LE = 0.2). Pairwise LD’s are about 0.07 after recombination. Frequency dependent selection needs to be moderately strong so that the distribution is stable, around Σ2c £ 0.45. Figure 5 Hz ± LL ExpA- 2 E
2Σ Next, I move to a bimodal distribution of resources, K(z) = . I show that !!!!!!!!!! 2
k
2
2 Π Σk
even diverged phenotypic distributions are stable under moderate frequency dependent selection. With bimodal carrying capacity, the impact of frequency−dependent selection on the overall disruptive selection can vanish as the fitness is almost zero near the edges of the phenotypic range. Therefore, no phenotype benefits from decreased competition 15
near the margins of the trait range. In contrast to previous examples, the limitation of trait range here (coupled with frequency−dependent selection) does not play a role in
Hz ± LL ExpA- 2 E
2Σ Next, I move to a bimodal distribution of resources, K(z) = . I show that !!!!!!!!!! 2
k
2
2 Π Σk
even diverged phenotypic distributions are stable under moderate frequency dependent selection. With bimodal carrying capacity, the impact of frequency−dependent selection on the overall disruptive selection can vanish as the fitness is almost zero near the edges of the phenotypic range. Therefore, no phenotype benefits from decreased competition near the margins of the trait range. In contrast to previous examples, the limitation of trait range here (coupled with frequency−dependent selection) does not play a role in generating disruptive selection. Figure 6 shows the fitnesses and phenotypic distributions for a bimodal resource with optima zopt = ± 4, Σ2c = 0.5, Σ2k = 1, and range . At equilibrium, disruptive selection leads to the increase of the variance to about Σ2z = 6.42 over one generation (∆t = 1, r0 = 1) from the variance after recombination: Σ2z = 4.01. The difference is about 1.6 fold. The solution is stable towards all fluctuations (within the symmetric model): maintenance of variance (p = 1, Λ1,1 = 0.97) is allowed by frequency−dependent 1 selection of strength = 2, and disruptive selection stabilizes the solution towards Σ2 c
fixation of particular genotypes for a given phenotypic value (p = 0, Λ0,1 = 0.96). (See Turelli and Barton (1994), p. 918−919 and Felsenstein (1979) for analysis of a bimodal Gaussian distribution without frequency−dependent selection.) Associations amongst loci for the trait under disruptive selection are still stable towards fluctuations − the highest eigenvalue Λ2,1 is 0.47. Pairwise linkage disequilibria are 0.034 (after recombination). Figure 6
16
Discussion Competition for a single resource can clearly lead to disruptive selection due to frequency−dependent competition if the phenotypic range is restricted. That is because marginal phenotypes experience a disproportionally lower impact of competition. However, this disruptive selection is most likely to be too weak to drive divergence in a single population (contrary to the suggestion of Dieckmann and Doebeli, 1999). It is, though, often strong enough to stabilize the phenotypic distribution under the symmetric model if genes determining the trait under selection have equal and additive effects and their frequencies are equal; Figs 3 and 4. That is important because as direct calculations are intractable for many genes, we ought to use some simplifying assumptions to assess the evolution of quantitative traits. Also disruptive selection under density dependence will strengthen if frequency−dependent selection operates (see Bolnick 2004 for experimental evidence.) Frequency−dependent selection is well documented in natural populations: causes include predator−prey interactions, response to pathogens, competition for a common resource (for a review, see Clarke, 2004). It is clearly not the only force which could maintain variation: other causes include varying or diversifying selection in parapatry (Dickinson and Antonovics, 1973), other forms of balancing selection (e.g. heterozygote advantage), or mutation−selection balance (Keightley and Hill, 1992; see Barton and Keightley, 2002). The distribution of a quantitative trait determined as in the symmetric model is unstable under pure stabilizing selection. Disruptive selection stabilizes the assumed equal ("symmetric") frequencies of the alleles and frequency−dependent selection is necessary to maintain the variance. This analysis shows that rather weak disruptive selection is sufficient to fulfill the assumptions of the symmetric model and can be driven by the frequency−dependent selection when the range is restricted. I show that the symmetry assumption of the symmetric model is fulfilled when variance favoured by selection is greater than the variance at linkage equilibrium, so that the selection is disruptive. For the Roughgarden model (1972) of logistic regulation with frequency−dependent 17 selection, this occurs if Zm < 2 n (Σ2k − Σ2c ). Frequency−dependent selection arising
from the competition between similar phenotypes (with variance Σ2c ) stabilizes the
sufficient to fulfill the assumptions of the symmetric model and can be driven by the frequency−dependent selection when the range is restricted. I show that the symmetry assumption of the symmetric model is fulfilled when variance favoured by selection is greater than the variance at linkage equilibrium, so that the selection is disruptive. For the Roughgarden model (1972) of logistic regulation with frequency−dependent selection, this occurs if Zm < 2 n (Σ2k − Σ2c ). Frequency−dependent selection arising from the competition between similar phenotypes (with variance Σ2c ) stabilizes the solution towards the shift of the mean phenotype. I show that weak to moderate disruptive selection (fitness of intermediate phenotypes smaller than about half of fitness of marginal phenotypes) arising on a limited trait range is unlikely to drive speciation in sympatry via evolution of association between the assortment trait and trait under disruptive selection: even a minute cost of assortment prevents the divergence. The effect of cost−free assortment is well known (Moore 1979, Gavrilets 2004 Chap. 9 and 10); and for the Dieckmann and Doebeli 1999 model, the same conclusion is supported by Gavrilets and Vose in the independent analysis of the cost of choosiness and mutation rate on the waiting time to speciation (Waxman and Gavrilets, 2005, Table 1). This conclusion is hardly surprising: from a simple analysis (e.g. by Gavrilets, 2003), we can see that the combined strength of assortment (1 locus) and disruptive selection (via another locus) has to be rather high so that associations between the loci can evolve and stable coexistence of prezygotically isolated populations differing in the other locus can be maintained. With more loci of additive effect, the threshold is even higher if fitness decreases towards zero only when all loci have the least fit allele, as all the associations between the loci would have to build up (unpublished analysis, JP)). Clearly, in nature, unless there is strict monogamy, there is likely to be a cost of assortment for similarity−based matching: extreme phenotypes are less frequent, and therefore the probability of them finding a similar partner is lower; and an additional cost may arise from the choice itself. I note that divergence can occur readily under weaker disruptive selection when (low cost) assortative mating is based directly on the similarity in the ecological trait, as in Dan Bolnick’s analysis of adaptive radiation under Dieckmann and Doebeli (1999) model. Plainly, as numerical analysis of divergence of polygenic trait under direct disruptive selection is quite computationally intensive, simpler models of disruptive selection (e.g., the Levene (1953) model), are more suitable for detailed assessment: implications of frequency−dependent selection will be easier to understand when overall strength of disruptive selection is determined first.
18
radiation under Dieckmann and Doebeli (1999) model. Plainly, as numerical analysis of divergence of polygenic trait under direct disruptive selection is quite computationally intensive, simpler models of disruptive selection (e.g., the Levene (1953) model), are more suitable for detailed assessment: implications of frequency−dependent selection will be easier to understand when overall strength of disruptive selection is determined first. The symmetric model serves only as a crude but tractable approximation of a quantitative trait. There is no doubt that the assumptions of symmetric model of pure additivity of loci, and their equal effect, are never fulfilled in nature. However, as it allows numerical solution for a large number of loci, it can help us to understand results from individual based simulations, where the range of important parameters is more difficult to explore, and provide a test robustness of analytical results obtained under other assumptions, e.g. of linkage equilibrium between loci or weak selection. Last, I emphasise that the conditions on stability of the symmetric model are likely to be fulfilled under selection regimes favourable for sympatric speciation − i.e., disruptive selection driving the divergence, and strong frequency−dependent selection maintaining the variance (e.g. Kondrashov and Kondrashov, 1999). However, as fitness functions vary between the studies, analysis of stability is always desirable if fluctuations may in principle lead to violation of the assumptions of the model used.
Acknowledgements I am very grateful to Nick Barton for extensive constructive comments and discussions and Sergey Gavrilets for the valuable suggestions on the MS. I would like to acknowledge Denis Roze and Max Shpak for the comments on the MS and Jay Taylor for the advice on related analysis. The work was supported by Grant Agency of the Czech Republic (206/05/H012) and by the grant from the National Environment Research Council, UK (NER/A/S/2002/00857).
19
References
Barton, N.H. 1992. On the spread of new gene combinations in the third phase of Wright’s shifting balance. Evolution 46, 551−557. Barton, N.H., Keightley, P.D. 2002. Understanding quantitative genetic variation. Nature Reviews Genetics 3, 11−21. Barton, N.H., Polechová, J. 2005. The limitations of adaptive dynamics as a model of evolution. J. Evol. Biol. 18, 1186−1190. Barton, N.H., Shpak, M. 2000. The stability of symmetrical solutions to polygenic models. Theor. Popul. Biol. 57, 249−264. Barton, N.H., Turelli, M. 1991. Natural and sexual selection on many loci. Genetics 127, 229−255. Bolnick, D. 2004. Can intraspecific competition drive disruptive selection? An experimetnal test in natural populations of sticklebacks. Evolution 58, 608−618. Bulmer, M.G. 1980. The mathematical theory of quantitative genetics. Clarendon Press, Oxford. Bürger, R., Gimelfarb, A. 2004. The effects of intraspecific competition and stabilizing selection on a polygenic trait. Genetics 167, 1425−1443. Clarke, B. 2004. Nonsynonymous polymorphisms and frequency−dependent selection. Pp. 156−177. In: Singh, R. & Uyenoyama, N., (Eds.), The evolution of population biology. Cambridge university press. Darwin, C. 1859. On the origin of species by means of natural selection. John Murray, London. Dickinson, H., Antonovics, A. 1973. Theoretical considerations of sympatric divergence. Am. Nat. 107, 256−274. Dieckmann, U., Doebeli, M. 1999. On the origin of species by sympatric speciation. Nature 400, 354−357. Doebeli, M. 1996a. A quantitative genetic competition model for sympatric speciation. J. Evol. Biol. 9, 893−910. Doebeli, M. 1996b. Quantitative genetics and population dynamics. Evolution 50, 532−546.
20
Felsenstein, J. 1979. Excursions along the interface between disruptive and stabilising
Dieckmann, U., Doebeli, M. 1999. On the origin of species by sympatric speciation. Nature 400, 354−357. Doebeli, M. 1996a. A quantitative genetic competition model for sympatric speciation. J. Evol. Biol. 9, 893−910. Doebeli, M. 1996b. Quantitative genetics and population dynamics. Evolution 50, 532−546. Felsenstein, J. 1979. Excursions along the interface between disruptive and stabilising selection. Genetics 93, 773−795. Felsenstein, J. 1981. Skepticism towards Santa Rosalia, or why are there so few kinds of animals? Evolution 35, 124−138. Gavrilets, S. 2003. Models of speciation: what we have learned in 40 years? Evolution 57, 2197−2215. Gavrilets, S. 2004. Fitness landscapes and the origin of species. Princeton University Press, Princeton and Oxford. Gyllenberg, M., Meszéna, G. 2005. On the impossibility of coexistence of infinitely many strategies. Journal of Mathematical Biology 50, 133−160. Keightley, P.D., Hill, W.G. 1992. Quantitative genetic variation in body size of mice from new mutations. Genetics 131, 693−700. Kirkpatrick, M., Johnson, T., Barton, N.H. 2002. General models of multilocus evolution. Genetics 161, 1727−1750. Kondrashov, A.S. 1983a. Multilocus model of sympatric speciation I One character. Theor. Popul. Biol. 24, 121−135. Kondrashov, A.S. 1983b. Multilocus model of sympatric speciation II Two characters. Theor. Popul. Biol. 24, 136−144. Kondrashov, A.S. 1984. On the intensity of selection for reproductive isolation at the beginnings of sympatric speciation. Genetika 20, 408−415. Kondrashov, A.S., Kondrashov, F.A. 1999. Interactions among quantitative traits in the course of sympatric speciation. Nature 400, 351−354. Lack, D. 1947. Darwin’s finches. Cambridge University Press, Cambridge. Levene, H. 1953. Genetic equilibrium when more than one niche is available. Am. Nat. 87, 331−333. Maynard Smith, J. 1966. Sympatric speciation. Am. Nat. 100, 637−650. Moore, W.S. 1979. A single locus mass action model of assortative mating, with comments on the process of speciation. Heredity 42, 173−186. Nagylaki, T. 1993. The evolution of multilocus 21 systems under weak selection. Genetics 134, 627−647.
Levene, H. 1953. Genetic equilibrium when more than one niche is available. Am. Nat. 87, 331−333. Maynard Smith, J. 1966. Sympatric speciation. Am. Nat. 100, 637−650. Moore, W.S. 1979. A single locus mass action model of assortative mating, with comments on the process of speciation. Heredity 42, 173−186. Nagylaki, T. 1993. The evolution of multilocus systems under weak selection. Genetics 134, 627−647. Polechová, J., Barton, N.H. 2005. Speciation through competition: a critical review. Evolution 59, 1194−1210. Roughgarden, J. 1972. The evolution of niche width. Am. Nat. 106, 683−718. Schluter, D. 2000. The ecology of adaptive radiation. Oxford University Press, New York. Slatkin, M. 1979. Frequency− and density−dependent selection on a quantitative character. Genetics 93, 755−771. Turelli, M., Barton, N.H. 1990. Dynamics of polygenic characters under selection. Theor. Popul. Biol. 38, 1−57. Turelli, M., Barton, N.H. 1994. Genetic and statistical analyses of strong selection on polygenic traits: What, me normal? Genetics 138, 913−941. Udovic, D. 1980. Frequency−dependent selection, disruptive selection, and the evolution of reproductive isolation. Am. Nat. 116, 621−641. Waxman, D., Gavrilets, S. 2005. Issues of terminology, gradient dynamics and the ease of sympatric speciation in Adaptive Dynamics. J. Evol. Biol. 18, 1214−1219. Zhivotovsky, L.A., Gavrilets, S. 1992. Quantitative variability and multilocus polymorphism under epistatic selection. Theor. Popul. Biol. 42, 254−283.
22
Table 1 Notation table: r
intrinsic growth rate of the population
Ψ Ψ Ψa
density of diploid population
G
Gaussian competition function with variance Σ2c , G is Gaussian G~N(z−y,Σc )
effective density of diploid population Ψ = (Ψ * G) effective density of similar individuals with respect to the assortment trait, a: Ψa = (Ψ * A)
K
carrying capacity: if unimodal, K is Gaussian G~ NHz - z0 , Σk L.
Σ2c
variance of the competition function
Σ2k
variance of carrying capacity, Σ2k º 1
A
assortment function with variance Σ2a , A is Gaussian A~N(x−x’,Σa )
z, y
diploid phenotype, z = i + j
i, j
phenotypes of haploids (gametes)
z0
optimum (unless stated otherwise, we analyse symmetric optimum, z0 = n)
r0
maximum growth rate
L
distance between optima
n
number of loci
Α
Zm allelic effect, Α = n
Zm
range of possible phenotypes
x
assortment trait
23
Figure Labels Figure 1: The figure shows how the mean fitness decreases when the mean phenotype 2 and variance departs from the optimum (z` = 0, Σ` = Σ2 − Σ2 ). Mean fitnesses are 0.288, z
k
c
0.28, 0.27 for the first three contours (from the center), then the fitness decreases by 0.02. Σ2k º 1. Figure 2: (A) On a limited range, frequency−dependent competition leads to disruptive selection via increased fitness of marginal phenotypes, whose competitors out of the range are ΨHz, ΨL absent (solid curve, W Hz, ΨL = r0 I1 - KHzL M). The dashed curve shows the
ΨHz, ΨL fitness without frequency−dependence: W Hz, ΨL = r0 H1 - KHzL L. (B) The
corresponding equilibrium distribution of diploid phenotypes is unimodal, and selection during the life cycle causes only a minute increase of the phenotypic variance (about 1.06) over one generation (∆t = 1, r0 = 1). Parameters: 10 loci, Σ2c = 0.5, phenotypic range Z = , Σ2k º 1, r0 = 1. Figure 3: The selection on variance Svar changes from stabilizing to disruptive as the number of *
*
loci increases (Svar = Σ2z - Σ2z , where Σ2z is the variance after selection). If the 2 equilibrium variance on the infinite range Σ` = Σ2 − Σ2 is greater than the variance at z
linkage equilibrium
Σ2LE
=
12
k
c
n Α2 , disruptive selection arises on the limited trait range
model, selection is stabilizing if Zm > "########################## 2 n H1 - Σ2c L# . In this example (Zm = 2, Σ2c =
via frequency−dependent competition between similar phenotypes. Specifically for the
Zm 0.5, Α = , Σk º 1), selection is stabilizing for 4 and less loci. For two loci, the n
selection on the variance is lower than shown (Svar = -0.144). The pattern for Λ1,1 , the largest eigenvalue which describes the stability of the solution towards fluctuations in genotype frequencies, is analogous: solution is unstable when selection is stabilizing, i.e. for 4 and less loci. Figure 4: (A) With minute cost for rare phenotypes (c = 0.001), weakly bimodal distribution evolves for the "ecological" trait z under assortment (trait x) at 5000 generations. 24
Covariance between the traits stays low, 0.02. (B) Distribution for the "ecological" trait z shows a weak bimodality (transect at the margins of the assortment trait, x = ± 0.5).
Figure 4: (A) With minute cost for rare phenotypes (c = 0.001), weakly bimodal distribution evolves for the "ecological" trait z under assortment (trait x) at 5000 generations. Covariance between the traits stays low, 0.02. (B) Distribution for the "ecological" trait z shows a weak bimodality (transect at the margins of the assortment trait, x = ± 0.5). Parameters: 10 loci, Σ2c = 0.16, Σ2a = 0.025, Σ2k º 1, r0 = 1, T = 5000, phenotypic range for "assortment" trait (X) and "ecological" trait (Z) are . After covariance evolves from zero, the divergence is considerably faster: after 15.000 generations the phenotypes are distinct and the covariance between the assortment and ecological trait converges to 0.205. However, with slight cost of assortment (c=0.01; other parameters kept same), the distributions stays unimodal (not shown). Figure 5: (A) Strong disruptive selection arises when weak disruptive selection is magnified by frequency−dependence on a limited range (solid line); dashed line shows the fitness S z2
2 −1 + without the frequency−dependent selection. Approximate growth rate r(z) > e
k, range , k = 0.05, Σ2k = −5. Then S U 0.2 with the frequency−dependent competition (Σ2c = 0.45) and S U 0.07 with absolute density regulation (Σ2c = ¥). Fitness of intermediates relative to marginal phenotypes is about 0.1. (B) Phenotypic distribution of diploids (after random union) is unimodal, Σ2z = 0.8. (C) Selection over one generation (∆t = 1, r0 = 1) substantially increases the variance: Σ2z = 1.48. Solution is just stable towards fluctuations in the mean (Λ0,1 = 0.999), stable towards discordant fluctuations (Λ1,1 = 0.91) and fluctuations in the associations (Λ2,1 = 0.42). With weaker frequency−dependent selection, the mean would shift to one of the alternative optima − the solution is unstable towards fluctuations of the mean. (Without frequency dependent selection, Λ0,1 = 1.3 > 1). Figure 6: (A) Strong disruptive selection readily arises under bimodal distribution of resources (solid line): the relative fitness of intermediate phenotypes is 0.047 (zopt = ± 4, range , Σ2c = 0.5, Σ2k = 1). Dashed line is without frequency−dependent selection. B: Phenotypic distribution is unimodal after random union. C: Phenotypic distribution in the middle of the life cycle, after selection for25one generation, is strongly bimodal (r0 = 1, ∆t = 1).) Solution is stable within the symmetric model. Without
Figure 6: (A) Strong disruptive selection readily arises under bimodal distribution of resources (solid line): the relative fitness of intermediate phenotypes is 0.047 (zopt = ± 4, range , Σ2c = 0.5, Σ2k = 1). Dashed line is without frequency−dependent selection. B: Phenotypic distribution is unimodal after random union. C: Phenotypic distribution in the middle of the life cycle, after selection for one generation, is strongly bimodal (r0 = 1, ∆t = 1).) Solution is stable within the symmetric model. Without frequency−dependent selection, the solution would be unstable (Λ0,1 = 0.96 vs Λ0,1 = 1.7) towards shifts of the mean phenotype, and the variation would be lost. Figure 7: The chart shows the simple linear relation between strength of frequency−dependent selection and limitation of the range which is sufficient to stabilize the phenotypic 2 distribution. Dots describe numerical solutions, lines the prediction Z 2 = 2 n Σ` for n m
= 10 loci (upper line) and n = 6 loci (lower line).
Appendix Figure 7
26
z
3 2.5 2 Σ2z 1.5 1 0.5 0
-3 -2 -1
0 z
1
2
3
Figure 1
0.75 W 0.5
0.2 Ψ 0.1
0.25 -2
0
2
z
-2
A
0 B
Figure 2
2
z
Svar 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04
2 3 4 5 6 7 8 10
15
20
# loci
25
Figure 3
Ψ 0.15 Ψ 0.1 0.05 0 -0.5
0.5
0.12 0.09
0Z X
0.06 0.03
0 0.5 -0.5
0 -0.5 -0.25
A
0 B
Figure 4
0.25
0.5
Z
0.5
0.1 Ψ 0.05
0.1 Ψ 0.05
W 0.25
-2
0
2
z
-2
0
A
2
z
-2
B
0
2
z
C
Figure 5
0.75 W 0.5
0.15 Ψ 0.1
0.15 Ψ 0.1
0.25
0.05
0.05
-6
0
6
z
-6
0
A
6
B
Z2m 20 15 10 5 0.4
0.6
0.8
0
10 C
Figure 6
0.2
z
1
Σ2c
Figure 7
20
z
