(b) Ecological divergence at a polygenic trait

Ecological divergence is often driven by disruptive selection acting on polygenic traits. Therefore, we first set out to analyse how increasing the number of loci influences the ability of the model to maintain polymorphism under disruptive selection. We study two to three loci analytically and up to 16 loci using individual-based simulations. In particular, we focus on violation of symmetries in niche proportions, which are defined as c=cI=1−cII for niche I and II, respectively (c∈(0,1)). In order to be able to compare the two- (neco=2) and three-(n_eco = 3) locus models, we normalize the strength of selection (s) and epistasis (ϵ) in the models with more than two ecological loci such that the mean per-trait selection and epistasis remain the same (sneco∗=2 s2loci/neco and ϵneco∗=ϵ2loci/(neco2)). This normalization assures that fitnesses of the specialists are always equal to 1−2s−ϵ, while preserving the scaling of selection independently of epistasis. This leads to a slight weakening of the trade-off as the number of loci increases (see also electronic supplementary material, figure S1).

For simplicity, we assume that all pairwise epistatic effects have the same value, and neglect higher-order interactions between alleles. Fitnesses of the individual genotypes for the two- and three-locus model are defined in table 1a and b, respectively. Graphical representation of the trade-offs between fitnesses of individual genotypes in niche I and niche II is shown and discussed in electronic supplementary material, figure S1 and the corresponding text.

First, we show that the region of parameter space where polymorphism is maintained is highly sensitive to violation of symmetry in niche proportions, c, even if loci are completely linked. Also, increasing convexity of the trade-off (i.e. more negative ϵ, more disruptive selection) further reduces the parameter space with maintained polymorphism (figure 1a). Both when the loci are completely linked (r = 0) and when they are freely recombining (r = 0.5), there is no difference between two and three loci when mean per-trait selection and epistasis remain the same (see above). The regions of parameter space with maintained polymorphism coincide and shift towards stronger selection as epistasis increases (figure 1c). When recombination between loci is low (r = 0.01, i.e. 1 cM), the regions of parameter space with maintained polymorphism shift closer towards those of the free recombination regime in the three-locus model than in the two-locus model, as in the three-locus case the per-locus strength of selection is lower (figure 1b). Interestingly, there is a threshold for recombination rate, above which the parameter space where polymorphism is maintained is independent of the number of loci, which we elaborate on more in the electronic supplementary material. Below we provide the analytical expressions for the boundaries of the stable regions, and the recombination threshold. The conditions are for protected polymorphism in the ecological loci, obtained by assessing local instability of the monomorphic equilibria. Note that as we have disruptive selection with symmetric selection coefficients, all ecological loci, when polymorphic, evolve to the same allele frequencies.

Range of niche proportions where polymorphism is maintained: where ecological divergence towards two specialists is stable. The upper half of the graphs show conditions for the two-locus model, the lower half for the three-locus model. On the x-axis is the normalized strength of selection (symmetric across loci) and on the y-axis are niche proportions. A polymorphic equilibrium is achieved for the parameter combinations of s∗ and c between the black curves and the white axis at c = 0.5 (shaded). Note that we are showing only one half of the parameter space for each model as the conditions are symmetrical. Therefore, every condition at a value of c has its symmetric counterpart at 1−c. The outer solid curves represent linear trade-offs (ϵ = 0), the dashed curves are for ϵ=−0.25s, and the dash-dotted curves for ϵ=−0.5s. (a) No recombination; (b) low recombination (r = 0.01) and (c) free recombination (r = 0.5). It is noteworthy that in the regime with low recombination (b), the region of parameter space where polymorphism is maintained decreases with increasing number of loci if selection and epistasis is normalized as described in the main text.

In both models with normalized selection and ϵ=0 a polymorphic equilibrium is stable if

In the case with no recombination (figure 1a) and ϵ<0, polymorphism is maintained if

and with free recombination (figure 1c) if

These conditions hold for the two-locus model for recombination rates r>(−ϵ/(1−s−ϵ)). If recombination rate is low (0<r<(−ϵ/(1−s−ϵ)); figure 1b), the conditions above do not hold anymore and maintenance of polymorphism is determined by

Since the equations describing the conditions for the three-locus model with low recombination and all other models presented from this point on exceed the width of a page, we confine them to the electronic supplementary material, where we also give more details on the stability analysis.