Probability to reach complete isolation
Mutational stepsize−−>
Mutation rate 04*10^−54*10^−44*10^−3
0.05 0.1 0.25
0 0 0
0 0.1
(48,000) 0.7
(38,000)
0 0.9
(33,000)
1
(6,000)
Figure 4.9: (Preliminary results) The probability to reach the “complete isolation“ equi-librium within 100,000 generations, depending on the mutation rate and the mutational stepsize. Mutations have a fixed stepsize. The number of steps between random mating and complete isolation is 1/stepsize, and ranges from 4 to 20. In brackets is the mean number of generations until the complete isolation equilibrium was reached. The population was as-sumed to be in complete isolation when there were no heterozygotes left. Other parameters:
σc= 0.4,x= 0.5,K0= 500.
and complete isolation. We find, as expected, that with higher mutation rate and larger stepsize, the complete isolation equilibrium is reached more often and faster.
too strong (small x or 1−k in Figures 4.1 and 4.4). A necessary condition can be seen from our model without sexual selection: Competition between homozygotes of opposite type must be substantially smaller than competi-tion between homozygotes and heterozygotes, such that the net reduccompeti-tion in competition due to a split into isolated clusters outweighs stabilizing selection from the resource distribution (a0 ¿a). If the niche width is too broad (weak frequency dependence of competition, σc large), the disruptive force due to competition is too weak to generate multiple niches. If the niche width is too narrow (frequency dependence of competition too strong), a third niche at an intermediate phenotype opens up that is filled by the heterozygotes. Finally, even for an optimal range of competition, that is, full competition between homozygotes and heterozygotes (a= 1), but no competition between homozy-gotes of opposite type (a0 = 0), complete isolation does not evolve if stabilizing selection is too strong (from equation 4.16, we see that complete isolation is always unstable fork <1/4). In our model without sexual selection, this neces-sary condition is also sufficient for speciation to happen. However, with sexual selection, complete isolation is not always reached in the described parameter range, for reasons which we explain in the next section.
In addition to sympatric speciation, we find a large parameter region in both our models where assortative mating always evolves to an intermediate equilibrium value (partial isolation regime in Figure 4.1 and 4.4). Partial re-productive isolation evolves, in particular, if competition is strongly frequency-dependent (short ranged), such that a third niche for heterozygotes emerges.
The size of this niche determines the proportion of heterozygotes that is sus-tained and the appropriate level of choosiness. If the niche for heterozygotes gets sufficiently large so that the (optimal) ratio of heterozygotes and homozy-gotes n exceeds 2, random mating becomes stable against increased levels of assortativeness (for n > 2, dis-assortative mating would evolve if this was possible). Note that partial reproductive isolation in natural populations is frequently interpreted as indicating “incipient speciation”. Our model shows that this is not necessarily true: Frequency-dependent selection can allow co-existence of three phenotypes, but withn <2, partial isolation can be a stable outcome of evolution.
Natural and sexual selection
Our results can be explained by the interplay of natural and sexual selection (see also Kirkpatrickand Nuismer 2004; Gourbiere2004). We analyzed the roles of the two selective forces by comparing the behavior of the model
with and without sexual selection. Without sexual selection, we find only three regimes: complete isolation, partial isolation and random mating. Each regime has only one stable equilibrium, and the ecological polymorphism is always stable.
Adding sexual selection against rare males makes the behavior of the model more complex. In addition to the three regimes mentioned above, there are now two regimes – the alternative extremes regime and the Matessi et al.
regime – where the evolutionary outcome depends on initial conditions. In these regimes, sexual selection favors heterozygotes when they are common, and this can stop (further) increase of assortativeness. If elevated levels of assortative mating already exist in a population, sexual selection can promote the evolution of complete isolation. Therefore, sexual selection causes a big increase in the parameter range in which complete isolation is stable (shown by the area left of the blue line, in Figure 4.1 and 4.4). This effect is particularly striking in the alternative extremes regime, where natural selection alone favors random mating. In addition, in some parts of the parameter space, sexual selection can cause a loss of the ecological polymorphism. This is because males carrying a rare allele will have difficulties finding a mate.
The region where complete isolation can evolve in small steps in an infinite population (complete isolation regime in Figure 4.4) is considerably reduced relative to the model without sexual selection. In a larger range (corresponding to the Matessi et al. regime in Figure 4.4), complete isolation is stable, but there is another stable equilibrium at intermediate assortativeness. Evolution in small steps will always stop at the intermediate equilibrium with partial isolation. However, our simulations show that evolution of complete isolation is still possible by “jumping” the intermediate optimum if mutation rates and mutational effects are sufficiently high. In the extreme case that a single mutation results in complete assortativeness, simulations show that in about 80% of the parameter range corresponding to the Matessi et al. regime such a mutation has positive invasion fitness if the population is currently at the intermediate equilibrium.
The evolution of complete isolation can also be inhibited by the loss of the ecological polymorphism, as previously described byB¨urgerandSchneider (2006a). For weak frequency dependence of competition and weak stabilizing selection (the lower left corner of Figure 4.4) the polymorphic equilibrium be-comes unstable for intermediate values of assortativeness. Our simulations show that in this case almost always a monomorphic equilibrium is reached.
In B¨urger and Schneider (2006a), the loss of the polymorphism was de-scribed as a consequence of the evolution to an intermediate optimum for
assortativeness in the Matessi et al. regime. However, our model shows that these phenomena are two different mechanisms that can both prevent speci-ation. An infinite population that takes small mutational steps will always evolve to partial isolation without loosing the polymorphism (see Figure 4.3).
A finite population, however, can loose the polymorphism if it moves past the stable intermediate equilibrium by drift (Figure 4.8).
Discussion of the modelling approach
Our approach in this study was to analyze a simplified version of the Dieck-mann and Doebeli (1999) model. This approach allowed us to (1) analyze the model in the entire ecological parameter space, (2) to gain a detailed and intuitive understanding of the interaction between the various selective forces and (3) to unify, in a single model, a large number of phenomena which have previously been studied or described individually in separate models. The latter include (1) the role of natural versus sexual selection (see Gourbiere 2004; Kirkpatrick and Nuismer 2004), (2) conditions for the maintenance or loss of the ecological polymorphism (seeKirkpatrick andNuismer2004;
B¨urger and Schneider 2006a,b), (3) potential evolutionary stability of in-complete isolation (Doebeli 1996; Matessi et al. 2001), and (4) the im-portance of ecological niches and a resulting non-linear relationship between niche width and the likelihood of speciation (Gourbiere2004;Bolnick2006;
B¨urgerand Schneider2006b).
Our work can be seen as an extension of the study by Matessi et al.
(2001). These authors used a quadratic approximation to our fitness function, which is valid if overall selection is weak. More precisely, the approximation guarantees that selection is always purely disruptive and heterozygotes always have lowest viability, as is the case to the left of the blue line in Figure 4.1. In accordance with our results for this area, Matessi et al. (2001) found two of or five regimes – the Matessi et al. regime and the complete isolation regime.
It is in the regions where their approximation is not valid that we find the other three regimes.
The key simplification in our model is the assumption that the ecological trait is determined by a single locus with two alleles. How general are our re-sults with regard to the genetic architecture of the trait? – On the one hand, it seems reasonable to expect that the five regimes described in this paper are generic also for other genetic architectures, because the interplay of natural and sexual selection, which we have described here, should be qualitatively independent of genetic details. This intuition is supported by the observation
that a behavior similar to the Matessi et al. regime was also found in a multilo-cus model byDoebeli (1996), and complete isolation was, of course, reached in the initial multilocus model by Dieckmannand Doebeli (1999).
On the other hand, the one-locus assumption has the obvious consequence that intermediate phenotypes can only exist as heterozygotes. Therefore, whenever more than two phenotypes can potentially coexist, natural selection tends to move the population toward partial isolation or random mating. In a model with a different genetic architecture, the evolution of assortative mating might instead lead to more than two reproductively isolated species (Bolnick 2006; B¨urger and Schneider 2006b). Thus, the behavior of the model in the partial isolation, random mating and alternative extremes regimes might be different in a model with more than one ecological locus. In particular, speciation may be possible for a larger range of parameters.
Another important assumption of our model is that the allelic effect of the ecological locus (x) is constant, whereas, in principle it might also be subject to selection (Geritz and Kisdi 2000; Kopp and Hermisson 2006; Kristan Schneider, unpublished manuscript). The coevolution of genetic architecture and assortative mating seems to be an interesting avenue for future studies.