Analysis of disruptive selection in subdivided populations

Abstract

Background: Analytical methods have been proposed to determine whether there are evolutionarily stable strategies (ESS) for a trait of ecological significance, or whether there is disruptive selection in a population approaching a candidate ESS. These criteria do not take into account all consequences of small patch size in populations with limited dispersal.

Results: We derive local stability conditions which account for the consequences of small and constant patch size. All results are derived from considering Rm, the overall production of successful emigrants from a patch initially colonized by a single mutant immigrant. Further, the results are interpreted in term of concepts of inclusive fitness theory. The condition for convergence to an evolutionarily stable strategy is proportional to some previous expressions for inclusive fitness. The condition for evolutionary stability stricto sensu takes into account effects of selection on relatedness, which cannot be neglected. It is function of the relatedness between pairs of genes in a neutral model and also of a three-genes relationship. Based on these results, I analyze basic models of dispersal and of competition for resources. In the latter scenario there are cases of global instability despite local stability. The results are developed for haploid island models with constant patch size, but the techniques demonstrated here would apply to more general scenarios with an island mode of dispersal.

Conclusions: The results allow to identity and to analyze the relative importance of the different selective pressures involved. They bridge the gap between the modelling frameworks that have led to the Rm concept and to inclusive fitness.

Figure 1

Threshold combinations of values of / and d for disruptive selection. The lines are the thresholds for different values of N. Branching is favored for sets of values in the top right corner delimited by these lines. The dashed line shows the thresholds for N = 12 obtained with the approximation K = 0.

Figure 2

Cases of global instability. The shaded area is the set of parameter values for which some mutants invade despite local non-invasibility of the candidate ESS z*. This figure is obtained from computation of R_m for δ up to 2.187.