Global Stability of Spatially Homogeneous Equilibria in Migration-Selection Models

Abstract

We investigate the evolution of the gene frequencies at a multiallelic locus under the joint action of migration and viability selection. The population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of their genotype. If the selection pattern is the same in every colony and such that $\hat{p}$ is a globally asymptotically stable equilibrium under pure selection, then can migration change the (global) stability of $\hat{p}$? When $\hat{p}$ is a complete polymorphism, the answer is no, which means the ultimate state of the population is unaffected by geographical structure. However, if not every allele is present in $\hat{p}$, this problem remains largely open. In this paper we resolve the latter case for three alleles. The situation when the population occupies a finite continuous habitat of arbitrary dimensionality and shape is similar and also addressed.