The island model with stochastic migration

Abstract

The island model with stochastically variable migration rate and immigrant gene frequency is investigated. It is supposed that the migration rate and the immigrant gene frequency are independent of each other in each generation, and each of them is independently and identically distributed in every generation. The treatment is confined to a single diallelic locus without mutation. If the diploid population is infinite, selection is absent and the immigrant gene frequency is fixed, then the gene frequency on the island converges to the immigrant frequency, and the logarithm of the absolute value of its deviation from it is asymptotically normally distributed. Assuming only neutrality, the evolution of the exact mean and variance of the gene frequency are derived for an island with finite populaton. Selection is included in the diffusion approximation: if all evolutionary forces have comparable roles, the gene frequency will be normally distributed at all times. All results in the paper are completely explicit.