Migration and mutation in stochastic models of gene frequency change

Summary

Migration has in the past been introduced deterministically into stochastic gene frequency models. Migration at rate m then reduces the between-population variability by a factor of (1 − m)² each generation. We show that with stochastic migration, whether of fixed or variable numbers of individuals, a positive term Δ_m is added to the variance. As a result of the Δ _m term, the equilibrium value of the between-population variability is increased compared to the corresponding value for deterministic migration by a factor of approximately (1 − m)⁻² for small m. An equivalent result is derived for mutation, using the infinite allele model for a single population. We show in addition that these results may be derived much more simply by use of identity-by-descent probability methods, but only if a modified definition of the probability of identity-by-descent is used, involving the sampling with instead of without replacement of pairs of genes from the population.