Asymptotic analysis of stochastic models in population genetics

Abstract

Stochastic models in population genetics, which lead to diffusion equations, are considered. A method for obtaining asymptotic expansions of the solutions of these equations is presented. The expansions are valid for t/ N small, where t is the time in generations and N is the population size. The method permits the analysis of models including selection, mutation, migration, etc. The case of two alleles at one locus is considered. Formulas and numerical results are presented. In the few special cases for which exact solutions are known, comparison shows that the asymptotic solution is good for values of t from zero to 2 N generations. The method can be applied to cases of more than two alleles at one locus and to linkage of genes at different loci.