Asymptotic Analysis of Diffusion Equations in Population Genetics

A variety of stochastic models in population genetics, which lead to diffusion equations in several dimensions, are described. Because these equations are difficult to solve, a ray method is presented for obtaining short time asymptotic solutions of them. The solutions are valid for $t \ll N$ generation times, where t is time and N is the population size. The method is applied to a general two dimensional boundary value problem with densities on the boundaries and at the corners. Then the resulting asymptotic solution is specialized to cases of independent traits. For a particular equation, this asymptotic solution is shown to agree with the asymptotic expansion of the exact solution. The method permits the analysis of models with more than two alleles at a locus, and with many loci. It was previously used by Voronka and Kelley [20] on problems in one dimension, and the results were in good agreement with some known exact solutions for t as large as N generation times