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Asymptotic Analysis of Diffusion Equations in Population Genetics

1978; Society for Industrial and Applied Mathematics; Volume: 34; Issue: 3 Linguagem: Inglês

10.1137/0134044

1095-712X

Charles Tier, Joseph B. Keller,

Genetic diversity and population structure

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