Some Theoretical Aspects of Diffusion Theory in Population Genetics

Abstract

This paper considers the problem of approximating a discrete time, discrete states, non-Markov process by a continuous diffusion process. The problem is set in the context of population genetics but may have more general application. In the genetic situation, most authors have treated the stochastic behavior of a gene frequency subject to evolutionary factors by assuming its probability distribution may be approximated by the solution of the Fokker-Planck diffusion equation. It is here shown that under certain sufficient conditions such an approximation is valid, even for genetic models in which the gene frequency does not necessarily form a Markov process. A summary of some old and new results concerning the asymptotic behavior of gene frequency is given, with special emphasis on the case when mutation is absent so that an absorbing state will ultimately be reached.