Fisher’s Nonlinear Diffusion Equation and Selection-Migration Models

Abstract

A large number of studies have been made of the statics and dynamics of gene frequencies in a population subject to selection pressures and whose individuals may migrate geographically. In the case of deterministic models with space and time continuous, and selection restricted to a single locus with two alleles, the problem may sometimes be reduced to a single nonlinear diffusion equation. Such a reduction is advantageous, as it permits many qualitative properties of the frequency distribution to be obtained without a great deal of difficulty. This is especially true in the investigation of dines when the geographic variation in the selection characteristics is of a simple nature. It should also be brought out that the relevance of nonlinear diffusion equations in population genetics has provided part of the motivation behind the considerable mathematical work which has been devoted to these equations in recent years. This chapter is concerned with the justification for the use of single diffusion equations in modeling selection- migration phenomena. We begin with an overview of arguments which have been used in the past to model the problem by a single equation. We shall then approach the problem from the point of view of the previous chapter, considering selection to occur at a single gene locus and writing the corresponding reaction-diffusion equations for the three genotypes (which will be our “species”). Asymptotic methods, operating under explicit assumptions, will be used to reduce the three equations to a single one. Throughout the chapter we restrict attention, for simplicity, to populations in a one-dimensional habitat.