On the persistence of spatially homogeneous solutions of a population genetics model with slow selection

Abstract

We consider a model of a nonconstant population with a genetic trait determined by one locus and two alleles. Natural selection as reflected in small differences among the genotype birth and death rates acts slowly and depends on the density. This model was first considered by F. Hoppensteadt for the density independent case. The assumption of slow selection allows the construction of solutions of the model using matched asymptotic expansions. It is found that to leading order the allele frequency is the solution of a logistic type differential equation, so that its ultimate value can be determined once the birth rates, death rates, and population size to leading order are given. In Sec. 2 solutions of the corresponding diffusion model and a related system which leads to Fisher's equation are constructed under two assumptions: first that dispersal is fast relative to selection, and then that it is comparable. With the first assumption, and to leading order, the genotype frequencies are slowly varying functions of time t for large t. In the second case the allele frequency is the solution of a generalization of Fisher's equation.