Classification of large-time behaviors in a reaction–diffusion system modeling diallelic selection

Abstract

We study a mathematical model from population genetics, describing a single-locus diallelic (A/a) selection-migration process. The model consists of a coupled system of three reaction–diffusion equations, one for the density of each genotype, posed in a bounded domain or in the whole space Rn. The genotype AA is advantageous, due to a smaller death rate, and the main concern is to determine whether or not the disadvantageous gene a is eliminated in the large time limit.This model was studied in the celebrated work of Aronson and Weinberger (1975,1977), where they derived a simplified scalar model as an approximation of the full system and studied the asymptotic behavior for the scalar model. In particular they showed that, in the fully recessive case (same death rate for the heterozygote and inferior homozygote), the behavior crucially depends on the space dimension. In a previous paper, we were able to prove that their results concerning the scalar model in the fully recessive case remain valid in a certain sense for the full system.In this paper, we reconsider the general case (all possible values of the death and birth rates). We succeed to give a complete picture of whether or not the disadvantageous gene a can survive as t→∞, according to the values of the death and birth rates and of the space dimension. We find distinctive behaviors according to whether the homozygote is superior, intermediate, or inferior and, in the latter case, to whether the common birth rate is smaller or higher than the difference of the death rates of the two heterozygotes. In cases when the disadvantageous gene disappears, the decay rate of its frequency is estimated as well.