The influence of space dimension on the large-time behavior 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 the whole space ${\mathbb R^n}$. The genotype AA is advantageous, due to a smaller death rate, and we consider the fully recessive case where the other two genotypes aa and Aa have the same (higher) death rate. In the nondiffusive (spatially homogeneous) case, the disadvantageous gene a is always eliminated in the large time limit. In the presence of diffusion, when the birth rate exceeds a certain threshold value, we prove that this conclusion is still true for dimensions n ≤ 2, whereas for n ≥ 3 there exist initial distributions for which the advantageous gene A ultimately disappears. This is the first rigorous result of this type for the full system, and it solves a problem which seems to have been open since the celebrated work of Aronson and Weinberger (Lecture notes in mathematics, vol 446, Springer, New York, pp 5–49, 1975; Adv Math 30, 33–76, 1977), where similar results had been obtained for a simplified scalar model, that they derived as an approximation of the full system. Interestingly, we moreover show that, at the threshold value of the birth rate, the cut-off dimension shifts from n = 2 to n = 6.