On a singular perturbation problem arising in the theory of Evolutionary Distributions

Abstract

Evolution by Natural Selection is a process by which progeny inherit some properties from their progenitors with small variation. These properties are subject to Natural Selection and are called adaptive traits and carriers of the latter are called phenotypes. The distribution of the density of phenotypes in a population is called Evolutionary Distributions (ED). We analyze mathematical models of the dynamics of a system of ED. Such systems are anisotropic in that diffusion of phenotypes in each population (equation) remains positive in the directions of their own adaptive space and vanishes in the directions of the other’s adaptive space. We prove that solutions to such systems exist in a sense weaker than the usual. We develop an algorithm for numerical solutions of such systems. Finally, we conduct numerical experiments—with a model in which populations compete—that allow us to observe salient attributes of a specific system of ED.