Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result

TL;DR

This paper proves that solutions of two Lotka-Volterra type models with mutations, one involving Laplace terms and the other integral kernels, converge to moving Dirac masses described by a Hamilton-Jacobi equation, generalizing previous results.

Contribution

It extends convergence results to integro-differential equations with more general initial data assumptions, broadening the applicability of the theory.

Findings

Solutions converge to sums of moving Dirac masses

Limit described by a constrained Hamilton-Jacobi equation

Generalization to integro-differential mutation models

Abstract

We study two equations of Lotka-Volterra type that describe the Darwinian evolution of a population density. In the first model a Laplace term represents the mutations. In the second one we model the mutations by an integral kernel. In both cases, we use a nonlinear birth-death term that corresponds to the competition between the traits leading to selection. In the limit of rare or small mutations, we prove that the solution converges to a sum of moving Dirac masses. This limit is described by a constrained Hamilton-Jacobi equation. This was already proved by B. Perthame and G. Barles for the case with a Laplace term. Here we generalize the assumptions on the initial data and prove the same result for the integro-differential equation.