Dirac concentrations in Lotka-Volterra parabolic PDEs

TL;DR

This paper studies how solutions to Lotka-Volterra parabolic PDEs with mutations concentrate into moving Dirac masses, revealing complex dynamics described by constrained Hamilton-Jacobi equations and new technical insights.

Contribution

It extends previous results to the parabolic case with general nonlinearities, introducing new techniques and counterexamples for Dirac concentration behavior.

Findings

Solutions converge to a moving Dirac mass as diffusion vanishes

Velocity and weights are characterized by a constrained Hamilton-Jacobi equation

Counterexamples show possible jumps at Dirac locations

Abstract

We consider parabolic partial differential equations of Lotka-Volterra type, with a non-local nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the nonlinear birth/death term represents competition leading to selection. Once rescaled with a small diffusion, we prove that the solutions converge to a moving Dirac mass. The velocity and weights cannot be obtained by a simple expression, e.g., an ordinary differential equation. We show that they are given by a constrained Hamilton-Jacobi equation. This extends several earlier results to the parabolic case and to general nonlinearities. Technical new ingredients are a $BV$ estimate in time on the non-local nonlinearity, a characterization of the concentration point (in a monomorphic situation) and, surprisingly, some counter-examples showing that jumps on the Dirac locations are indeed possible.