The homogenized equation of a heterogenous Reaction-Diffusion model involving pulsating traveling fronts

TL;DR

This paper derives a homogenized reaction-diffusion equation from a heterogeneous Fisher-KPP model with pulsating traveling fronts, revealing the limiting behavior and minimal propagation speed in a periodic medium.

Contribution

It introduces a method to find the homogenized limit of pulsating fronts and characterizes their minimal speed in a periodic heterogeneous environment.

Findings

Homogenized limit of stationary states depends on space variables.

Pulsating fronts converge to classical traveling fronts in the homogenized equation.

Minimal speed of the homogenized front is explicitly characterized.

Abstract

The goal of this paper is to find the homogenized equation of a heterogenous Fisher-KPP model in a periodic medium. The solutions of this model are pulsating travelling fronts whose \emph{speeds} are superior to a parametric minimal speed $c^*_L$. We first find the homogenized limit of the stationary states which depend on the space variable in many cases. Then, we prove that the pulsating travelling fronts converge to a classical $u_0:=u_0(t,x)$ of a homogenous reaction-diffusion equation. The homogenized limit $u_0$ is also a travelling front whose minimal speed of propagation is given in terms of the coefficients of the problem.