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

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.