Asymptotic analysis of a monostable equation in periodic media

TL;DR

This paper analyzes the asymptotic behavior of a multidimensional monostable reaction-diffusion equation with periodic heterogeneity, showing convergence to a limit interface governed by pulsating front speeds, highlighting differences from homogeneous media.

Contribution

It provides a rigorous proof of the interface convergence and characterizes its motion in heterogeneous media, extending classical results to periodic environments.

Findings

Convergence to a limit interface as the rescaling parameter tends to zero.

Interface motion governed by minimal pulsating front speeds in each direction.

Distinct behavior compared to homogeneous media due to speed dependence on normal direction.

Abstract

We consider a multidimensional monostable reaction-diffusion equation whose nonlinearity involves periodic heterogeneity. This serves as a model of invasion for a population facing spatial heterogeneities. As a rescaling parameter tends to zero, we prove the convergence to a limit interface, whose motion is governed by the minimal speed (in each direction) of the underlying pulsating fronts. This dependance of the speed on the (moving) normal direction is in contrast with the homogeneous case and makes the analysis quite involved. Key ingredients are the recent improvement \cite{A-Gil} of the well-known spreading properties \cite{Wein02}, \cite{Ber-Ham-02}, and the solution of a Hamilton-Jacobi equation.