Blocking and invasion for reaction-diffusion equations in periodic media

TL;DR

This paper studies the long-term behavior of reaction-diffusion equations in periodic media, identifying conditions for invasion, the influence of domain geometry on propagation, and asymmetric invasion phenomena.

Contribution

It introduces new conditions for invasion in periodic media, analyzes the impact of domain geometry on propagation, and demonstrates asymmetric invasion in specific periodic domains.

Findings

Conditions guaranteeing invasion with compact initial data

Domain geometry can block or permit invasion

Existence of asymmetric invasion in certain periodic domains

Abstract

We investigate the large time behavior of solutions of reaction-diffusion equations with general reaction terms in periodic media. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous equation set in a domain with periodic holes, and specifically in the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion. We finally exhibit a periodic domain on which the propagation takes place in an asymmetric fashion, in the sense that the invasion occurs in a direction but is blocked in the opposite one.