Propagation and blocking in periodically hostile environments

TL;DR

This paper investigates how species propagate or are blocked in environments that are periodically hostile, using reaction-diffusion models, and provides conditions for propagation or blocking based on minimal speeds and geometric criteria.

Contribution

It introduces a new analysis of propagation and blocking phenomena in periodically hostile environments through reaction-diffusion equations, including conditions for propagation based on minimal speeds.

Findings

Existence of a minimal nonnegative stationary solution.

Geometric conditions for propagation blocking.

Asymptotic behavior of pulsating traveling fronts.

Abstract

We study the persistence and propagation (or blocking) phenomena for a species in periodically hostile environments. The problem is described by a reaction-diffusion equation with zero Dirichlet boundary condition. We first derive the existence of a minimal nonnegative nontrivial stationary solution and study the large-time behavior of the solution of the initial boundary value problem. To the main goal, we then study a sequence of approximated problems in the whole space with reaction terms which are with very negative growth rates outside the domain under investigation. Finally, for a given unit vector, by using the information of the minimal speeds of approximated problems, we provide a simple geometric condition for the blocking of propagation and we derive the asymptotic behavior of the approximated pulsating travelling fronts. Moreover, for the case of constant diffusion matrix, we provide two conditions for which the limit of approximated minimal speeds is positive.