Propagation phenomena in periodic patchy landscapes with interface conditions

TL;DR

This paper studies the spread of a species in a one-dimensional environment with periodic patchy landscapes, establishing well-posedness, steady states, and analyzing spreading speeds and traveling waves.

Contribution

It introduces a comprehensive analysis of propagation phenomena in periodic patchy landscapes, including existence of pulsating traveling waves and a variational formula for spreading speed.

Findings

Existence of positive steady states

Asymptotic spreading speed equals minimal wave speed

Variational formula for spreading speed

Abstract

This paper is concerned with a model for the dynamics of a single species in a one-dimensional heterogeneous environment. The environment consists of two kinds of patches, which are periodically alternately arranged along the spatial axis. We first establish the well-posedness for the Cauchy problem. Next, we give existence and uniqueness results for the positive steady state and we analyze the long-time behavior of the solutions to the evolution problem. Afterwards, based on dynamical systems methods, we investigate the spreading properties and the existence of pulsating traveling waves in the positive and negative directions. It is shown that the asymptotic spreading speed, c * , exists and coincides with the minimal wave speed of pulsating traveling waves in positive and negative directions. In particular, we give a variational formula for c * by using the principal eigenvalues of certain linear periodic eigenvalue problems.