Spreading speeds and pulsating fronts for a field-road model in a spatially periodic habitat

TL;DR

This paper analyzes a reaction-diffusion field-road model in a spatially periodic habitat, establishing the existence of a spreading speed and pulsating fronts, combining PDE and dynamical systems methods in heterogeneous landscapes.

Contribution

It extends the field-road model to heterogeneous environments, proving the existence of a spreading speed and pulsating fronts in such complex habitats.

Findings

Existence of asymptotic spreading speed c* in heterogeneous landscapes

Coincidence of spreading speed with minimal wave speed of pulsating fronts

Development of a method combining PDE and dynamical systems approaches

Abstract

A reaction-diffusion model which is called the field-road model was introduced by Berestycki, Roquejoffre and Rossi [9] to describe biological invasion with fast diffusion on a line. In this paper, we investigate this model in a heterogeneous landscape and establish the existence of the asymptotic spreading speed c * as well as its coincidence with the minimal wave speed of pulsating fronts along the road. We start with a truncated problem with an imposed Dirichlet boundary condition. We prove the existence of spreading speed c * R which coincides with the minimal speed of pulsating fronts for the truncated problem in the direction of the road. The arguments combine the dynamical system method with PDE's approach. Finally, we turn back to the original problem in the half-plane via generalized principal eigenvalue approach as well as an asymptotic method.