A KPP road-field system with spatially periodic exchange terms

TL;DR

This paper extends the analysis of a reaction-diffusion model with a road-field system to a spatially periodic setting, identifying conditions under which the road accelerates propagation and exploring eigenvalue-based methods for further applications.

Contribution

It generalizes previous models by incorporating spatial periodicity and establishes a threshold for road diffusion affecting spreading speed.

Findings

Existence of a diffusion threshold for accelerated spreading

Global speed is increased when road diffusion exceeds the threshold

Application of generalized principal eigenvalues to analyze spreading

Abstract

We take interest in a reaction-diffusion system which has been recently proposed [11] as a model for the effect of a road on propagation phenomena arising in epidemiology and ecology. This system consists in coupling a classical Fisher-KPP equation in a half-plane with a line with fast diffusion accounting for a straight road. The effect of the line on spreading properties of solutions (with compactly supported initial data) was investigated in a series of works starting from [11]. We recover these earlier results in a more general spatially periodic framework by exhibiting a threshold for road diffusion above which the propagation is driven by the road and the global speed is accelerated. We also discuss further applications of our approach, which will rely on the construction of a suitable generalized principal eigenvalue, and investigate in particular the spreading of solutions with exponentially decaying initial data.