The effect of a line with non-local diffusion on Fisher-KPP propagation

TL;DR

This paper introduces a new model with non-local diffusion on a line coupled to a reaction-diffusion system in a half-plane, demonstrating how transportation networks can accelerate biological invasions and alter propagation dynamics.

Contribution

It extends previous local diffusion models to include non-local diffusion on a line, providing a comprehensive analysis of invasion speeds and patterns.

Findings

Line accelerates propagation in the direction of the line

Overall propagation in the plane is enhanced by the line

Propagation on the line is exponentially fast in time

Abstract

We propose here a new model of accelerating fronts, consisting of one equation with non-local diffusion on a line, coupled via the boundary condition with a reaction-diffusion equation in the upper half-plane. The underlying biological question is to understand how transportation networks may enhance biological invasions. We show that the line accelerates the propagation in the direction of the line and enhances the overall propagation in the plane and that the propagation is directed by diffusion on the line, where it is exponentially fast in time. We also describe completely the invasion in the upper half-plane. This work is a non-local version of the model introduced in [Berestycki-Roquejoffre-Rossi 2013], where the line had a strong but local diffusion described by the classical Laplace operator.