The influence of a line with fast diffusion on Fisher-KPP propagation

TL;DR

This paper introduces a new model for biological invasions that incorporates a line with fast diffusion, showing how such a line can significantly accelerate the spread of invasive species when diffusion exceeds a certain threshold.

Contribution

The paper develops a novel mathematical model capturing the effect of a fast diffusion line on invasion speed and characterizes the asymptotic spreading speed in the plane.

Findings

Fast diffusion lines can enhance invasion speed when diffusion exceeds a threshold.

The spreading speed grows proportionally to the square root of the diffusion coefficient on the line.

Below the threshold, the line has no significant effect on invasion dynamics.

Abstract

We propose here a new model to describe biological invasions in the plane when a strong diffusion takes place on a line. We establish the main properties of the system, and also derive the asymptotic speed of spreading in the direction of the line. For low diffusion, the line has no effect, whereas, past a threshold, the line enhances global diffusion in the plane and the propagation is directed by diffusion on the line. It is shown here that the global asymptotic speed of spreading in the plane, in the direction of the line, grows as the square root of the diffusion on the line. The model is much relevant to account for the effects of fast diffusion lines such as roads on spreading of invasive species.