How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?

TL;DR

This paper investigates how the spreading speed of Fisher-KPP reaction-diffusion equations is affected by random stationary heterogeneity in diffusion and reaction terms, revealing that heterogeneity can increase invasion speed.

Contribution

It introduces a new characterization of spreading speed to analyze the impact of heterogeneity on propagation speed in random media.

Findings

Adding a reaction term with null average speeds up propagation.

Rescaling coefficients by x/L with L>1 increases spreading speed.

Heterogeneity in the medium generally enhances invasion speed.

Abstract

We consider one-dimensional reaction-diffusion equations of Fisher-KPP type with random stationary ergodic coefficients. A classical result of Freidlin and Gartner [16] yields that the solutions of the initial value problems associated with compactly supported initial data admit a linear spreading speed almost surely. We use in this paper a new characterization of this spreading speed recently proved in [8] in order to investigate the dependence of this speed with respect to the heterogeneity of the diffusion and reaction terms. We prove in particular that adding a reaction term with null average or rescaling the coefficients by the change of variables x $\rightarrow$ x/L, with L \textgreater{} 1, speeds up the propagation. From a modelling point of view, these results mean that adding some heterogeneity in the medium gives a higher invasion speed, while fragmentation of the medium slows down the invasion.