The influence of fractional diffusion in Fisher-KPP equations

TL;DR

This paper investigates how replacing the standard Laplacian with a fractional diffusion operator in the Fisher-KPP equation causes the invasion front to move exponentially fast, contrasting with the constant speed in classical models.

Contribution

It provides a rigorous mathematical analysis of fractional diffusion effects in Fisher-KPP equations, confirming heuristics about exponential front propagation.

Findings

Front position grows exponentially in time with fractional diffusion.

Standard Laplacian leads to constant speed invasion.

Results justify heuristic predictions about fractional Fisher-KPP models.

Abstract

We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the stan- dard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable L\'evy process, the front position is exponential in time. Our results provide a mathe- matically rigorous justification of numerous heuristics about this model.