Explicit pulsating fronts and minimal speeds in periodic Fisher-KPP equations

TL;DR

This paper derives explicit pulsating traveling fronts and minimal speeds for a class of periodic Fisher-KPP equations by transforming the problem into a homogeneous case, providing exact solutions and extending delay results.

Contribution

It introduces a method to explicitly construct pulsating fronts in periodic media via a nonlinear change of variables, extending known results to heterogeneous diffusion cases.

Findings

Explicit pulsating front solutions in certain periodic media

Closed-form expression for asymptotic spreading speed

Extension of Bramson-type delay results to heterogeneous diffusion

Abstract

We study a Fisher-KPP equation with spatially periodic diffusion and reaction terms. We identify a class of periodic media for which the equation admits an explicit, closed-form solution. Through a nonlinear change of variables, the problem is reduced to the homogeneous Fisher-KPP equation, allowing us to construct an exact pulsating traveling front that connects the positive periodic stationary state to 0. We also derive an explicit expression for the asymptotic spreading speed and establish new asymptotic and comparison results. Finally, combining our change of variables and eigenvalue transform with existing results on KPP fronts in periodic media, we extend Bramson-type logarithmic delay results to the case of heterogeneous periodic diffusion.