The Bramson logarithmic delay in the cane toads equations

TL;DR

This paper proves a Bramson logarithmic delay in the spreading of cane toads modeled by a nonlocal reaction-diffusion equation, showing the lag grows logarithmically over time and using a novel Harnack inequality.

Contribution

It establishes a Bramson-type delay result for a nonlocal cane toads equation, extending understanding of spreading speeds in structured populations.

Findings

The delay grows as (3/(2λ)) log t over time.

A new Harnack inequality compares nonlocal and local Fisher-KPP equations.

Traveling wave solutions are characterized in bounded trait spaces.

Abstract

We study a nonlocal reaction-diffusion-mutation equation modeling the spreading of a cane toads population structured by a phenotypical trait responsible for the spatial diffusion rate. When the trait space is bounded, the cane toads equation admits traveling wave solutions [7]. Here, we prove a Bramson type spreading result: the lag between the position of solutions with localized initial data and that of the traveling waves grows as (3/(2$\lambda$ *)) log t. This result relies on a present-time Harnack inequality which allows to compare solutions of the cane toads equation to those of a Fisher-KPP type equation that is local in the trait variable.