The Bramson logarithmic delay in the cane toads equations

Emeric Bouin, Christopher Henderson, Lenya Ryzhik

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

We study a non-local reaction-diffusion-mutation equation modelling 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 travelling wave solutions as shown in an earlier work of the first author and V. Calvez. Here, we prove a Bramson type spreading result: the lag between the position of solutions with localized initial data and that of the travelling waves grows as (3/(2λ*)) log t. This result relies on a present-time Harnack inequality which allows one to compare solutions of the cane toads equation to those of a Fisher-KPP type equation that is local in the trait variable.

Original languageEnglish (US)
Pages (from-to)599-634
Number of pages36
JournalQuarterly of Applied Mathematics
Volume75
Issue number4
DOIs
StatePublished - 2017
Externally publishedYes

ASJC Scopus subject areas

  • Applied Mathematics

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