Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The Bramson logarithmic delay in the cane toads equations

Authors: Emeric Bouin, Christopher Henderson and Lenya Ryzhik
Journal: Quart. Appl. Math. 75 (2017), 599-634
MSC (2010): Primary 35K57, 45K05, 35C07, 35Q92
DOI: https://doi.org/10.1090/qam/1470
Published electronically: April 18, 2017
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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/(2\lambda ^*))\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.

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Emeric Bouin
Affiliation: CEREMADE - Université Paris-Dauphine, UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
Email: bouin@ceremade.dauphine.fr

Christopher Henderson
Affiliation: Department of Mathematics, the University of Chicago, Chicago, Illinois 60637
Email: henderson@math.uchicago.edu

Lenya Ryzhik
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: ryzhik@math.stanford.edu

DOI: https://doi.org/10.1090/qam/1470
Received by editor(s): October 12, 2016
Received by editor(s) in revised form: March 17, 2017
Published electronically: April 18, 2017
Article copyright: © Copyright 2017 Brown University

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