The Bramson logarithmic delay in the cane toads equations

Bouin, Emeric; Henderson, Christopher; Ryzhik, Lenya

doi:10.1090/qam/1470

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
MathSciNet review: 3686514
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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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