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arXiv · 1710.03628

The Bramson delay in the non-local Fisher-KPP equation

Emeric Bouin·Christopher Henderson·Lenya Ryzhik
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Abstract

We consider the non-local Fisher-KPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either $2t - ({3}/2)\log t + O(1)$, as in the local case, or $2t - O(t^β)$ for some explicit $β\in (0,1)$. Our main tools here are alocal-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Dirichlet boundary condition. Our analysis also yields, for any $β\in(0,1)$, examples of Fisher-KPP type non-linearities $f\_β$ such that the front for the localFisher-KPP equation with reaction term $f\_β$ is at $2t - O(t^β)$.

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Emeric Bouin, Christopher Henderson, Lenya Ryzhik. 2019-11-27. The Bramson delay in the non-local Fisher-KPP equation. https://arxiv.org/abs/1710.03628

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