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

Emeric Bouin, Christopher Henderson, Lenya Ryzhik

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

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(tβ) for some explicit β∈(0,1). Our main tools here are a local-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∈(0,1), examples of Fisher-KPP type non-linearities fβ such that the front for the local Fisher-KPP equation with reaction term fβ is at 2t−O(tβ).

Original languageEnglish (US)
Pages (from-to)51-77
Number of pages27
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume37
Issue number1
DOIs
StatePublished - Jan 1 2020
Externally publishedYes

Keywords

  • Logarithmic delay
  • Parabolic Harnack inequality
  • Reaction-diffusion equations

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

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