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

TL;DR

This paper analyzes the invasion front behavior in a non-local Fisher-KPP equation, revealing different asymptotic positions depending on the competition kernel, and introduces new analytical tools for such problems.

Contribution

It provides the first detailed asymptotic analysis of the invasion front in non-local Fisher-KPP equations with varying kernels, using novel analytical techniques.

Findings

Front position is either $2t - (3/2) \log t + O(1)$ or $2t - O(t^eta)$ depending on the kernel.

Develops a local-in-time Harnack inequality for non-local equations.

Constructs examples of nonlinearities with fronts at $2t - O(t^eta)$.

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 $2t - O(t^\beta)$ for some explicit $\beta \in (0,1)$. 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 $\beta\in(0,1)$, examples of Fisher-KPP type non-linearities $f\_\beta$ such that the front for the localFisher-KPP equation with reaction term $f\_\beta$ is at $2t - O(t^\beta)$.