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

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 is an element of (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 beta is an element of (0, 1), examples of Fisher-KPP type non-linearities f(beta) such that the front for the local Fisher-KPP equation with reaction term f(beta) is at 2t - O(t(beta)). (C) 2019 Published by Elsevier Masson SAS.