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^β) 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 f_β such that the front for the local Fisher-KPP equation with reaction term f_β is at 2t−O(t^β).