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 2 t − (3 / 2) log ⁡ t + O (1) , as in the local case, or 2 t − 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 2 t − O (t β). [ABSTRACT FROM AUTHOR]