The non-local Fisher-KPP equation: travelling waves and steady states

We consider the Fisher-KPP equation with a non-local saturation effect defined through an interaction kernel phi(x) and investigate the possible differences with the standard Fisher-KPP equation. Our first concern is the existence of steady states. We prove that if the Fourier transform (phi) over cap(xi) is positive or if the length sigma of the non-local interaction is short enough, then the only steady states are u equivalent to 0 and u equivalent to 1. Next, we study existence of the travelling waves. We prove that this equation admits travelling wave solutions that connect u = 0 to an unknown positive steady state u(infinity)(x), for all speeds c >= c*. The travelling wave connects to the standard state u(infinity)(x) = 1 under the aforementioned conditions: (phi) over cap(xi) > 0 or sigma is sufficiently small. However, the wave is not monotonic for sigma large.