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 ph(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 \hat\phi(\xi) is positive or if the length s of the non-local interaction is short enough, then the only steady states are u [?] 0 and u [?] 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[?](x), for all speeds c [?] c*. The travelling wave connects to the standard state u[?](x) [?] 1 under the aforementioned conditions: \hat\phi(\xi)>0 or s is sufficiently small. However, the wave is not monotonic for s large. [ABSTRACT FROM AUTHOR]