Abstract. We consider the Fisher–KPP equation with a non-local saturation effect defined through an interaction kernel (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 is positive or if the length σ 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: or σ is sufficiently small. However, the wave is not monotonic for σ large.

Mathematics Subject Classification: 35C07, 35Q92

Print publication: Issue 12 (December 2009)
Received 20 March 2009, in final form 27 August 2009
Published 30 October 2009