Traveling waves in the nonlocal KPP-Fisher equation: different roles of the right and the left interactions

Abstract

We consider the nonlocal KPP-Fisher equation $u_t(t,x)=u_{xx}(t,x)+u(t,x)(1-(K\ast u)(t,x))$ which describes the evolution of population density $u(t,x)$ with respect to time $t$ and location $x$. The non-locality is expressed in terms of the convolution of $u(t,\cdot )$ with kernel $K(\cdot )\ge 0,$ ${\int }_{\mathbb{R}}K(s)ds=1$. The restrictions $K(s),s\ge 0,$ and $K(s),s\le 0,$ are responsible for interactions of an individual with his left and right neighbors, respectively. We show that these two parts of $K$ play quite different roles as for the existence and uniqueness of traveling fronts to the KPP-Fisher equation. In particular, if the left interaction is dominant, the uniqueness of fronts can be proved, while the dominance of the right interaction can induce the co-existence of monotone and oscillating fronts. We also present a short proof of the existence of traveling waves without assuming various technical restrictions usually imposed on $K$.

38 pages, submitted

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