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

TL;DR

This paper investigates the nonlocal KPP-Fisher equation, revealing how interactions from the left and right sides differently influence the existence, uniqueness, and nature of traveling wave solutions.

Contribution

It demonstrates the distinct roles of left and right interactions in the existence and uniqueness of traveling fronts, and provides a simplified proof of wave existence without restrictive assumptions.

Findings

Left dominance ensures uniqueness of traveling fronts.

Right dominance can lead to coexistence of monotone and oscillating fronts.

Traveling waves exist under broad conditions without technical restrictions.

Abstract

We consider the nonlocal KPP-Fisher equation $u_t(t,x) = u_{xx}(t,x) + u(t,x)(1-(K *u)(t,x))$ 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 $K(\cdot) \geq 0,$ $\int_{\mathbb{R}} K(s)ds =1$. The restrictions $K(s), s \geq 0,$ and $K(s), s \leq 0,$ 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 $K$.