Slow travelling wave solutions of the nonlocal Fisher-KPP equation

Abstract

We study travelling wave solutions, u = U(x - ct), of the nonlocal Fisher- KPP equation in one spatial dimension, dimension, (Display equation presented), with D = 1 and c = 1, where = = u is the spatial convolution of the population density, u(x, t), with a continuous, symmetric, strictly positive kernel, =(x), which is decreasing for x > 0 and has a finite derivative as x = 0+, normalized so that = = -= =(x)dx = 1. In addition, we restrict our attention to kernels for which the spatially-uniform steady state u = 1 is stable, so that travelling wave solutions have U = 1 as x - ct → - and U = 0 as x - ct→ for c > 0. We use the formal method of matched asymptotic expansions and numerical methods to solve the travelling wave equation for various kernels, =(x), when c = 1. The most interesting feature of the leading order solution behind the wavefront is a sequence of tall, narrow spikes with O(1) weight, separated by regions where U is exponentially small. The regularity of =(x) at x = 0 is a key factor in determining the number and spacing of the spikes, and the spatial extent of the region where spikes exist.