Abstract
We study travelling wave solutions, , of the nonlocal Fisher-KPP equation in one spatial dimension, with and , where is the spatial convolution of the population density, , with a continuous, symmetric, strictly positive kernel, , which is decreasing for x > 0 and has a finite derivative as , normalized so that . 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 as and as for c > 0.We use the formal method of matched asymptotic expansions and numerical methods to solve the travelling wave equation for various kernels, , when . The most interesting feature of the leading order solution behind the wavefront is a sequence of tall, narrow spikes with weight, separated by regions where U is exponentially small. The regularity of 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.
Highlights
0.2 n zm w m n=7 n=6 n=5 n=4 10-1 n=3 n=2
(x) = 0.001 2(x) + 0.999 (x), c = 10-12 5
Summary
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