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Slow travelling wave solutions of the nonlocal Fisher-KPP equation

Billingham, John


Professor of Theoretical Mechanics


© 2020 IOP Publishing Ltd & London Mathematical Society. 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.


Billingham, J. (2020). Slow travelling wave solutions of the nonlocal Fisher-KPP equation. Nonlinearity, 33(5), 2106-2142.

Journal Article Type Article
Acceptance Date Jan 23, 2020
Online Publication Date Mar 16, 2020
Publication Date May 1, 2020
Deposit Date Jul 5, 2019
Publicly Available Date Mar 17, 2021
Journal Nonlinearity
Print ISSN 0951-7715
Electronic ISSN 1361-6544
Publisher IOP Publishing
Peer Reviewed Peer Reviewed
Volume 33
Issue 5
Pages 2106-2142
Keywords nonlocal differential equation, travelling wave solution, matched asymptotic expansions
Public URL
Additional Information This is the accepted version of the following article: John Billingham, Slow travelling wave solutions of the nonlocal Fisher-KPP equatiion, 2020 Nonlinearity 33 2106, which has been published in final form at


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