Slow travelling wave solutions of the nonlocal Fisher-KPP equation

Billingham, John

Abstract

© 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.

Citation

Billingham, J. (2020). Slow travelling wave solutions of the nonlocal Fisher-KPP equation. Nonlinearity, 33(5), 2106-2142. https://doi.org/10.1088/1361-6544/ab6f4f

• 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
• DOI: https://doi.org/10.1088/1361-6544/ab6f4f
• Keywords: nonlocal differential equation, travelling wave solution, matched asymptotic expansions
• Public URL: https://nottingham-repository.worktribe.com/output/2273665
• 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 https://iopscience.iop.org/article/10.1088/1361-6544/ab6f4f