JOHN BILLINGHAM john.billingham@nottingham.ac.uk
Professor of Theoretical Mechanics
Travelling wave solutions of the cubic nonlocal Fisher-KPP equation: I. General theory and the near local limit
Billingham, J; Needham, D J
Authors
D J Needham
Abstract
We study non-negative travelling wave solutions, u ≡ U (x − ct) with constant wavespeed c > 0, of the cubic nonlocal Fisher-KPP equation in one spatial dimension, namely, ∂u ∂t = ∂ 2 u ∂x 2 + u 2 1 − 1 λ ∞ −∞ ϕ y − x λ u (y, t) dy , for (x, t) ∈ R × R + , where u(x, t) is the population density. Here ϕ(y) is a prescribed, piecewise continuous, symmetric, nonnegative and nontrivial, integrable kernel, which is nonincreasing for y > 0, has a finite derivative as y → 0 + and is normalized so that ∞ −∞ ϕ(y)dy = 1. The parameter λ is the ratio of the lengthscale of the kernel to the diffusion lengthscale. The quadratic version of the equation, with reaction term u(1 − ϕ * u), has a unique travelling wave solution (up to translation) for all c ≥ c min = 2. This minimum wavespeed is determined locally in the region where u ≪ 1, [1]. For the cubic equation, we find that a minimum wavespeed also exists, but that the numerical value of the minimum wavespeed is determined globally, just as it is for the local version of the equation, [2]. We also consider the asymptotic solution in the limit of a spatially-localised kernel, λ ≪ 1, for which the travelling wave solutions are close to those of the cubic Fisher-KPP equation, u t = u xx + u 2 (1 − u). We find that when ϕ = o(y −3) as y → ∞, the minimum wavespeed is 1 √ 2 + O(λ 4), but that when ϕ = O(y −n) with 1 < n ≤ 3, the minimum wavespeed is 1 √ 2 + O(λ 2(n−1)). In each case we determine the correction terms. We also compare these asymptotic solutions to numerical solutions and find excellent agreement for some specific choices of kernel.
Citation
Billingham, J., & Needham, D. J. (2022). Travelling wave solutions of the cubic nonlocal Fisher-KPP equation: I. General theory and the near local limit. Nonlinearity, 35(12), 6098-6123. https://doi.org/10.1088/1361-6544/ac98ea
Journal Article Type | Article |
---|---|
Acceptance Date | Oct 6, 2022 |
Online Publication Date | Oct 26, 2022 |
Publication Date | Dec 1, 2022 |
Deposit Date | Oct 6, 2022 |
Publicly Available Date | Mar 28, 2024 |
Journal | Nonlinearity |
Print ISSN | 0951-7715 |
Electronic ISSN | 1361-6544 |
Publisher | IOP Publishing |
Peer Reviewed | Peer Reviewed |
Volume | 35 |
Issue | 12 |
Pages | 6098-6123 |
DOI | https://doi.org/10.1088/1361-6544/ac98ea |
Keywords | Applied Mathematics; General Physics and Astronomy; Mathematical Physics; Statistical and Nonlinear Physics |
Public URL | https://nottingham-repository.worktribe.com/output/12032417 |
Publisher URL | https://iopscience.iop.org/article/10.1088/1361-6544/ac98ea |
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Publisher Licence URL
https://creativecommons.org/licenses/by/3.0/
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