Skip to main content

Research Repository

Advanced Search

Travelling wave solutions of the cubic nonlocal Fisher-KPP equation: I. General theory and the near local limit

Billingham, J; Needham, D J


Professor of Theoretical Mechanics

D J Needham


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.


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.

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 Oct 27, 2023
Journal Nonlinearity
Print ISSN 0951-7715
Electronic ISSN 1361-6544
Publisher IOP Publishing
Peer Reviewed Peer Reviewed
Volume 35
Issue 12
Pages 6098-6123
Keywords Applied Mathematics; General Physics and Astronomy; Mathematical Physics; Statistical and Nonlinear Physics
Public URL
Publisher URL


You might also like

Downloadable Citations