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, . 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, . 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. https://doi.org/10.1088/1361-6544/ac98ea