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

Abstract

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 d y , 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 normalised so that ∫ − ∞ ∞ ϕ ( y ) d y = 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, (Berestycki et al 2009 Nonlinearity 22 2813–44). 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, (Billingham and Needham 1991 Dynam. Stabil. Syst. 6 33–49). 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.