The non-local Lotka–Volterra system with a top hat kernel—Part 1: dynamics and steady states with small diffusivity

Abstract

We study the dynamics of the non-local Lotka–Volterra system u t = D u u x x + u ( 1 − ϕ ∗ u − α v ) , v t = D v v x x + v ( 1 − ϕ ∗ v − β u ) , where a star denotes the spatial convolution and the kernel ϕ is a top hat function. We initially focus on the case of small, equal diffusivities ( D = D u = D v ≪ 1 ) together with weak interspecies interaction ( α , β ≪ 1 ), and specifically α , β ≪ D . This can then be extended to consider small, but unequal, diffusivities and weak interactions, with now α , β ≪ D u , D v ≪ 1 . Finally, we are able to develop the theory for the situation when the diffusivities remain small, but the interactions become stronger. In each case, we find that u and v independently develop into periodic spatial patterns that consist of separated humps on an O ( 1 ) time scale, and that these patterns become quasi-steady on a time scale proportional to the inverse diffusivity. These then interact on a longer time scale proportional to the inverse interaction scale, and approach a meta-stable state. Finally, a stable steady state is achieved on a much longer timescale, which is exponentially large relative to the preceding time scales. We are able to quantify this interaction process by determining a planar dynamical system that governs the temporal evolution of the separation between the two periodic arrays of humps on these sequentially algebraically and then exponentially long time scales. We find that, once the humps no longer overlap, the subsequent dynamics lead to a symmetric disposition of the humps, occurring on the exponentially long time scale. Numerical solutions of the full evolution problem cannot access the behaviour on this final extreme time scale, but it can be fully explored through the dynamical system.