Competition instabilities of spike patterns for the 1D Gierer–Meinhardt and Schnakenberg models are subcritical

Abstract

Spatially localized 1D spike patterns occur for various two-component reaction–diffusion (RD) systems in the singular limit of a large diffusivity ratio. A competition instability of a steady-state spike pattern is a linear instability that locally preserves the sum of the heights of the spikes. This instability, which results from a zero-eigenvalue crossing of a nonlocal eigenvalue problem at a certain critical value of the inhibitor diffusivity, has been implicated from full PDE numerical simulations of various RD systems of triggering a nonlinear event leading to spike annihilation. As a result, this linear instability is believed to be a key mechanism for initiating a coarsening process of 1D spike patterns. As an extension of the linear theory, we develop and implement a weakly nonlinear theory to analyse competition instabilities associated with symmetric two-boundary spike equilibria on a finite 1D domain for the Gierer–Meinhardt and Schnakenberg RD models. Two symmetric boundary spikes interacting through a long-range bulk diffusion field is the simplest spatial configuration of interacting localized spikes that can undergo a competition instability. Within a neighborhood of the parameter value for the competition instability threshold, a multi-scale asymptotic expansion is used to derive an explicit amplitude equation for the heights of the boundary spikes. This amplitude equation confirms that the competition instability is subcritical and, moreover, it shows that the competition instability threshold corresponds to a symmetry-breaking bifurcation point where an unstable branch of asymmetric two-boundary spike equilibria emerges from the symmetric branch. Results from our weakly nonlinear analysis are confirmed from full numerical solutions of the steady-state problem using numerical bifurcation software.