Oscillatory translational instabilities of localized spot patterns in the Schnakenberg reaction-diffusion system on general 2-D domains

TL;DR

This paper analyzes how the shape of a 2D domain influences oscillatory instabilities of localized spot patterns in a reaction-diffusion system, revealing geometric effects on stability thresholds and oscillation modes through asymptotic and numerical methods.

Contribution

It introduces a nonlinear matrix-eigenvalue framework to determine stability of spot patterns considering domain geometry, with explicit results for perturbed disk domains.

Findings

Stability thresholds depend on domain shape via Green's function terms.

Oscillation modes are influenced by specific Fourier modes of domain perturbation.

Numerical simulations confirm asymptotic predictions and illustrate geometric effects.

Abstract

For a bounded 2-D planar domain $\Omega$, we investigate the impact of domain geometry on oscillatory translational instabilities of $N$-spot equilibrium solutions for a singularly perturbed Schnakenberg reaction-diffusion system with $\mO(\eps^2) \ll \mO(1)$ activator-inhibitor diffusivity ratio. An $N$-spot equilibrium is characterized by an activator concentration that is exponentially small everywhere in $\Omega$ except in $N$ well-separated localized regions of $\mO(\eps)$ extent. We use the method of matched asymptotic analysis to analyze Hopf bifurcation thresholds above which the equilibrium becomes unstable to translational perturbations, which result in $\mO(\eps^2)$-frequency oscillations in the locations of the spots. We find that stability to these perturbations is governed by a nonlinear matrix-eigenvalue problem, the eigenvector of which is a $2N$-vector that characterizes the possible modes (directions) of oscillation. The $2N\times 2N$ matrix contains terms associated with a certain Green's function on $\Omega$, which encodes geometric effects. For the special case of a perturbed disk with radius in polar coordinates $r = 1 + \sigma f(\theta)$ with \red{$0< \varepsilon \ll \sigma \ll 1$}, $\theta \in [0,2\pi)$, and $f(\theta)$ $2\pi$-periodic, we show that only the mode-$2$ coefficients of the Fourier series of $f$ impact the bifurcation threshold at leading order in $\sigma$. We further show that when $f(\theta) = \cos2\theta$, the dominant mode of oscillation is in the direction parallel to the longer axis of the perturbed disk. Numerical investigations on the full Schnakenberg PDE are performed for various domains $\Omega$ and $N$-spot equilibria to confirm asymptotic results and also to demonstrate how domain geometry impacts thresholds and dominant oscillation modes.