A ring of spikes

TL;DR

This paper analyzes the stability of symmetric spike configurations in the Schnakenberg model, revealing conditions under which rings of spikes are stable or unstable inside disks and annuli, depending on the number of spikes and feed-rate.

Contribution

It provides a detailed stability analysis of symmetric spike rings in the Schnakenberg model, including the effects of geometry, spike number, and feed-rate, which was not previously characterized.

Findings

Rings of 9 or more spikes are always unstable inside a disk.

Rings of 8 or fewer spikes can be stable with high feed-rate.

Instability leads to deformation into square-like patterns or spike death.

Abstract

For the Schnakenberg model, we consider a highly symmetric configuration of N spikes whose locations are located at the vertices of a regular N-gon inside either a unit disk or an annulus. We call such configuration a ring of spikes. The ring radius is characterized in terms of the modified Green's function. For a disk, we find that a ring of 9 or more spikes is always unstable with respect to small eigenvalues. Conversely, a ring of 8 or less spikes is stable inside a disk provided that the feed-rate $A$ is sufficiently large. More generally, for sufficiently high feed-rate, a ring of $N$ spikes can be stabilized provided that the annulus is thin enough. As $A$ is decreased, we show that the ring is destabilized due to small eigenvalues first, and then due to large eigenvalues, although both of these thresholds are separated by an asymptotically small amount. For a ring of 8 spikes inside a disk, the instability appears to be supercritical, and deforms the ring into a square-like configuration. For less than 8 spikes, this instability is subcritical and results in spike death.