A ring of spikes for the Schnakenberg model

Consider N spikes on located along a ring inside a unit disk. This highly symmetric configuration corresponds to an equilibrium of a two-dimensional Schnakenberg model; the ring radius can be characterized in terms of the modified Green’s function. We study the stability of such a ring with respect to both small and large eigenvalues (corresponding to spike position and spike height perturbations, respectively), and characterize the instability thresholds. For sufficiently large feed rate, we find that a ring of 8 or less spikes is stable with respect to both small and large eigenvalues, whereas a ring of 9 spikes is unstable with respect to small eigenvalues. For 8 spikes or less, as the feed rate is decreased, a small eigen-value instability is triggered first, followed by large eigenvalue instability. For 8 spikes, this instability appears to be supercritical, and deforms a ring into a square-type configuration. The main tool we use is circulant matrices and an analogue of the floquet theory.