# A ring of spikes for the Schnakenberg model

Date: Mon, May 10, 2021

Location: UBC, Online

Conference: PIMS Workshop on New Trends in Localized Patterns in PDES

Subject: Mathematics

Class: Scientific

### Abstract:

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.