Existence, Stability, and Dynamics of Ring and Near-Ring Solutions to the Saturated Gierer--Meinhardt Model in the Semistrong Regime

Abstract

We analyze a singularly perturbed reaction-diffusion system in the semistrong diffusion regime in two spatial dimensions where an activator species is localized to a closed curve, while the inhibitor species exhibits long-range behavior over the domain. In the limit of small activator diffusivity, we derive a new moving boundary problem characterizing the slow time evolution of the curve, which is defined in terms of a quasi-steady state inhibitor diffusion field and its properties on the curve. Numerical results from this curve evolution problem are illustrated for the Gierer--Meinhardt model with saturation (GMS) in the activator kinetics. A detailed analysis of the existence, stability, and dynamics of ring and near-ring solutions for the GMS model is given, whereby the activator concentrates on a thin ring concentric within a circular domain. A key new result for this ring geometry is that by including activator saturation there is a qualitative change in the phase portrait of ring equilibria, in that there is an S-shaped bifurcation diagram for ring equilibria, which allows for hysteresis behavior. In contrast, without saturation, it is well known that there is a saddle-node bifurcation for the ring equilibria. For a near-circular ring, we develop an asymptotic expansion up to quadratic order to fully characterize the normal velocity perturbations from our curve evolution problem. In addition, we also analyze the linear stability of the ring solution to both breakup instabilities, leading to the disintegration of a ring into localized spots, and zig-zag instabilities, leading to the slow shape deformation of the ring. We show from a nonlocal eigenvalue problem that activator saturation can stabilize breakup patterns that otherwise would be unstable. Through a detailed matched asymptotic analysis, we derive a new explicit formula for the small eigenvalues associated with zig-zag instabilities, and we show that they are equivalent to the velocity perturbations induced by the near-circular ring geometry. Finally, we present full numerical simulations from the GMS PDE system that confirm the predictions of the analysis.