Axisymmetric ring solutions of the 2D Gray–Scott model and their destabilization into spots

This article concerns annular ring solutions of the Gray–Scott model. In the monostable regime, annular rings are far-from-equilibrium patterns supported on annuli inside of which the activator is concentrated. The diffusive flux of inhibitor over long length scales toward such an annulus feeds the production of activator there, and the interaction is semi-strong. Numerical and experimental observations show that annular rings often split into spots, and the main result presented in this article is a method to predict the number of spots that an annular ring, unstable to angular disturbances, will split into. This method is an extension to 2D circular geometries of the nonlocal eigenvalue problem (NLEP) method developed for pulse solutions of the 1D Gray–Scott problem, in which the full eigenvalue problem—a pair of second-order, nonautonomous coupled equations—is recast as a single, second-order equation with a nonlocal term. We also continue the results for the monostable regime into the bistable regime of the Gray–Scott model, where target patterns exist and their rings are observed to destabilize into rings of spots, as may be shown using a classical Turing/Ginzburg–Landau analysis. Thus, for these 2D circular geometries, the NLEP method is to the instability of annular rings in the monostable regime what the Turing analysis is to the instability of target patterns in the bistable regime near criticality.