The Stability of Localized Spot Patterns for the Brusselator on the Sphere

In the singularly perturbed limit of an asymptotically small diffusivity ratio ε, the existence and stability of localized quasi-equilibrium multi-spot patterns is analyzed for the Brusselator reaction-diffusion model on the unit sphere. Formal asymptotic methods are used to derive a nonlinear algebraic system that characterizes quasi-equilibrium spot patterns, and to formulate eigenvalue problems governing the stability of spot patterns to three types of “fast” O(1) time-scale instabilities; self-replication, competition, or oscillatory instabilities of the spot amplitudes. The nonlinear algebraic system and the spectral problems are then studied using simple numerical methods, with emphasis on the special case where the spots have a common amplitude. Overall, the theoretical framework provides a hybrid asymptotic-numerical characterization of the existence and stability of spot patterns that is asymptotically correct to within all logarithmic correction terms in powers of ν = −1/ log ε. From a leading-order-in-ν analysis, and with an asymptotically large inhibitor diffusivity, some rigorous results for competition and oscillatory instabilities are obtained from an analysis of a new class of nonlocal eigenvalue problem (NLEP). Theoretical results for the stability of spot patterns are confirmed with full numerical computations of the Brusselator PDE system on the sphere using the Closest Point Method.