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

Abstract

In the singularly perturbed limit of an asymptotically small diffusivity ratio ${\varepsilon}^2$, the existence and stability of localized quasi-equilibrium multispot 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” ${\mathcal O}(1)$ time-scale instabilities: self-replication, competition, and 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 $\n...