Effects of Open Systems on the Existence, Dynamics, and Stability of Spot Patterns in the 2D Brusselator Model

Spot patterns, whereby the activator field becomes spatially localized near certain dynamically-evolving discrete spatial locations in a bounded multi-dimensional domain, is a common occurrence for two-component reaction-diffusion (RD) systems in the singular limit of a large diffusivity ratio. In previous studies of 2-D localized spot patterns for various specific well-known RD systems, the domain boundary was assumed to be impermeable to both the activator and inhibitor. We extend this theory by developing a hybrid asymptotic-numerical method to analyze spot patterns for the singularly perturbed 2-D Brusselator model in the case where the domain boundary is either only partially impermeable, as modeled by an inhomogeneous Robin boundary condition, or when there is an influx of inhibitor across the domain boundary. By applying our theory to the special case of one-spot patterns and ring patterns of spots inside the unit disk, we provide a detailed analysis of the effect on these two types of generalized boundary conditions on the existence, linear stability, and slow dynamics of quasi-equilibrium spot patterns. In particular, when there is an influx of inhibitor across the boundary of the unit disk, ring patterns of spots become pinned to a ringradius closer to the domain boundary. Under a Robin condition, a quasi-equilibrium ring patterns of spots is shown to exhibit a novel saddle-node bifurcation behavior in terms of either the inhibitor diffusivity, the Robin constant, or the ambient background concentration. Additional new solution behavior regarding spot patterns, which differs from that under standard no-flux boundary conditions, is illustrated. Results from our asymptotic theory are validated from full numerical simulations of the Brusselator model with either type of generalized boundary condition.