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A hybrid asymptotic-numerical theory is developed to analyze the effect ofdifferent types of localized heterogeneities on the existence, linearstability, and slow dynamics of localized spot patterns for the two-componentSchnakenberg reaction-diffusion model in a 2-D domain. Two distinct types oflocalized heterogeneities are considered: a strong localized perturbation of aspatially uniform feed rate and the effect of removing a small hole in thedomain, through which the chemical species can leak out. Our hybrid theoryreveals a wide range of novel phenomena such as, saddle-node bifurcations forquasi-equilibrium spot patterns that otherwise would not occur for ahomogeneous medium, a new type of spot solution pinned at the concentrationpoint of the feed rate, spot self-replication behavior leading to the creationof more than two new spots, and the existence of a creation-annihilationattractor with at most three spots. Depending on the type of localizedheterogeneity introduced, localized spots are either repelled or attractedtowards the localized defect on asymptotically long time scales. Results forslow spot dynamics and detailed predictions of various instabilities ofquasi-equilibrium spot patterns, all based on our hybrid asymptotic-numericaltheory, are illustrated and confirmed through extensive full PDE numericalsimulations.

Spot Patterns in the 2-D Schnakenberg Model with Localized Heterogeneities