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Originating from the pioneering study of Alan Turing, the bifurcationanalysis predicting spatial pattern formation from a spatially uniform statefor diffusing morphogens or chemical species that interact through nonlinearreactions is a central problem in many chemical and biological systems. From amathematical viewpoint, one key challenge with this theory for two componentsystems is that stable spatial patterns can typically only occur from aspatially uniform state when a slowly diffusing "activator" species reacts witha much faster diffusing "inhibitor" species. However, from a modelingperspective, this large diffusivity ratio requirement for pattern formation isoften unrealistic in biological settings since different molecules tend todiffuse with similar rates in extracellular spaces. As a result, one keylong-standing question is how to robustly obtain pattern formation in thebiologically realistic case where the time scales for diffusion of theinteracting species are comparable. For a coupled 1-D bulk-compartmenttheoretical model, we investigate the emergence of spatial patterns for thescenario where two bulk diffusing species with comparable diffusivities arecoupled to nonlinear reactions that occur only in localized "compartments",such as on the boundaries of a 1-D domain. The exchange between the bulk mediumand the spatially localized compartments is modeled by a Robin boundarycondition with certain binding rates. As regulated by these binding rates, weshow for various specific nonlinearities that our 1-D coupled PDE-ODE modeladmits symmetry-breaking bifurcations, leading to linearly stable asymmetricsteady-state patterns, even when the bulk diffusing species have equaldiffusivities. Depending on the form of the nonlinear kinetics, oscillatoryinstabilities can also be triggered. Moreover, the analysis is extended totreat a periodic chain of compartments.

The Emergence of Spatial Patterns for Compartmental Reaction Kinetics Coupled by Two Bulk Diffusing Species with Comparable Diffusivities