  Originating from the pioneering study of Alan Turing, the bifurcation
analysis predicting spatial pattern formation from a spatially uniform state
for diffusing morphogens or chemical species that interact through nonlinear
reactions is a central problem in many chemical and biological systems. From a
mathematical viewpoint, one key challenge with this theory for two component
systems is that stable spatial patterns can typically only occur from a
spatially uniform state when a slowly diffusing "activator" species reacts with
a much faster diffusing "inhibitor" species. However, from a modeling
perspective, this large diffusivity ratio requirement for pattern formation is
often unrealistic in biological settings since different molecules tend to
diffuse with similar rates in extracellular spaces. As a result, one key
long-standing question is how to robustly obtain pattern formation in the
biologically realistic case where the time scales for diffusion of the
interacting species are comparable. For a coupled 1-D bulk-compartment
theoretical model, we investigate the emergence of spatial patterns for the
scenario where two bulk diffusing species with comparable diffusivities are
coupled to nonlinear reactions that occur only in localized "compartments",
such as on the boundaries of a 1-D domain. The exchange between the bulk medium
and the spatially localized compartments is modeled by a Robin boundary
condition with certain binding rates. As regulated by these binding rates, we
show for various specific nonlinearities that our 1-D coupled PDE-ODE model
admits symmetry-breaking bifurcations, leading to linearly stable asymmetric
steady-state patterns, even when the bulk diffusing species have equal
diffusivities. Depending on the form of the nonlinear kinetics, oscillatory
instabilities can also be triggered. Moreover, the analysis is extended to
treat a periodic chain of compartments.
