Localized patterns and semi-strong interaction, a unifying framework for reaction–diffusion systems

Abstract Systems of activator–inhibitor reaction–diffusion equations posed on an infinite line are studied using a variety analytical and numerical methods. A canonical form is considered, which contains all known models with simple cubic autocatalytic nonlinearity arbitrary constant linear kinetics. Restricting attention to that have unique homogeneous equilibrium, this class includes the classical Schnakenberg Brusselator models, as well other systems proposed in literature model morphogenesis. Such feature Turing instability, when activator diffuses more slowly than inhibitor, leading stable spatially periodic patterns. Conversely limit small feed rates, semi-strong interaction asymptotic analysis shows existence isolated spike-like This paper describes broad bifurcation structures connect these two regimes. certain universal two-parameter state diagram revealed becomes sub-critical, onset homoclinic snaking. regime then morphs into spike regime, outer-fold being predicted by asymptotics. rescaling parameters field concentrations how can be independently diffusion rates. Temporal dynamics found strongly depend ratio though. Hopf occurs along branch spikes, subcritical for ratio, collapse state. As increases, typically supercritical interacts snaking also bifurcation, complex spatio-temporal dynamics. The details worked out number different fit theory mixture weakly nonlinear analysis, asymptotics continuation algorithms.