Anomalous Scaling of Hopf Bifurcation Thresholds for the Stability of Localized Spot Patterns for Reaction-Diffusion Systems in 2-D

For three specific singularly perturbed two-component reaction diffusion systems in a bounded 2-D domain admitting localized multi-spot patterns, we provide a detailed analysis of the parameter values for the onset of temporal oscillations of the spot amplitudes. The two key bifurcation parameters in each of the RD systems are the reaction-time parameter τ and the inhibitor diffusivity D. In the limit of large diffusivity D = D0/ν ≫ 1, with D0 = O(1), ν ≡ −1/ log ε and ε denoting the activator diffusivity, a leading-order-in-ν analysis shows that the linear stability of multi-spot patterns is determined by the spectrum of a class of nonlocal eigenvalue problems (NLEPs). The specific form for these NLEPs depends on whether τ = O(1) or τ ≫ 1. For D0 < D0c, where D0c > 0 is some critical threshold, we show from a new parameterization of the NLEP that no Hopf bifurcations leading to temporal oscillations in the spot amplitudes can occur for any O(1) value of the reaction-time parameter τ . This resolves a long-standing open problem in NLEP theory (see [33] [J. Wei, M. Winter Mathematical aspects of pattern formation in biological systems, Applied Mathematical Science Series, Vol. 189, Springer, (2014)]). Instead, by deriving a new modified NLEP appropriate to the regime τ ≫ 1, we show for the range D0 < D0c that a Hopf bifurcation will occur at some τ = τH ≫ 1, where τH has the anomalous scaling law τH ∼ νεc ≫ 1, for some τc satisfying 0 < τc < 2. The anomalous exponent τc is calculated from the modified NLEP for each of the three RD systems.