Anomalous scaling of Hopf bifurcation thresholds for the stability of localized spot patterns for reaction-diffusion systems in two dimensions

For three specific singularly perturbed two-component reaction diffusion systems in a bounded two-dimensional domain admitting localized multispot 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 ² denoting the activator diffusivity, a leading-order-in-ν analysis shows that the linear stability of multispot 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 [J. Wei and M. Winter, Mathematical aspects of pattern formation in biological systems, Appl. Math. Sci. 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 ∼ ν ^{− 1}ε ⁻τ ^c ≫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.