On Accurately Estimating Stability Thresholds for Periodic Spot Patterns of Reaction-Diffusion Systems in R

In the limit of an asymptotically large diffusivity ratio of order O(ε) ≫ 1, steady-state spatially periodic patterns of localized spots, where the spots are centred at lattice points of a Bravais lattice, are well-known to exist for certain twocomponent reaction-diffusion systems (RD) in R. For the Schnakenberg RD model, such a localized periodic spot pattern is linearly unstable when the diffusivity ratio exceeds a certain critical threshold. However, since this critical threshold has an infinite order logarithmic series in powers of the logarithmic gauge ν ≡ −1/ log ε, a low-order truncation of this series is expected to be in rather poor agreement with the true stability threshold unless ε is very small. To overcome this difficulty, a hybrid asymptotic-numerical method is formulated and implemented that has the effect of summing this infinite order logarithmic expansion for the stability threshold. The numerical implementation of this hybrid method relies critically on obtaining a rapidly converging infinite series representation of the regular part of the Bloch Green’s function for the reduced-wave operator. Numerical results from the hybrid method for the stability threshold associated with a periodic spot pattern on a regular hexagonal lattice are compared with the two-term asymptotic results of [10] [Iron et al., J. Nonlinear Science, 2014]. As expected, the difference between the two-term and hybrid results is rather large when ε is only moderately small. A related hybrid method is devised for accurately approximating the stability threshold associated with a periodic pattern of localized spots for the Gray-Scott RD system in R.