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On accurately estimating stability thresholds for periodic spot patterns of reaction-diffusion systems in $\mathbb{R}$ 2

  • D. IRON (a1), J. RUMSEY (a2), M. J. WARD (a3) and J. WEI (a3)

Abstract

In the limit of an asymptotically large diffusivity ratio of order $\mathcal{O}$ −2) ≫ 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 two-component reaction–diffusion systems (RD) in $\mathbb{R}$ 2. 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 $\mathbb{R}$ 2.

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Keywords

On accurately estimating stability thresholds for periodic spot patterns of reaction-diffusion systems in $\mathbb{R}$ 2

  • D. IRON (a1), J. RUMSEY (a2), M. J. WARD (a3) and J. WEI (a3)

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