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

Published online by Cambridge University Press:  04 March 2015

D. IRON ,
J. RUMSEY ,
M. J. WARD  and
J. WEI
D. IRON
Affiliation:
Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada
J. RUMSEY
Affiliation:
Faculty of Management, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada
M. J. WARD
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada
J. WEI
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada
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Abstract

In the limit of an asymptotically large diffusivity ratio of order $\mathcal{O}$(ϵ−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.

Keywords

singular perturbationslocalized spotssumming logarithmic expansionsBravais latticeFloquet–Bloch theoryGreen's function
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 3 , June 2015 , pp. 325 - 353
Copyright
Copyright © Cambridge University Press 2015 

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