Refined stability thresholds for localized spot patterns for the Brusselator model in R2

Y. Chang, J. C. Tzou, M. Ward, J. C. Wei

Research output: Contribution to journalArticleResearchpeer-review

Abstract

In the singular perturbation limit ε → 0, we analyse the linear stability of multi-spot patterns on a bounded 2-D domain, with Neumann boundary conditions, as well as periodic patterns of spots centred at the lattice points of a Bravais lattice in  , for the Brusselator reaction–diffusion model

where the parameters satisfy 0 < f < 1, τ > 0 and D > 0. A previous leading-order linear stability theory characterizing the onset of spot amplitude instabilities for the parameter regime D =  (ν−1), where ν = −1/log ϵ, based on a rigorous analysis of a non-local eigenvalue problem (NLEP), predicts that zero-eigenvalue crossings are degenerate. To unfold this degeneracy, the conventional leading-order-in-ν NLEP linear stability theory for spot amplitude instabilities is extended to one higher order in the logarithmic gauge ν. For a multi-spot pattern on a finite domain under a certain symmetry condition on the spot configuration, or for a periodic pattern of spots centred at the lattice points of a Bravais lattice in  , our extended NLEP theory provides explicit and improved analytical predictions for the critical value of the inhibitor diffusivity D at which a competition instability, due to a zero-eigenvalue crossing, will occur. Finally, when D is below the competition stability threshold, a different extension of conventional NLEP theory is used to determine an explicit scaling law, with anomalous dependence on ϵ, for the Hopf bifurcation threshold value of τ that characterizes temporal oscillations in the spot amplitudes.

LanguageEnglish
Pages791-828
Number of pages38
JournalEuropean Journal of Applied Mathematics
Volume30
Issue number4
Early online date30 Jul 2018
DOIs
Publication statusPublished - Aug 2019

Fingerprint

Nonlocal Problems
Eigenvalue Problem
Linear Stability
Lattice Points
Stability Theory
Eigenvalue
Zero
Reaction-diffusion
Scaling Laws
Singular Perturbation
Diffusivity
Threshold Value
Neumann Boundary Conditions
Degeneracy
Model
Hopf Bifurcation
Inhibitor
Anomalous
Critical value
Gauge

Keywords

  • Bloch Green's function
  • Hopf and competition stability thresholds
  • Spot patterns
  • nonlocal eigenvalue problem

Cite this

@article{bb656e4fe3514d028c09a78ecdb563ca,
title = "Refined stability thresholds for localized spot patterns for the Brusselator model in R2",
abstract = "In the singular perturbation limit ε → 0, we analyse the linear stability of multi-spot patterns on a bounded 2-D domain, with Neumann boundary conditions, as well as periodic patterns of spots centred at the lattice points of a Bravais lattice in  , for the Brusselator reaction–diffusion modelwhere the parameters satisfy 0 < f < 1, τ > 0 and D > 0. A previous leading-order linear stability theory characterizing the onset of spot amplitude instabilities for the parameter regime D =  (ν−1), where ν = −1/log ϵ, based on a rigorous analysis of a non-local eigenvalue problem (NLEP), predicts that zero-eigenvalue crossings are degenerate. To unfold this degeneracy, the conventional leading-order-in-ν NLEP linear stability theory for spot amplitude instabilities is extended to one higher order in the logarithmic gauge ν. For a multi-spot pattern on a finite domain under a certain symmetry condition on the spot configuration, or for a periodic pattern of spots centred at the lattice points of a Bravais lattice in  , our extended NLEP theory provides explicit and improved analytical predictions for the critical value of the inhibitor diffusivity D at which a competition instability, due to a zero-eigenvalue crossing, will occur. Finally, when D is below the competition stability threshold, a different extension of conventional NLEP theory is used to determine an explicit scaling law, with anomalous dependence on ϵ, for the Hopf bifurcation threshold value of τ that characterizes temporal oscillations in the spot amplitudes.",
keywords = "Bloch Green's function, Hopf and competition stability thresholds, Spot patterns, nonlocal eigenvalue problem",
author = "Y. Chang and Tzou, {J. C.} and M. Ward and Wei, {J. C.}",
year = "2019",
month = "8",
doi = "10.1017/S0956792518000426",
language = "English",
volume = "30",
pages = "791--828",
journal = "European Journal of Applied Mathematics",
issn = "0956-7925",
publisher = "Cambridge University Press",
number = "4",

}

Refined stability thresholds for localized spot patterns for the Brusselator model in R2. / Chang, Y.; Tzou, J. C.; Ward, M.; Wei, J. C.

In: European Journal of Applied Mathematics, Vol. 30, No. 4, 08.2019, p. 791-828.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - Refined stability thresholds for localized spot patterns for the Brusselator model in R2

AU - Chang,Y.

AU - Tzou,J. C.

AU - Ward,M.

AU - Wei,J. C.

PY - 2019/8

Y1 - 2019/8

N2 - In the singular perturbation limit ε → 0, we analyse the linear stability of multi-spot patterns on a bounded 2-D domain, with Neumann boundary conditions, as well as periodic patterns of spots centred at the lattice points of a Bravais lattice in  , for the Brusselator reaction–diffusion modelwhere the parameters satisfy 0 < f < 1, τ > 0 and D > 0. A previous leading-order linear stability theory characterizing the onset of spot amplitude instabilities for the parameter regime D =  (ν−1), where ν = −1/log ϵ, based on a rigorous analysis of a non-local eigenvalue problem (NLEP), predicts that zero-eigenvalue crossings are degenerate. To unfold this degeneracy, the conventional leading-order-in-ν NLEP linear stability theory for spot amplitude instabilities is extended to one higher order in the logarithmic gauge ν. For a multi-spot pattern on a finite domain under a certain symmetry condition on the spot configuration, or for a periodic pattern of spots centred at the lattice points of a Bravais lattice in  , our extended NLEP theory provides explicit and improved analytical predictions for the critical value of the inhibitor diffusivity D at which a competition instability, due to a zero-eigenvalue crossing, will occur. Finally, when D is below the competition stability threshold, a different extension of conventional NLEP theory is used to determine an explicit scaling law, with anomalous dependence on ϵ, for the Hopf bifurcation threshold value of τ that characterizes temporal oscillations in the spot amplitudes.

AB - In the singular perturbation limit ε → 0, we analyse the linear stability of multi-spot patterns on a bounded 2-D domain, with Neumann boundary conditions, as well as periodic patterns of spots centred at the lattice points of a Bravais lattice in  , for the Brusselator reaction–diffusion modelwhere the parameters satisfy 0 < f < 1, τ > 0 and D > 0. A previous leading-order linear stability theory characterizing the onset of spot amplitude instabilities for the parameter regime D =  (ν−1), where ν = −1/log ϵ, based on a rigorous analysis of a non-local eigenvalue problem (NLEP), predicts that zero-eigenvalue crossings are degenerate. To unfold this degeneracy, the conventional leading-order-in-ν NLEP linear stability theory for spot amplitude instabilities is extended to one higher order in the logarithmic gauge ν. For a multi-spot pattern on a finite domain under a certain symmetry condition on the spot configuration, or for a periodic pattern of spots centred at the lattice points of a Bravais lattice in  , our extended NLEP theory provides explicit and improved analytical predictions for the critical value of the inhibitor diffusivity D at which a competition instability, due to a zero-eigenvalue crossing, will occur. Finally, when D is below the competition stability threshold, a different extension of conventional NLEP theory is used to determine an explicit scaling law, with anomalous dependence on ϵ, for the Hopf bifurcation threshold value of τ that characterizes temporal oscillations in the spot amplitudes.

KW - Bloch Green's function

KW - Hopf and competition stability thresholds

KW - Spot patterns

KW - nonlocal eigenvalue problem

UR - http://www.scopus.com/inward/record.url?scp=85052588111&partnerID=8YFLogxK

U2 - 10.1017/S0956792518000426

DO - 10.1017/S0956792518000426

M3 - Article

VL - 30

SP - 791

EP - 828

JO - European Journal of Applied Mathematics

T2 - European Journal of Applied Mathematics

JF - European Journal of Applied Mathematics

SN - 0956-7925

IS - 4

ER -