### 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.

Language | English |
---|---|

Pages | 791-828 |

Number of pages | 38 |

Journal | European Journal of Applied Mathematics |

Volume | 30 |

Issue number | 4 |

Early online date | 30 Jul 2018 |

DOIs | |

Publication status | Published - Aug 2019 |

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### Keywords

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

### Cite this

^{2}.

*European Journal of Applied Mathematics*,

*30*(4), 791-828. https://doi.org/10.1017/S0956792518000426

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^{2}'

*European Journal of Applied Mathematics*, vol. 30, no. 4, pp. 791-828. https://doi.org/10.1017/S0956792518000426

**Refined stability thresholds for localized spot patterns for the Brusselator model in R ^{2}.** / Chang, Y.; Tzou, J. C.; Ward, M.; Wei, J. C.

Research output: Contribution to journal › Article › Research › peer-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 -

^{2}. European Journal of Applied Mathematics. 2019 Aug;30(4):791-828. https://doi.org/10.1017/S0956792518000426