### Abstract

For three specific singularly perturbed two-component reaction diffusion systems in a bounded two-dimensional domain admitting localized multispot patterns, we provide a detailed analysis of the parameter values for the onset of temporal oscillations of the spot amplitudes. The two key bifurcation parameters in each of the RD systems are the reaction-time parameter τ and the inhibitor diffusivity D. In the limit of large diffusivity D = D0/ν ≫1 with D0 = O(1), ν ≡ −1/log ε, and ε
^{2} denoting the activator diffusivity, a leading-order-in-ν analysis shows that the linear stability of multispot patterns is determined by the spectrum of a class of nonlocal eigenvalue problems (NLEPs). The specific form for these NLEPs depends on whether τ = O(1) or τ ≫1. For D0 < D0c, where D0c > 0 is some critical threshold, we show from a new parameterization of the NLEP that no Hopf bifurcations leading to temporal oscillations in the spot amplitudes can occur for any O(1) value of the reaction-time parameter τ. This resolves a long-standing open problem in NLEP theory (see [J. Wei and M. Winter, Mathematical aspects of pattern formation in biological systems, Appl. Math. Sci. 189, Springer, 2014]). Instead, by deriving a new modified NLEP appropriate to the regime τ ≫1, we show for the range D0 < D0c that a Hopf bifurcation will occur at some τ = τH ≫1, where τH has the anomalous scaling law τH ∼ ν
^{−}
^{1}ε
^{−}τ
^{c} ≫1 for some τc satisfying 0 < τc < 2. The anomalous exponent τc is calculated from the modified NLEP for each of the three RD systems.

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

Pages | 982–1022 |

Number of pages | 41 |

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 17 |

Issue number | 1 |

DOIs | |

Publication status | Published - 29 Mar 2018 |

Externally published | Yes |

### Fingerprint

### Keywords

- spot patterns
- Hopf bifurcation
- nonlocal eigenvalue problem
- anomalous scaling
- Green’s matrix

### Cite this

*SIAM Journal on Applied Dynamical Systems*,

*17*(1), 982–1022. https://doi.org/10.1137/17M1137759

}

*SIAM Journal on Applied Dynamical Systems*, vol. 17, no. 1, pp. 982–1022. https://doi.org/10.1137/17M1137759

**Anomalous scaling of Hopf bifurcation thresholds for the stability of localized spot patterns for reaction-diffusion systems in two dimensions.** / Tzou, J. C.; Ward, M. J.; Wei, J. C.

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

TY - JOUR

T1 - Anomalous scaling of Hopf bifurcation thresholds for the stability of localized spot patterns for reaction-diffusion systems in two dimensions

AU - Tzou,J. C.

AU - Ward,M. J.

AU - Wei,J. C.

PY - 2018/3/29

Y1 - 2018/3/29

N2 - For three specific singularly perturbed two-component reaction diffusion systems in a bounded two-dimensional domain admitting localized multispot patterns, we provide a detailed analysis of the parameter values for the onset of temporal oscillations of the spot amplitudes. The two key bifurcation parameters in each of the RD systems are the reaction-time parameter τ and the inhibitor diffusivity D. In the limit of large diffusivity D = D0/ν ≫1 with D0 = O(1), ν ≡ −1/log ε, and ε 2 denoting the activator diffusivity, a leading-order-in-ν analysis shows that the linear stability of multispot patterns is determined by the spectrum of a class of nonlocal eigenvalue problems (NLEPs). The specific form for these NLEPs depends on whether τ = O(1) or τ ≫1. For D0 < D0c, where D0c > 0 is some critical threshold, we show from a new parameterization of the NLEP that no Hopf bifurcations leading to temporal oscillations in the spot amplitudes can occur for any O(1) value of the reaction-time parameter τ. This resolves a long-standing open problem in NLEP theory (see [J. Wei and M. Winter, Mathematical aspects of pattern formation in biological systems, Appl. Math. Sci. 189, Springer, 2014]). Instead, by deriving a new modified NLEP appropriate to the regime τ ≫1, we show for the range D0 < D0c that a Hopf bifurcation will occur at some τ = τH ≫1, where τH has the anomalous scaling law τH ∼ ν − 1ε −τ c ≫1 for some τc satisfying 0 < τc < 2. The anomalous exponent τc is calculated from the modified NLEP for each of the three RD systems.

AB - For three specific singularly perturbed two-component reaction diffusion systems in a bounded two-dimensional domain admitting localized multispot patterns, we provide a detailed analysis of the parameter values for the onset of temporal oscillations of the spot amplitudes. The two key bifurcation parameters in each of the RD systems are the reaction-time parameter τ and the inhibitor diffusivity D. In the limit of large diffusivity D = D0/ν ≫1 with D0 = O(1), ν ≡ −1/log ε, and ε 2 denoting the activator diffusivity, a leading-order-in-ν analysis shows that the linear stability of multispot patterns is determined by the spectrum of a class of nonlocal eigenvalue problems (NLEPs). The specific form for these NLEPs depends on whether τ = O(1) or τ ≫1. For D0 < D0c, where D0c > 0 is some critical threshold, we show from a new parameterization of the NLEP that no Hopf bifurcations leading to temporal oscillations in the spot amplitudes can occur for any O(1) value of the reaction-time parameter τ. This resolves a long-standing open problem in NLEP theory (see [J. Wei and M. Winter, Mathematical aspects of pattern formation in biological systems, Appl. Math. Sci. 189, Springer, 2014]). Instead, by deriving a new modified NLEP appropriate to the regime τ ≫1, we show for the range D0 < D0c that a Hopf bifurcation will occur at some τ = τH ≫1, where τH has the anomalous scaling law τH ∼ ν − 1ε −τ c ≫1 for some τc satisfying 0 < τc < 2. The anomalous exponent τc is calculated from the modified NLEP for each of the three RD systems.

KW - spot patterns

KW - Hopf bifurcation

KW - nonlocal eigenvalue problem

KW - anomalous scaling

KW - Green’s matrix

UR - http://web.science.mq.edu.au/~jtzou/PUBLICATIONS/hopf2d.pdf

UR - http://www.scopus.com/inward/record.url?scp=85046691378&partnerID=8YFLogxK

U2 - 10.1137/17M1137759

DO - 10.1137/17M1137759

M3 - Article

VL - 17

SP - 982

EP - 1022

JO - SIAM Journal on Applied Dynamical Systems

T2 - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

SN - 1536-0040

IS - 1

ER -