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

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

Research output: Contribution to journalArticleResearchpeer-review

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 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.

LanguageEnglish
Pages982–1022
Number of pages41
JournalSIAM Journal on Applied Dynamical Systems
Volume17
Issue number1
DOIs
Publication statusPublished - 29 Mar 2018
Externally publishedYes

Fingerprint

Anomalous Scaling
Nonlocal Problems
Hopf bifurcation
Reaction-diffusion System
Hopf Bifurcation
Eigenvalue Problem
Two Dimensions
Diffusivity
Reaction Time
Scaling laws
Biological systems
Parameterization
Oscillation
Critical Threshold
Linear Stability
Scaling Laws
Pattern Formation
Singularly Perturbed
Biological Systems
Inhibitor

Keywords

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

Cite this

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title = "Anomalous scaling of Hopf bifurcation thresholds for the stability of localized spot patterns for reaction-diffusion systems in two dimensions",
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 − 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.",
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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.

In: SIAM Journal on Applied Dynamical Systems, Vol. 17, No. 1, 29.03.2018, p. 982–1022.

Research output: Contribution to journalArticleResearchpeer-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

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T2 - SIAM Journal on Applied Dynamical Systems

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