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