Anomalous Scaling of Hopf Bifurcation Thresholds for the Stability of Localized Spot Patterns for Reaction-Diffusion Systems in Two Dimensions
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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 $\tau$ and the inhibitor diffusivity $D$. In the limit of large diffusivity $D={D_0/\nu}\gg 1$ with $D_0={\mathcal O}(1)$, $\nu\equiv {-1/\log\varepsilon}$, and $\varepsilon^2$ denoting the activator diffusivity, a leading-order-in-$\nu$ 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 $\tau={\mathcal O}(1)$ or $\tau\gg 1$. For $D_0<D_{0c}$, where $D_{0c}>0$ 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 ${\mathcal O}(1)$ value of the reaction-time parameter $\tau$. 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 $\tau\gg 1$, we show for the range $D_0<D_{0c}$ that a Hopf bifurcation will occur at some $\tau=\tau_H\gg 1$, where $\tau_H$ has the anomalous scaling law $\tau_H\sim \nu^{-1}\varepsilon^{-\tau_c}\gg 1$ for some $\tau_c$ satisfying $0<\tau_c<2$. The anomalous exponent $\tau_c$ is calculated from the modified NLEP for each of the three RD systems.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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