二维Gray-Scott模型中局域斑图的稳定性和动力学
The Stability and Dynamics of Localized Spot Patterns in the Two-Dimensional Gray-Scott Model
摘要：
研究了二维区域中Gray-Scott（GS）反应扩散模型在其中一个解的小扩散系数$ε$的奇摄动极限下的多斑图的动力学和稳定性.将匹配渐近展开法与某些特征值问题的详细数值研究相结合，采用混合渐近-数值方法预测了GS模型在极限$ε\to 0$下的多点拟平衡斑图的动力学行为和不稳定机制.导出了集体坐标$S_j$和$j=1,...,k$的${\mathbf x}_j$的微分代数常微分方程组，它表征了光斑图案的慢动力学。多光斑图案的不稳定性，由于三个不同的机制，点自复制，点振荡，和现场湮灭，研究了通过使用奇异摄动技术首先导出某些相关的本征值问题。从这些特征值问题的频谱的数值计算，相图代表在GS参数空间对应的发病点不稳定性得到各种简单的空间配置的多斑点图案。此外，它示出，有一个很宽的参数范围内，一个点的不稳定性可以被触发，只有作为一个结果的固有的慢动作的收集点。斑点动力学和斑点不稳定性的混合渐近数值结果进行了验证，从GS模型计算的各种空间配置的斑点的完整的数值结果。
英文摘要：
The dynamics and stability of multi-spot patterns to the Gray-Scott (GS) reaction-diffusion model in a two-dimensional domain is studied in the singularly perturbed limit of small diffusivity $ε$ of one of the two solution components. A hybrid asymptotic-numerical approach based on combining the method of matched asymptotic expansions with the detailed numerical study of certain eigenvalue problems is used to predict the dynamical behavior and instability mechanisms of multi-spot quasi-equilibrium patterns for the GS model in the limit $ε\to 0$. A differential algebraic ODE system for the collective coordinates $S_j$ and ${\mathbf x}_j$ for $j=1,...,k$ is derived, which characterizes the slow dynamics of a spot pattern. Instabilities of the multi-spot pattern due to the three distinct mechanisms of spot self-replication, spot oscillation, and spot annihilation, are studied by first deriving certain associated eigenvalue problems by using singular perturbation techniques. From a numerical computation of the spectrum of these eigenvalue problems, phase diagrams representing in the GS parameter space corresponding to the onset of spot instabilities are obtained for various simple spatial configurations of multi-spot patterns. In addition, it is shown that there is a wide parameter range where a spot instability can be triggered only as a result of the intrinsic slow motion of the collection of spots. The hybrid asymptotic-numerical results for spot dynamics and spot instabilities are validated from full numerical results computed from the GS model for various spatial configurations of spots.
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