A hybrid asymptotic‐numerical theory is developed to analyze the effect of different types of localized heterogeneities on the existence, linear stability, and slow dynamics of localized spot patterns for the two‐component Schnakenberg reaction‐diffusion model in a 2‐D domain. Two distinct types of localized heterogeneities are considered: a strong localized perturbation of a spatially uniform feed rate and the effect of removing a small hole in the domain, through which the chemical species can leak out. Our hybrid theory reveals a wide range of novel phenomena such as saddle‐node bifurcations for quasi‐equilibrium spot patterns that otherwise would not occur for a homogeneous medium, a new type of spot solution pinned at the concentration point of the feed rate, spot self‐replication behavior leading to the creation of more than two new spots, and the existence of a creation‐annihilation attractor with at most three spots. Depending on the type of localized heterogeneity introduced, localized spots are either repelled or attracted toward the localized defect on asymptotically long time scales. Results for slow spot dynamics and detailed predictions of various instabilities of quasi‐equilibrium spot patterns, all based on our hybrid asymptotic‐numerical theory, are illustrated and confirmed through extensive full PDE numerical simulations.

中文翻译：

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具有局部异质性的二维Schnakenberg模型中的斑点模式

提出了一种混合渐近数值理论，用于分析二维域中两组分Schnakenberg反应扩散模型的不同类型的局部异质性对局部斑点模式的存在，线性稳定性和慢动力学的影响。考虑了两种不同类型的局部异质性：对空间均匀进给速度的强烈局部扰动，以及去除域中小孔的影响，化学物质可通过该孔泄漏出去。我们的混合理论揭示了许多新颖的现象，例如准平衡斑点图案的鞍节点分叉，否则对于均匀介质不会发生，一种新型斑点溶液固定在进料速率的集中点，斑点自我复制行为导致创建了两个以上的新点，以及存在最多三个位置的创造灭绝吸引子。根据引入的局部异质性的类型，在渐近的长时间尺度上，局部斑点被排斥或被吸引到局部缺陷。均基于我们的混合渐近数值理论，对慢点动力学的结果以及准平衡点模式各种不稳定性的详细预测进行了说明，并通过广泛的完整PDE数值模拟进行了证实。