标题： 局部模式形成和三组分Gierer Meinhardt模型中的振荡不稳定性 显示英文标题

标题： Localized Pattern Formation and Oscillatory Instabilities in a Three-component Gierer Meinhardt Model

摘要： 在本文中，我们引入了一个三组分的Gierer-Meinhardt模型，在半强相互作用区域，其特征是扩散率比渐近地大。 该模型的一个关键特点是内部尖峰可以在振幅和位置上经历Hopf分支，导致经典两组分系统中不存在的丰富振荡动力学。 通过渐近分析和数值路径跟踪，我们构建了局部尖峰平衡态，并分析了通过鞍点分支之后的缓慢通过而发生的尖峰生成。 此外，通过引入时间缩放参数来分析尖峰平衡的稳定性，揭示了两种不同的机制：由大特征值不稳定性触发的振幅振荡以及与小特征值相关的振荡尖峰运动。 数值模拟展示了这些动力学及其过渡区域。 这种双重机制突显了三组分系统中更丰富的尖峰行为，并提出了几个未来研究的开放问题。 显示英文摘要

摘要： In this paper, we introduce a three-component Gierer-Meinhardt model in the semi-strong interaction regime, characterized by an asymptotically large diffusivity ratio. A key feature of this model is that the interior spike can undergo Hopf bifurcations in both amplitude and position, leading to rich oscillatory dynamics not present in classical two-component systems. Using asymptotic analysis and numerical path-following, we construct localized spike equilibria and analyze spike nucleation that occurs through slow passage beyond a saddle-node bifurcation. Moreover, stability of spike equilibrium is analyzed by introducing time-scaling parameters, which reveal two distinct mechanisms: amplitude oscillations triggered by large-eigenvalue instabilities and oscillatory spike motion associated with small eigenvalues. Numerical simulations illustrate these dynamics and their transition regimes. This dual mechanism highlights richer spike behavior in three-component systems and suggests several open problems for future study.