SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 1053-1089, January 2021.
Quasi-stationary states consisting of localized spots in a reaction-diffusion system are considered on the surface of a torus with major radius $R$ and minor radius $r$. Under the assumption that these localized spots persist stably, the evolution equation of the spot cores is derived analytically based on the higher-order matched asymptotic expansion with the analytic expression of the Green's function of the Laplace--Beltrami operator on the toroidal surface. Owing to the analytic representation, one can investigate the existence of equilibria with a single spot, two spots, and the ring configuration where $N$ localized spots are equally spaced along a latitudinal line with mathematical rigor. We show that localized spots at the innermost/outermost locations of the torus are equilibria for any aspect ratio $\alpha=\frac{R}{r}$. In addition, we find that there exists a range of the aspect ratio in which localized spots stay at a special location of the torus. The theoretical results and the linear stability of these spot equilibria are confirmed by solving the nonlinear evolution of the Brusselator reaction-diffusion model by numerical means. We also compare the spot dynamics with the point vortex dynamics, which is another model of spot structures.

中文翻译：

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环面反应扩散系统的光斑动力学

SIAM Journal on Applied Dynamical Systems，第 20 卷，第 2 期，第 1053-1089 页，2021 年 1 月。
在具有大半径 $R$ 和小半径 $r$ 的环面表面上考虑由反应扩散系统中的局部点组成的准静止状态。在这些局部光点稳定存在的假设下，基于高阶匹配渐近展开式和环面上拉普拉斯-贝尔特拉米算子的格林函数的解析表达式，解析导出了光点核的演化方程。由于解析表示，我们可以用数学严谨性来研究具有单个点、两个点以及 $N$ 局部点沿纬度线等距分布的环配置的平衡的存在。我们表明，对于任何纵横比 $\alpha=\frac{R}{r}$，环面最内/最外位置的局部点都是平衡的。此外，我们发现存在一定范围的纵横比，其中局部斑点停留在环面的特殊位置。通过数值方法求解Brusselator反应-扩散模型的非线性演化，证实了这些点平衡的理论结果和线性稳定性。我们还将斑点动力学与点涡旋动力学进行比较，后者是斑点结构的另一种模型。