Spot Dynamics of a Reaction-Diffusion System on the Surface of a Torus

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