For the Schnakenberg activator-inhibitor model on a torus, in the singularly perturbed regime of small activator to inhibitor diffusivity ratio ε2 « 1, we derive a reduced ODE describing the influence of curvature on the the slow drift dynamics of a single localised spot, and also stability thresholds to fast amplitude instabilities of one- and two-spot patterns. By way of a hybrid asymptotic-numerical analysis, we obtain the results in terms of certain quantities associated with the Green's function for both the Laplace–Beltrami (Δg) and Helmholtz (Δg - V) operators on the torus. To this end, we introduce a new analytic-numerical method for computing Green's functions on surfaces that requires only the numerical solution of a problem that is as regular as is desired. This allows properties of Green's functions at the location of the singularity to be determined to a high degree of accuracy. The method is applicable to operators of the form Δg + X - V for any metric tensor g, first order differential operator X, and smooth potential . It centers on a microlocal approach for analytically determining the coefficients of all singular terms of the local behavior of a Green's function inside a region around the singular point. Remaining terms of the Green's function are solved for numerically using finite differences. The primary purpose of this paper is to both introduce the theoretical underpinnings of this technique and to numerically demonstrate its ability to accurately yield properties of a Green's function on a curved surface. All results are confirmed by numerical finite element solutions of the Schnakenberg reaction–diffusion system on the torus.
- reaction–diffusion system
- localized spot patterns
- microlocal analysis
- Hadamard parametrix
- Green’s functions on curved surfaces