Stability of patterns on thin curved surfaces

We consider reaction-diffusion equations on a thin curved surface and obtain a set of effective reaction-diffusion (R-D) equations to O(ε^{2}), where ε is the surface thickness. We observe that the R-D systems on these curved surfaces can have space-dependent reaction kinetics. Further, we use linear stability analysis to study the Schnakenberg model on spherical and cylindrical geometries. The dependence of the steady state on the thickness is determined for both cases, and we find that a change in the thickness can stabilize the unstable modes, and vice versa. The combined effect of thickness and curvature can play an important role in the rearrangement of spatial patterns on thin curved surfaces.