Role of curvature and domain shape on Turing patterns

We consider pattern formation using reaction-diffusion equation on various non-uniformly curved surfaces. We explore how, in general, curvature and, in particular the domain shape would affect the pattern formation in these geometries. As examples, we study stripe and spot patterns on a torus, and on an ellipsoid. Our results show that the curvature and domain shape can control the orientation of stripe pattern as well as the size and number of spots. Our results also indicate that by controlling the curvature and shape, one can drive the chemicals to a preferred region. Specifically on a torus, curvature and shape can guide the chemicals more on to outer side than inner side. This result may prove important in the studies of self-organization of molecules in biological membranes.