  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.
