This talk presents a robust, e_cient and accurate numerical method for solving reaction-

di_usion systems on stationary and evolving spheroidal surfaces. These surfaces are the

deformation of sphere such as ellipsoids, dumbbell, and hearth-shape surface. Reaction-

di_usion models are routinely used in the areas of developmental biology, cancer research,

wound healing, tissue regeneration, and cell motility. The advantages of the generalized

radially projected _nite element method are that it is easy to implement and that it

provides a conforming _nite element discretization which is \logically" rectangular. To

demonstrate the robustness, applicability and generality of this numerical method, we

present solutions of reaction-di_usion systems on various stationary and evolving sur-

faces. We show that the method preserves positivity of the solutions of reaction-di_usion

equations which is not true for the Galerkin type methods. We conclude that surface

geometry plays a pivotal role in pattern formation. For a _xed set of model parame-

ter values, di_erent surfaces give rise to di_erent pattern generation sequences of either

spots or stripes or a combination (observed as circular spot-stripe patterns). These re-

sults clearly demonstrate the need to carry out detailed theoretical analytical studies to

understand how surface geometry and curvature inuence pattern formation on complex

surfaces.