Diffusion-Driven Instability on a Curved Surface: Spherical Case Revisited

In this manuscript, we review the reaction-diffusion systems when these processes occur on curved surfaces. We show a general overview, from the original manuscripts by Turing, to the most recent developments with thick curved surfaces. We use the classical Schnakenberg model to present in a self-contained way the instability conditions of pattern formation in a flat surface; next, we give the basic elements of differential geometry of surfaces. With these tools, we study the reaction-diffusion system on a curved surface particularly on the sphere. When comparing the dispersion relations of both geometries, we found a modification in the range of the wavenumber due solely to the geometry of the substrate where the species diffuses.