Gray-Scott reaction-diffusion on a sphere. The code will work on arbitrary meshes, since it uses the mesh Laplacian in the diffusion term. Pretty simple, but for a little while I thought I might be on to something, since it took a bit of digging before I found a paper which addressed reaction diffusion on other than regular grids. There's paper from 1991 which discusses reaction-diffusion on arbitrary surfaces, but it's very complicated and expensive. Finally, I did find a 2007 paper which uses much the same approach as I'm using here, which is super simple.
Slightly more complicated geometry:
Denser mesh (327,680 vertices):