Pinning of Diffusional Instabilities by Non-Uniform Curvature

Abstract

Turing patterns emerge from a spatially uniform state following a linear instability driven by diffusion. Features of the eventual pattern (stabilized by non-linearities) are already present in the initial unstable modes. On a uniform flat surface or perfect sphere, the unstable modes and final patterns are degenerate, reflecting translational/rotational symmetry. This symmetry can be broken, e.g. by a bump on a flat substrate or by deforming a sphere. As the diffusion operator on a two dimensional manifold depends on the underlying curvature, the degeneracy of the initial unstable mode is similarly reduced. Different shapes can pin different modes. We adapt methods of conformal mapping and perturbation theory to analytically examine how bumps and ripples entrain modes of the diffusion operator on cylinders and spheres. We confirm these results numerically, and provide closed form expressions that describe how non-uniformities in curvature pin diffusion-driven instabilities and the resulting patterns.