Curvature effect on patterning dynamics on spherical membranes

Leonardo Pacciani, Enzo Orlandini

Published in Padua Digital University Archive, 2015



The aim of this thesis is to carry out an analytical study of the role that surface curvature may have on the evolution and selection of patterns seen as solutions of a set of reaction-diffusion equations defined on spheres. Such equations exhibit diffusion-driven instability of spatially uniform structures leading to spatially non-uniform textures such as coat markings of animals and pigmentation patterns on butterfly wings. While the case with planar domains has been thoroughly studied in the past, much less is known for reaction-diffusion equations defined on closed surfaces. Here we will consider the simpler case of spherical domains, and by describing the possible stationary solutions in terms of spherical harmonics, we will perform a linear stability analysis of the equations as a function of the radius $R$ of the sphere (i.e. its curvature) and look for the most stable set of patterns compatible with that radius.