Pattern formation in a nonlinear model for animal coats

Abstract

Several models have been proposed for describing the formation of animal coat patterns. We consider reaction–diffusion models due to Murray, which rely on a Turing instability for the pattern selection. In this paper, we describe the early stages of the pattern formation process for large domain sizes. This includes the selection mechanism and the geometry of the patterns generated by the nonlinear system on one-, two-, and three-dimensional base domains. These results are obtained by an adaptation of results explaining the occurrence of spinodal decomposition in materials science as modeled by the Cahn–Hilliard equation. We use techniques of dynamical systems, viewing solutions of the reaction–diffusion model in terms of nonlinear semiflows. Our results are applicable to any parabolic system exhibiting a Turing instability.