Several Models have been proposed for describing the fomation 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 staes of the pattern formation process for large domain sizeds. This includes the selection mechanism and the geometry of the patterns generated by the nonlienar system on one-, two-, and three-dimensional base domains. These results are obtained by an adaptation of results explaining the occurence 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 applicable to any parabolic system exhibiting a Turing instability.
We investigate bifurcations in the chain recurrent set for a particular class of one-parameter families of diffeomorphisms in the plane. We give necessary and sufficient conditions for a discontinuous change in the chain recurrent set to occur at a point of heteroclinic tangency. These are also necessary and sufficient conditions for an O-explosion to occur at that point.