UNRAVELING THE TURING BIFURCATION USING SPATIALLY VARYING DIFFUSION-COEFFICIENTS

The Turing bifurcation is the basic bifurcation generating spatial pat tern, and lies at the heart of almost all mathematical models for patt erning in biology and chemistry. in this paper the authors determine t he structure of this bifurcation for two coupled reaction diffusion eq uations on a two-dimensional square spatial domain when the diffusion coefficients have a small explicit variation in space across the domai n. In the case of homogeneous diffusivities, the Turing bifurcation is highly degenerate. Using a two variable perturbation method, the auth ors show that the small explicit spatial inhomogeneity splits the bifu rcation into two separate primary and two separate secondary bifurcati ons, with all solution branches distinct. This splitting of the bifurc ation is more effective than that given by making the domain slightly rectangular, and shows clearly the structure of the Turing bifurcation and the way in which the various solution branches collapse together as the spatial variation is reduced. The authors determine the stabili ty of the solution branches, which indicates that several new phenomen a are introduced by the spatial variation, including stable subcritica l striped patterns, and the possibility that stable stripes lose stabi lity supercritically to give stable spotted patterns.