A picture of the global bifurcation diagram in ecological interacting and diffusing systems

This paper concerns global phenomena of pattern formation in stationary reaction-diffusion equations, possessing Turing's diffusion-induced instability, which appear typically in mathematical biology. Global bifurcation diagrams with respect to two diffusion parameters are presented by integrating two complementary approaches-analytical and numerical. Simple and double bifurcation analysis using group theoretic methods for the compact Lie group D_x, results from singular perturbations when one diffusion constant is sufficiently small, and a global existence theorem on primary bifurcated branches when the other diffusion is sufficiently large, are the main analytical results. A new numerical method is used to trace all bifurcating branches. We examine interrelations between those local and semi-global results under the light of global pictures obtained by numerical studies. A variety of interesting and new diagrams near double eigenvalues are observed. Coexistence of multiple stable stationary states and global extension and deformation of local double structures are one of the main conclusions.