A competition system with nonlinear cross-diffusion: exact periodic patterns

Our concern in this paper is to shed some additional light on the mechanism and the effect caused by the so called cross-diffusion. We consider a two-species reaction-diffusion (RD) system. Both "fluxes" contain the gradients of both unknown solutions. We show that-for some parameter range- there exist two different type of periodic stationary solutions. Using them, we are able to divide into parts the (eight-dimensional) parameter space and indicate the so called Turing domains where our solutions exist. The boundaries of these domains, in analogy with "bifurcation point", called "bifurcation surfaces". As it is commonly believed, these solutions are limits as t goes to infinity of the solutions of corresponding evolution system. In a forthcoming paper we shall give a detailed account about our numerical results concerning different kind of stability. Here we also show some numerical calculations making plausible that our solutions are in fact attractors with a large domain of attraction in the space of initial functions.