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

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