Stationary solutions for some shadow system of the Keller-Segel model with logistic growth

From a viewpoint of the pattern formation, the Keller-Segel sys- tem with the growth term is studied. This model exhibited various static and dynamic patterns caused by the combination of three effects, chemotaxis, dif- fusion and growth. In a special case when chemotaxis effect is very strong, some numerical experiment in [1],[22] showed static and chaotic patterns. In this paper we consider the logistic source for the growth and a shadow system in the limiting case that a diffusion coefficient and chemotactic intensity grow to infinity. We obtain the global structure of stationary solutions of the shadow system in the one-dimensional case. Our proof is based on the bifurcation, sin- gular perturbation and a level set analysis. Moreover, we show some numerical results on the global bifurcation branch of solutions by using AUTO package.