Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

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

Pages: 1023 - 1034, Volume 8, Issue 5, October 2015      doi:10.3934/dcdss.2015.8.1023

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Tohru Tsujikawa - Faculty of Engineering, University of Miyazaki, Miyazaki, 889-2192, Japan (email)
Kousuke Kuto - Department of Communication Engineering and Informatics, The University of Electro-Communications, Tokyo, 182-8585, Japan (email)
Yasuhito Miyamoto - Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, 153-8914, Japan (email)
Hirofumi Izuhara - Faculty of Engineering, University of Miyazaki, Miyazaki, 889-2192, Japan (email)

Abstract: From a viewpoint of the pattern formation, the Keller-Segel system with the growth term is studied. This model exhibited various static and dynamic patterns caused by the combination of three effects, chemotaxis, diffusion 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, singular perturbation and a level set analysis. Moreover, we show some numerical results on the global bifurcation branch of solutions by using AUTO package.

Keywords:  Bifurcation, chemotaxis, pattern formation, shadow system.
Mathematics Subject Classification:  Primary: 35B32, 37G40, 35K57; Secondary: 35B36.

Received: December 2013;      Revised: March 2015;      Available Online: July 2015.