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

Tohru  Tsujikawa; Kousuke  Kuto; Yasuhito  Miyamoto; Hirofumi  Izuhara

doi:10.3934/dcdss.2015.8.1023

Article Contents

2015, Volume 8, Issue 5: 1023-1034. Doi: 10.3934/dcdss.2015.8.1023

This issue Previous Article Towards modelling spiral motion of open plane curves Next Article

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

1.
Faculty of Engineering, University of Miyazaki, Miyazaki, 889-2192
2.
Department of Communication Engineering and Informatics, The University of Electro-Communications, Tokyo, 182-8585
3.
Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, 153-8914

Received: November 30, 2013

Revised: February 28, 2015

Published: October 2015

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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.

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Mathematics Subject Classification: Primary: 35B32, 37G40, 35K57; Secondary: 35B36.

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