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

Tohru Tsujikawa, Kousuke Kuto, Yasuhito Miyamoto, Hirofumi Izuhara

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1023-1034
Number of pages12
JournalDiscrete and Continuous Dynamical Systems - Series S
Volume8
Issue number5
DOIs
Publication statusPublished - 2015 Oct 1

Keywords

  • Bifurcation
  • Chemotaxis
  • Pattern formation
  • Shadow system

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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