Dynamic transition and pattern formation for chemotactic systems

Tian Ma; Shouhong Wang

doi:10.3934/dcdsb.2014.19.2809

Article Contents

2014, Volume 19, Issue 9: 2809-2835. Doi: 10.3934/dcdsb.2014.19.2809

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Dynamic transition and pattern formation for chemotactic systems

Tian Ma^1, and
Shouhong Wang^2,

1.
Department of Mathematics, Sichuan University, Chengdu
2.
Department of Mathematics, Indiana University, Bloomington, IN 47405

Received: May 31, 2012

Revised: May 31, 2014

Published: October 2014

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Abstract

The main objective of this article is to study the dynamic transition and pattern formation for chemotactic systems modeled by the Keller-Segel equations. We study chemotactic systems with either rich or moderated stimulant supplies. For the rich stimulant chemotactic system, we show that the chemotactic system always undergoes a Type-I or Type-II dynamic transition from the homogeneous state to steady state solutions. The type of transition is dictated by the sign of a non dimensional parameter $b$, which is derived by incorporating the nonlinear interactions of both stable and unstable modes. For the general Keller-Segel model where the stimulant is moderately supplied, the system can undergo a dynamic transition to either steady state patterns or spatiotemporal oscillations. From the pattern formation point of view, the formation and the mechanism of both the lamella and rectangular patterns are derived.

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Mathematics Subject Classification: Primary: 92C17; Secondary: 35Q92.

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