Pattern formation for a volume-filling chemotaxis model with logistic growth

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:448

Article

Han, Yazhou¹  Li, Zhongfang¹  Tao, Jicheng¹  Ma, Manjun² 

[1] China Jiliang Univ, Coll Sci, Dept Math, Hangzhou 310018, Zhejiang, Peoples R China

[2] Zhejiang Sci Tech Univ, Sch Sci, Dept Math, Hangzhou 310018, Zhejiang, Peoples R China

关键词: Pattern formation;    Weakly nonlinear analysis;    Chemotaxis system;    Volume-filling;    Logistic growth;   

DOI  :  10.1016/j.jmaa.2016.11.040

来源: Elsevier

【 摘 要 】

This paper is devoted to investigate the pattern formation of a volume-filling chemotaxis model with logistic cell growth. We first apply the local stability analysis to establish sufficient conditions of destabilization for uniform steady-state solution. Then, weakly nonlinear analysis with multi-scales is used to deal with the emerging process of patterns near the bifurcation point. For the single unstable mode case, we derive the Stuart-Landau equations describing the evolution of the amplitude, and thus the asymptotic expressions of patterns are obtained in both supercritical case and subcritical case. While for the case of multiple unstable modes, we also derive coupled amplitude equations to study the competitive behavior between two unstable modes through the phase plane analysis. In particular, we find that the initial data play a dominant role in the competition. All the theoretical and numerical results are in excellently qualitative agreement and better quantitative agreement than that in [10]. Moreover, in the subcritical case, we confirm the existence of stationary patterns with larger amplitudes when the bifurcation parameter is less than the first bifurcation point, which gives an positive answer to the open problem proposed in [9]. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free