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Qualitative analysis of stationary Keller–Segel chemotaxis models with logistic growth

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Abstract

We study the stationary Keller–Segel chemotaxis models with logistic cellular growth over a one-dimensional region subject to the Neumann boundary condition. We show that nonconstant solutions emerge in the sense of Turing’s instability as the chemotaxis rate \({\chi}\) surpasses a threshold number. By taking the chemotaxis rate as the bifurcation parameter, we carry out bifurcation analysis on the system to obtain the explicit formulas of bifurcation values and small amplitude nonconstant positive solutions. Moreover, we show that solutions stay strictly positive in the continuum of each branch. The stabilities of these steady-state solutions are well studied when the creation and degradation rate of the chemical is assumed to be a linear function. Finally, we investigate the asymptotic behaviors of the monotone steady states. We construct solutions with interesting patterns such as a boundary spike when the chemotaxis rate is large enough and/or the cell motility is small.

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Authors and Affiliations

  1. Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, Sichuan, China

    Qi Wang

  2. Hanqing Advanced Institute of Economics and Finance, Renmin University of China, No. 59 Zhongguancun Street, Haidian District, Beijing, 100872, China

    Jingda Yan

  3. Department of Mathematics and Statistics, Dalhousie University, 6316 Coburg Road, Halifax, NS, B3H 4R2, Canada

    Chunyi Gai

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  1. Qi Wang
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  2. Jingda Yan
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  3. Chunyi Gai
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Correspondence to Qi Wang.

Additional information

QW is partially supported by NSF-China (Grant No. 11501460) and the Project (No. 15ZA0382) from Department of Education, Sichuan China.

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Wang, Q., Yan, J. & Gai, C. Qualitative analysis of stationary Keller–Segel chemotaxis models with logistic growth. Z. Angew. Math. Phys. 67, 51 (2016). https://doi.org/10.1007/s00033-016-0648-9

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  • DOI: https://doi.org/10.1007/s00033-016-0648-9

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