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Mathematical Biosciences and Engineering

2015, Volume 12, Issue 4: 717-737. doi: 10.3934/mbe.2015.12.717
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Traveling bands for the Keller-Segel model with population growth

  • 1.
    Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899
  • 2.
    Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong
  • Received: 01 April 2014 Accepted: 29 June 2018 Published: 01 April 2015

  • MSC : 35C07, 35K55, 46N60, 62P10, 92C17.

  • This paper is concerned with the existence of the traveling bands to the Keller-Segel model with cell population growth in the form of a chemical uptake kinetics. We find that when the cell growth is considered, the profile of traveling bands, the minimum wave speed and the range of the chemical consumption rate for the existence of traveling wave solutions will change. Our results reveal that collective interaction of cell growth and chemical consumption rate plays an essential role in the generation of traveling bands. The research in the paper provides new insights into the mechanisms underlying the chemotactic pattern formation of wave bands.

    Citation: Shangbing Ai, Zhian Wang. Traveling bands for the Keller-Segel model with population growth[J]. Mathematical Biosciences and Engineering, 2015, 12(4): 717-737. doi: 10.3934/mbe.2015.12.717

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  • This paper is concerned with the existence of the traveling bands to the Keller-Segel model with cell population growth in the form of a chemical uptake kinetics. We find that when the cell growth is considered, the profile of traveling bands, the minimum wave speed and the range of the chemical consumption rate for the existence of traveling wave solutions will change. Our results reveal that collective interaction of cell growth and chemical consumption rate plays an essential role in the generation of traveling bands. The research in the paper provides new insights into the mechanisms underlying the chemotactic pattern formation of wave bands.


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    6. Yizhuo Wang, Shangjiang Guo, Dynamics for a two-species competitive Keller-Segel chemotaxis system with a free boundary, 2021, 502, 0022247X, 125259, 10.1016/j.jmaa.2021.125259
    7. José Luis López, On nonstandard chemotactic dynamics with logistic growth induced by a modified complex Ginzburg–Landau equation, 2022, 148, 0022-2526, 248, 10.1111/sapm.12440
    8. Yizhuo Wang, Shangjiang Guo, Traveling wave solutions for a two-species competitive Keller–Segel chemotaxis system, 2023, 73, 14681218, 103900, 10.1016/j.nonrwa.2023.103900
    9. Shangbing Ai, Zengji Du, Traveling wave solutions for a Keller-Segel system with nonlinear chemical gradient, 2024, 0022247X, 129128, 10.1016/j.jmaa.2024.129128
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