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Travelling Waves in Hyperbolic Chemotaxis Equations

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Abstract

Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235–248, 1971). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically.

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Authors and Affiliations

  1. Mathematical Biosciences Institute, Ohio State University, Jennings Hall 3rd Floor, 1735 Neil Ave., Columbus, OH, 43210, USA

    Chuan Xue

  2. Department of Mathematics, Pohang University of Science and Technology, Pohang, 790-784, Republic of Korea

    Hyung Ju Hwang

  3. Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Kevin J. Painter

  4. Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK

    Radek Erban

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  1. Chuan Xue
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  2. Hyung Ju Hwang
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  3. Kevin J. Painter
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  4. Radek Erban
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Correspondence to Chuan Xue.

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Xue, C., Hwang, H.J., Painter, K.J. et al. Travelling Waves in Hyperbolic Chemotaxis Equations. Bull Math Biol 73, 1695–1733 (2011). https://doi.org/10.1007/s11538-010-9586-4

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  • DOI: https://doi.org/10.1007/s11538-010-9586-4

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