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Travelling Waves in Hybrid Chemotaxis Models

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Abstract

Hybrid models of chemotaxis combine agent-based models of cells with partial differential equation models of extracellular chemical signals. In this paper, travelling wave properties of hybrid models of bacterial chemotaxis are investigated. Bacteria are modelled using an agent-based (individual-based) approach with internal dynamics describing signal transduction. In addition to the chemotactic behaviour of the bacteria, the individual-based model also includes cell proliferation and death. Cells consume the extracellular nutrient field (chemoattractant), which is modelled using a partial differential equation. Mesoscopic and macroscopic equations representing the behaviour of the hybrid model are derived and the existence of travelling wave solutions for these models is established. It is shown that cell proliferation is necessary for the existence of non-transient (stationary) travelling waves in hybrid models. Additionally, a numerical comparison between the wave speeds of the continuum models and the hybrid models shows good agreement in the case of weak chemotaxis and qualitative agreement for the strong chemotaxis case. In the case of slow cell adaptation, we detect oscillating behaviour of the wave, which cannot be explained by mean-field approximations.

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Acknowledgements

The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No. 239870. This publication was based on work supported in part by Award No KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). Radek Erban would also like to thank the Royal Society for a University Research Fellowship; Brasenose College, University of Oxford, for a Nicholas Kurti Junior Fellowship, and the Leverhulme Trust for a Philip Leverhulme Prize. This prize money was used to support research visits of Chuan Xue and Kevin Painter in Oxford. Kevin Painter acknowledges a Leverhulme Trust Research Fellowship award (RF-2011-045). Chuan Xue is supported by the National Science Foundation in the United States through grant DMS-1312966 and the Mathematical Biosciences Institute at the Ohio State University as a long-term visitor.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    Benjamin Franz & Radek Erban

  2. Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, OH, 43210, USA

    Chuan Xue

  3. Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Kevin J. Painter

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  1. Benjamin Franz
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  2. Chuan Xue
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  4. Radek Erban
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Correspondence to Radek Erban.

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Franz, B., Xue, C., Painter, K.J. et al. Travelling Waves in Hybrid Chemotaxis Models. Bull Math Biol 76, 377–400 (2014). https://doi.org/10.1007/s11538-013-9924-4

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