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A Mixed-Culture Biofilm Model with Cross-Diffusion

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Abstract

We propose a deterministic continuum model for mixed-culture biofilms. A crucial aspect is that movement of one species is affected by the presence of the other. This leads to a degenerate cross-diffusion system that generalizes an earlier single-species biofilm model. Two derivations of this new model are given. One, like cellular automata biofilm models, starts from a discrete in space lattice differential equation where the spatial interaction is described by microscopic rules. The other one starts from the same continuous mass balances that are the basis of other deterministic biofilm models, but it gives up a simplifying assumption of these models that has recently been criticized as being too restrictive in terms of ecological structure. We show that both model derivations lead to the same PDE model, if corresponding closure assumptions are introduced. To investigate the role of cross-diffusion, we conduct numerical simulations of three biofilm systems: competition, allelopathy and a mixed system formed by an aerobic and an anaerobic species. In all cases, we find that accounting for cross-diffusion affects local distribution of biomass, but it does not affect overall lumped quantities such as the total amount of biomass in the system.

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Acknowledgments

This study was support by the Natural Science and Engineering Research Council of Canada (NSERC) under a Discovery Grant awarded to HJE and by the Canada Research Chairs Program. The computing equipment used in this study was a SGI Altix 450 funded by the Canada Foundation for Innovation through a New Leaders Opportunity Grant and operated by SHARCNET, an SGI Altix UV and custom-built Intel Xeon workstation funded by NSERC through a Research Tool and Infrastructure Grant and operated by the Department of Mathematics and Statistics. The authors thank Kaizaad Bilimorya (SHARCNET) and Larry Banks (Dept. Math & Stats) for the technical support. The authors thank Greg Kepka (Lakehead University) for providing the CLSM images.

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  1. Department Mathematics and Statistics, University of Guelph, Guelph, ON, N1G 2W1, Canada

    Kazi A. Rahman, Rangarajan Sudarsan & Hermann J. Eberl

  2. Biophysics Interdepartmental Program, University of Guelph, Guelph, ON, N1G 2W1, Canada

    Rangarajan Sudarsan & Hermann J. Eberl

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  1. Kazi A. Rahman
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Correspondence to Hermann J. Eberl.

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Rahman, K.A., Sudarsan, R. & Eberl, H.J. A Mixed-Culture Biofilm Model with Cross-Diffusion. Bull Math Biol 77, 2086–2124 (2015). https://doi.org/10.1007/s11538-015-0117-1

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