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The Diffusion Limit of Transport Equations in Biology

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Mathematical Models and Methods for Living Systems

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2167))

Abstract

Transport equations add a whole new level of modelling to our menu of mathematical models for spatial spread of populations. They are situated between individual based models, which act on the microscopic scale and reaction diffusion equations, which rank on the macro-scale. Transport equations are thus often associated with a meso scale. These equations use movement characteristics of individual particles (velocity, turning rate etc.), but they describe a population by a continuous density.In this chapter we introduce transport equations as a modelling tool for biological populations, where we outline the relations to biological measurements. The link to individual based random walk models and the relation to diffusion equations are discussed. In particular, the diffusion limit (or parabolic limit) forms the main part of this chapter. We present the detailed mathematical framework and we discuss isotropic versus non-isotropic diffusion. Throughout the manuscript we investigate a large variety of applications including bacterial movement, amoeboid movement, movement of myxobacteria, and pattern formation through chemotaxis, swarming or alignment. We hope to convince the reader that transport equations form a useful alternative to other models in certain situations. Their full strength arises in situations where directionality of movement plays an important role.

“A smart model is a good model.” -Tyra Banks

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Acknowledgements

We are grateful for CIME to support this interesting summer school and who invited us to contribute this book chapter. We also thank Dr. K. Painter for continued collaboration and his support of these lecture notes. The work of TH and AS is supported by NSERC grants.

K.P. Hadeler [22] has published a book chapter entitled Reaction Random Walk Systems where the results of our Sect. 2.2.6 are discussed in great detail. Hadeler’s book chapter arose as one of the CIME lecture notes from a workshop in 1997. We are honoured to be able to continue Hadeler’s work through these CIME notes on transport equations.

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Authors and Affiliations

  1. Centre for Mathematical Biology, University of Alberta, Edmonton, AB, T6G 2R3, Canada

    Thomas Hillen & Amanda Swan

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  1. Thomas Hillen
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Correspondence to Thomas Hillen .

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Editors and Affiliations

  1. Mathematical Sciences, Politecnico di Torino, Torino, Italy

    Luigi Preziosi

  2. School of Mathematics and Statistics, University of St. Andrews, St. Andrews, United Kingdom

    Mark Chaplain

  3. Department of Mathematics, Università di Trento, Povo, Italy

    Andrea Pugliese

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Hillen, T., Swan, A. (2016). The Diffusion Limit of Transport Equations in Biology. In: Preziosi, L., Chaplain, M., Pugliese, A. (eds) Mathematical Models and Methods for Living Systems. Lecture Notes in Mathematics(), vol 2167. Springer, Cham. https://doi.org/10.1007/978-3-319-42679-2_2

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