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Cross-Diffusion in Reaction-Diffusion Models: Analysis, Numerics, and Applications

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Progress in Industrial Mathematics at ECMI 2016 (ECMI 2016)

Part of the book series: Mathematics in Industry ((TECMI,volume 26))

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Abstract

Cross-diffusion terms are nowadays widely used in reaction-diffusion equations encountered in models from mathematical biology and in various engineering applications. In this contribution we review the basic model equations of such systems, give an overview of their mathematical analysis, with an emphasis on pattern formation and positivity preservation, and finally we present numerical simulations that highlight special features of reaction-cross-diffusion models.

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Acknowledgements

AM is a Royal Society Wolfson Research Merit Award Holder generously supported by the Wolfson Foundation. All the authors (AM, RB, AG) thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme Coupling Geometric PDEs with Physics for Cell Morphology, Motility and Pattern Formation; EPSRC EP/K032208/1.

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Authors and Affiliations

  1. School of Mathematical and Physical Sciences, Department of Mathematics, University of Sussex, Pevensey 3 5c15, Brighton, BN1 9QH, UK

    Anotida Madzvamuse

  2. Polytechnic Institute of Setubal, Barreiro School of Technology, Rua Américo da Silva Marinho-Lavradio, 2839-001, Barreiro, Portugal

    Raquel Barreira

  3. TU Darmstadt, Fachbereich Mathematik, AG Numerik und Wissenschaftliches Rechnen, Dolivostr. 15, D-64293, Darmstadt, Germany

    Alf Gerisch

Authors
  1. Anotida Madzvamuse
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  2. Raquel Barreira
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  3. Alf Gerisch
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Corresponding author

Correspondence to Anotida Madzvamuse .

Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Peregrina Quintela

  2. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Patricia Barral

  3. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Dolores Gómez

  4. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Francisco J. Pena

  5. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Jerónimo Rodríguez

  6. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Pilar Salgado

  7. Department of Applied Mathematics, University of Santiago de Compostela, Lugo, Spain, Spain

    Miguel E. Vázquez-Méndez

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Madzvamuse, A., Barreira, R., Gerisch, A. (2017). Cross-Diffusion in Reaction-Diffusion Models: Analysis, Numerics, and Applications. In: Quintela, P., et al. Progress in Industrial Mathematics at ECMI 2016. ECMI 2016. Mathematics in Industry(), vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-63082-3_61

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