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Non-Local Cell Adhesion Models

Symmetries and Bifurcations in 1-D

  • Book
  • © 2021

Overview

Authors:
  1. Andreas Buttenschön

    1. Department of Mathematics, University of British Columbia, Vancouver, Canada

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    You can also search for this author in PubMed   Google Scholar

  2. Thomas Hillen

    1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada

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  • Presents the first ever application of abstract bifurcation theory to a non-local problem
  • Includes leading research on pattern formation of non-local models
  • Describes in detail the development of basic properties of nonlocal adhesion models
  • Defines biological non-local boundary conditions

Part of the book series: CMS/CAIMS Books in Mathematics (CMS/CAIMS BM, volume 1)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-viii
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  2. Introduction

    1. Front Matter

      Pages 1-1
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    2. Introduction

      • Andreas Buttenschön, Thomas Hillen
      Pages 3-12

    3. Preliminaries

      • Andreas Buttenschön, Thomas Hillen
      Pages 13-29

  3. The Periodic Problem

    1. Front Matter

      Pages 31-31
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    2. Basic Properties

      • Andreas Buttenschön, Thomas Hillen
      Pages 33-53

    3. Local Bifurcation

      • Andreas Buttenschön, Thomas Hillen
      Pages 55-68

    4. Global Bifurcation

      • Andreas Buttenschön, Thomas Hillen
      Pages 69-106

  4. Non-local Equations with Boundary Conditions

    1. Front Matter

      Pages 107-107
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    2. No-Flux Boundary Conditions for Non-local Operators

      • Andreas Buttenschön, Thomas Hillen
      Pages 109-133

    3. Discussion and Future Directions

      • Andreas Buttenschön, Thomas Hillen
      Pages 135-138

  5. Back Matter

    Pages 139-152
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Keywords

About this book

This monograph considers the mathematical modeling of cellular adhesion, a key interaction force in cell biology. While deeply grounded in the biological application of cell adhesion and tissue formation, this monograph focuses on the mathematical analysis of non-local adhesion models. The novel aspect is the non-local term (an integral operator), which accounts for forces generated by long ranged cell interactions. The analysis of non-local models has started only recently, and it has become a vibrant area of applied mathematics. This monograph contributes a systematic analysis of steady states and their bifurcation structure, combining global bifurcation results pioneered by Rabinowitz, equivariant bifurcation theory, and the symmetries of the non-local term. These methods allow readers to analyze and understand cell adhesion on a deep level.

Reviews

“Modelers who wish to use similar approaches in their modeling will find this a good source of base information, as well as a valuable guide for initiating similar analyses for their own models. Analysts wishing to expand our understanding … will find this book a fine building block. It could also prove a useful resource for graduate students looking for potential projects … . this monograph is an admirable attempt … and hopefully will inspire significant further study.” (Kevin Painter, SIAM Review, Vol. 64 (1), March, 2022)



“The detailed analysis, as presented here, shows a stimulating interaction between model symmetries, mathematical analysis, and biological reality, which probably are inspired the authors and hopefully the readers of this book too.” (Andrey Zahariev, zbMATH 1473.92001, 2021)

Authors and Affiliations

  • Department of Mathematics, University of British Columbia, Vancouver, Canada

    Andreas Buttenschön

  • Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada

    Thomas Hillen

Bibliographic Information

  • Book Title: Non-Local Cell Adhesion Models

  • Book Subtitle: Symmetries and Bifurcations in 1-D

  • Authors: Andreas Buttenschön, Thomas Hillen

  • Series Title: CMS/CAIMS Books in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-67111-2

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

  • Hardcover ISBN: 978-3-030-67110-5Published: 10 June 2021

  • Softcover ISBN: 978-3-030-67113-6Published: 11 June 2022

  • eBook ISBN: 978-3-030-67111-2Published: 09 June 2021

  • Series ISSN: 2730-650X

  • Series E-ISSN: 2730-6518

  • Edition Number: 1

  • Number of Pages: VIII, 152

  • Number of Illustrations: 20 b/w illustrations, 15 illustrations in colour

  • Topics: Mathematical and Computational Biology, Mathematical Modeling and Industrial Mathematics

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