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2017 | OriginalPaper | Chapter

Continuum Modeling of Biological Network Formation

Authors : Giacomo Albi, Martin Burger, Jan Haskovec, Peter Markowich, Matthias Schlottbom

Published in: Active Particles, Volume 1

Publisher: Springer International Publishing

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Abstract

We present an overview of recent analytical and numerical results for the elliptic–parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transportation networks. The model describes the pressure field using a Darcy type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. We first introduce micro- and mesoscopic models and show how they are connected to the macroscopic PDE system. Then, we provide an overview of analytical results for the PDE model, focusing mainly on the existence of weak and mild solutions and analysis of the steady states. The analytical part is complemented by extensive numerical simulations. We propose a discretization based on finite elements and study the qualitative properties of network structures for various parameter values.

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Metadata
Title
Continuum Modeling of Biological Network Formation
Authors
Giacomo Albi
Martin Burger
Jan Haskovec
Peter Markowich
Matthias Schlottbom
Copyright Year
2017
Book
Active Particles, Volume 1

Print ISBN: 978-3-319-49994-9

Electronic ISBN: 978-3-319-49996-3

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-49996-3_1

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