Communications in Mathematical Sciences

Volume 17 (2019)

Number 5

Dedicated to the memory of Professor David Shen Ou Cai

A mesoscopic model of biological transportation networks

Pages: 1213 – 1234

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n5.a3

Authors

Martin Burger (Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg, Erlangen, Germany)

Jan Haskovec (Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia)

Peter Markowich (Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia; and Faculty of Mathematics, University of Vienna, Austria)

Helene Ranetbauer (Faculty of Mathematics, University of Vienna, Austria)

Abstract

We introduce a mesoscopic model for natural network formation processes, acting as a bridge between the discrete and continuous network approach proposed in [D. Hu and D. Cai, Phys. Rev. Lett., 111(13):138701, 2013]. The models are based on a common approach where the dynamics of the conductance network is subject to pressure force effects. We first study topological properties of the discrete model and we prove that if the metabolic energy consumption term is concave with respect to the conductivities, the optimal network structure is a tree (i.e., no loops are present). We then analyze various aspects of the mesoscopic modeling approach, in particular its relation to the discrete model and its stationary solutions, including discrete network solutions. Moreover, we present an alternative formulation of the mesoscopic model that avoids the explicit presence of the pressure in the energy functional.

Keywords

network formation, mesoscopic model, measure valued solutions, stationary solutions, optimal transport structure

2010 Mathematics Subject Classification

35B36, 35K55, 49J20, 92C42

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H.R. acknowledges support by the Austrian Science Fund (FWF) project F 65. M.B. acknowledges support by ERC via Grant EU FP 7 - ERC Consolidator Grant 615216 LifeInverse. M.B. would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Variational Methods for Imaging and Vision, where work on this paper was undertaken, supported by EPSRC grant no EP/K032208/1 and the Simons foundation.

Received 31 May 2018

Accepted 4 May 2019

Published 6 December 2019