A mesoscopic model of biological transportation networks

Martin Burger, Jan Haskovec, Peter A. Markowich, Helene Ranetbauer

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


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.
Original languageEnglish (US)
Pages (from-to)1213-1234
Number of pages22
JournalCommunications in Mathematical Sciences
Issue number5
StatePublished - Dec 6 2019

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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.


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