Communications in Mathematical Sciences
Volume 17 (2019)
Dedicated to the memory of Professor David Shen Ou Cai
A mesoscopic model of biological transportation networks
Pages: 1213 – 1234
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.
network formation, mesoscopic model, measure valued solutions, stationary solutions, optimal transport structure
2010 Mathematics Subject Classification
35B36, 35K55, 49J20, 92C42
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