Communications in Mathematical Sciences

Volume 20 (2022)

Number 4

Tensor PDE model of biological network formation

Pages: 1173 – 1191

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n4.a10

Authors

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)

Giulia Pilli (Faculty of Mathematics, University of Vienna, Austria)

Abstract

We study an elliptic-parabolic system of partial differential equations describing formation of biological network structures. The model takes into consideration the evolution of the permeability tensor under the influence of a diffusion term, representing randomness in the material structure, a decay term describing metabolic cost and a pressure force. A Darcy’s law type equation describes the pressure field. In the spatially two-dimensional setting, we present a constructive, formal derivation of the PDE system from the discrete network formation model in the refinement limit of a sequence of unstructured triangulations. Moreover, we show that the PDE system is a formal $L^2$-gradient flow of an energy functional with biological interpretation, and study its convexity properties. For the case when the energy functional is convex, we construct unique global weak solutions of the PDE system. Finally, we construct steady state solutions in one- and multi-dimensional settings and discuss their stability properties.

Keywords

biological transportation networks, tensor PDE model, continuum limit, gradient flow

2010 Mathematics Subject Classification

35D30, 35G61, 35K57, 92C42

Received 3 December 2020

Received revised 13 October 2021

Accepted 4 November 2021

Published 11 April 2022