Communications in Mathematical Sciences
Volume 20 (2022)
Tensor PDE model of biological network formation
Pages: 1173 – 1191
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.
biological transportation networks, tensor PDE model, continuum limit, gradient flow
2010 Mathematics Subject Classification
35D30, 35G61, 35K57, 92C42
G. P. acknowledges support from the Austrian Science Fund (FWF) through the grants F65 and W1245.
Received 3 December 2020
Received revised 13 October 2021
Accepted 4 November 2021
Published 11 April 2022