Tensor PDE model of biological network formation

TL;DR

This paper develops a PDE model for biological network formation, deriving it from discrete models, analyzing its mathematical properties, and constructing solutions and steady states with biological relevance.

Contribution

It provides a formal derivation of a PDE system from discrete network models and analyzes its gradient flow structure, convexity, and solution properties.

Findings

The PDE system is a formal $L^2$-gradient flow of an energy functional.

Convexity of the energy functional leads to unique global weak solutions.

Steady state solutions are constructed and their stability discussed.

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.