Notes on a PDE System for Biological Network Formation

TL;DR

This paper advances understanding of a PDE model for biological network formation by providing new analytical solutions, exploring solution behavior, and conducting extensive numerical simulations across different parameters and dimensions.

Contribution

It extends existence results for solutions, analyzes finite-time breakdown, constructs stationary solutions, and validates findings through numerical simulations in multiple dimensions.

Findings

Existence of weak and mild solutions for all relevant relaxation exponents.

Finite time extinction or breakdown of solutions in 1D for certain exponents.

Numerical evidence suggests analytical results may extend to higher dimensions.

Abstract

We present new analytical and numerical results for the elliptic-parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transport networks. The model describes the pressure field using a Darcy's type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. The analytical part extends the results of Haskovec, Markowich and Perthame regarding the existence of weak and mild solutions to the whole range of meaningful relaxation exponents. Moreover, we prove finite time extinction or break-down of solutions in the spatially onedimensional setting for certain ranges of the relaxation exponent. We also construct stationary solutions for the case of vanishing diffusion and critical value of the relaxation exponent, using a variational formulation and a penalty method. The analytical part is complemented by extensive numerical simulations. We propose a discretization based on mixed finite elements and study the qualitative properties of network structures for various parameters values. Furthermore, we indicate numerically that some analytical results proved for the spatially one-dimensional setting are likely to be valid also in several space dimensions.