A mesoscopic model of biological transportation networks

TL;DR

This paper introduces a mesoscopic model for biological network formation that bridges discrete and continuous approaches, analyzing its properties, optimal structures, and stationary solutions.

Contribution

It presents a novel mesoscopic modeling framework linking discrete and continuous network models, with analysis of optimal structures and alternative formulations.

Findings

Optimal network structures are trees when metabolic energy is concave.

The model's relation to discrete networks is analyzed.

Stationary solutions and alternative formulations are provided.

Abstract

We introduce a mesoscopic model for natural network formation processes, acting as a bridge between the discrete and continuous network approach proposed by Hu and Cai. 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.