ODE and PDE based modeling of biological transportation networks

TL;DR

This paper analyzes the formation of biological transportation networks using both discrete ODE models and their continuum PDE limits, proving solution existence and demonstrating convergence through numerical simulations.

Contribution

It introduces a PDE model as a continuum limit of a discrete ODE network model and proves the global existence of solutions for this PDE.

Findings

Numerical simulations show convergence to steady states.

Steady states are non-unique and depend on initial data.

The PDE model accurately captures the behavior of the discrete network model.

Abstract

We study the global existence of solutions of a discrete (ODE based) model on a graph describing the formation of biological transportation networks, introduced by Hu and Cai. We propose an adaptation of this model so that a macroscopic (PDE based) system can be obtained as its formal continuum limit. We prove the global existence of weak solutions of the macroscopic PDE model. Finally, we present results of numerical simulations of the discrete model, illustrating the convergence to steady states, their non-uniqueness as well as their dependence on initial data and model parameters.