Communications in Mathematical Sciences

Volume 17 (2019)

Number 5

Dedicated to the memory of Professor David Shen Ou Cai

ODE- and PDE-based modeling of biological transportation networks

Pages: 1235 – 1256

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n5.a4

Authors

Jan Haskovec (Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technology, Saudi Arabia)

Lisa Maria Kreusser (Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, United Kingdom)

Peter Markowich (Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia)

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.

Keywords

weak solutions, energy dissipation, continuum limit, pattern formation, numerical modeling

2010 Mathematics Subject Classification

35B32, 35B36, 35K55, 35Q92, 70F10, 92C42

The full text of this article is unavailable through your IP address: 18.206.238.77

LMK was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 and the German National Academic Foundation (Studienstiftung des Deutschen Volkes).

Received 22 May 2018

Accepted 4 May 2019

Published 6 December 2019