Communications in Mathematical Sciences
Volume 17 (2019)
Dedicated to the memory of Professor David Shen Ou Cai
ODE- and PDE-based modeling of biological transportation networks
Pages: 1235 – 1256
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.
weak solutions, energy dissipation, continuum limit, pattern formation, numerical modeling
2010 Mathematics Subject Classification
35B32, 35B36, 35K55, 35Q92, 70F10, 92C42
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