ODE-and PDE-based modeling of biological transportation networks

Jan Haskovec, Lisa Maria Kreusser, Peter Markowich

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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.

Original languageEnglish
Pages (from-to)1235-1256
Number of pages22
JournalCommunications in Mathematical Sciences
Issue number5
Publication statusPublished - 6 Dec 2019

Bibliographical note

Funding Information:
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).


  • Continuum limit
  • Energy dissipation
  • Numerical modeling
  • Pattern formation
  • Weak solutions

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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