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Researcher: Jan Haskovec, Lisa Maria Kreusser and Peter A. Markowich 

Network formation and transportation networks are fundamental processes in living systems. A new dynamic modeling approach on a graph has recently been introduced by Hu and Cai to describe the formation of biological transport networks. We study the existence of solutions to this model and propose an adaptation so that a macroscopic system can be obtained as its formal continuum limit. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. We also show the global existence of weak solutions of the macroscopic gradient flow. Results of numerical simulations of the discrete gradient flow illustrate the convergence to steady states, their non-uniqueness as well as their dependence on initial data and model parameters. Based on this model we propose an adapted model in the cellular context for leaf venation, investigate the model analytically and show numerically that it can produce branching vein patterns. 

Related Publications 

Rigorous Continuum Limit for the Discrete Network Formation Problem
J Haskovec, LM Kreusser, P Markowich
ODE and PDE based modeling of biological transportation networks
J Haskovec, LM Kreusser, P Markowich
Auxin transport model for leaf venation
J Haskovec, H Jönsson, LM Kreusser, P Markowich