We demonstrate the validity of Murray's law, which represents a scaling relation for branch conductivities in a transportation network, for discrete and continuum models of biological networks. We first consider discrete networks with general metabolic coefficient and multiple branching nodes and derive a generalization of the classical 3/4-law. Next we prove an analogue of the discrete Murray's law for the continuum system obtained in the continuum limit of the discrete model on a rectangular mesh. Finally, we consider a continuum model derived from phenomenological considerations and show the validity of the Murray's law for its linearly stable steady states.
|Number of pages
|Mathematical Models and Methods in Applied Sciences
|Published - Sep 9 2019
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: Giulia Pilli acknowledges support from the Austrian Science Fund (FWF) through the grants F 65 and W 1245.