Murray’s law for discrete and continuum models of biological networks

Abstract

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.

Tensor PDE model of biological network formation

“…Detailed mathematical analysis of the system (1.9)-(1.10) with M (s) := s γ was carried out in the series of papers [1,2,8,9] and in [16,20,21], while its various other aspects were studied in [4,[10][11][12][13].…”

Emergence of biological transportation networks as a self-regulated process

“…where D = D(t, x) and t is the time-like variable induced by the gradient flow. A particular case of this type of process in the context of biological applications (e.g., leaf venation in plants) is the network formation problem introduced in [26] and further analyzed in the series of papers [1,2,11,20,21,22,31,39,40]. Here the quantity D = D(t, x) represent the tensor-valued local conductivity of the network, which is understood as a continuous porous medium.…”