  The model by Hu and Cai [Phys. Rev. Lett., Vol. 111(13) (2013)1 ] describes
the self-organization of vascular networks for transport of fluids from source
to sinks. Diameters, and thereby conductances, of vessel segments evolve so as
to minimize a cost functional E. The cost is the trade-off between the power
required for pumping the fluid and the energy consumption for vessel
maintenance. The model has been used to show emergence of cyclic structures in
the presence of locally fluctuating demand, i.e. non-constant net flow at sink
nodes. Under rapid and sufficiently large fluctuations, the dynamics exhibits
bistability of tree-like and cyclic network structures. We compare these
solutions in terms of the cost functional E. Close to the saddle-node
bifurcation giving rise to the cyclic solutions, we find a parameter regime
where the tree-like solution rather than the cyclic solution is cost-optimal.
Further increase of fluctuation amplitude then leads to an additional
transition at which the cyclic solution becomes optimal. The findings hold both
in a small system of one source and two sinks and in an empirical vascular
network with hundreds of sinks. In the small system, we further analyze the
case of slower fluctuations, i.e., on the same time scale as network
adaptation. We find that the noisy dynamics settles around the cyclic
structures even when these structures are not cost-optimal.
