  Transport networks are crucial to the functioning of natural systems and
technological infrastructures. For flow networks in many scenarios, such as
rivers or blood vessels, acyclic networks (i.e., trees) are optimal structures
when assuming time-independent in- and outflow. Dropping this assumption,
fluctuations of net flow at source and/or sink nodes may render the pure tree
solutions unstable even under a simple local adaptation rule for conductances.
Here, we consider tree-like networks under the influence of spatially
heterogeneous distribution of fluctuations, where the root of the tree is
supplied by a constant source and the leaves at the bottom are equipped with
sinks with fluctuating loads. We find that the network divides into two regions
characterized by tree-like motifs and stable cycles. The cycles emerge through
transcritical bifurcations at a critical amplitude of fluctuation. For a simple
network structure, depending on parameters defining the local adaptation,
cycles first appear close to the leaves (or root) and then appear closer
towards the root (or the leaves). The interaction between topology and dynamics
gives rise to complex feedback mechanisms with many open questions in the
theory of network dynamics. A general understanding of the dynamics in adaptive
transport networks is essential in the study of mammalian vasculature, and
adaptive transport networks may find technological applications in
self-organizing piping systems.
