  The dynamics of flow within a material transport network is dependent upon
the dynamics of its power source. Responding to a change of these dynamics is
critical for the fitness of living flow networks, e.g. the animal vasculature,
which are subject to frequent and sudden shifts when the pump (the heart)
transitions between different steady states. The combination of flow
resistance, fluid inertia, and elasticity of the vessel walls causes the flow
and pressure of the fluid throughout the network to respond to these
transitions and adapt to the new power source operating profiles over a nonzero
time scale. We find that this response time can exist in one of two possible
regimes; one dominated by the decay rate of travelling wavefronts and
independent of system size, and one dominated by the diffusive nature of the
fluid mechanical energy over large length scales. These regimes are shown to
exist for both single vessels and hierarchically structured networks with
systems smaller than a critical size in the former and larger systems in the
later. Applying biologically relevant parameters to the model suggests that
animal vascular networks may have evolved to occupy a state within the minimal
response time regime but close to this critical system size.
