  Many biological, geophysical and technological systems involve the transport
of resource over a network. In this paper we present an algorithm for
calculating the exact concentration of resource at any point in space or time,
given that the resource in the network is lost or delivered out of the network
at a given rate, while being subject to advection and diffusion. We consider
the implications of advection, diffusion and delivery for simple models of
glucose delivery through a vascular network, and conclude that in certain
circumstances, increasing the volume of blood and the number of glucose
transporters can actually decrease the total rate of glucose delivery. We also
consider the case of empirically determined fungal networks, and analyze the
distribution of resource that emerges as such networks grow over time. Fungal
growth involves the expansion of fluid filled vessels, which necessarily
involves the movement of fluid. In three empirically determined fungal networks
we found that the minimum currents consistent with the observed growth would
effectively transport resource throughout the network over the time-scale of
growth. This suggests that in foraging fungi, the active transport mechanisms
observed in the growing tips may not be required for long range transport.
