  The paper investigates the throughput behavior of single-commodity dynamical
flow networks governed by monotone distributed routing policies. The networks
are modeled as systems of ODEs based on mass conversation laws on directed
graphs with limited flow capacities on the links and constant external inflows
at certain origin nodes. Under monotonicity assumptions on the routing
policies, it is proven that a globally asymptotically stable equilibrium exists
so that the network achieves maximal throughput, provided that no cut capacity
constraint is violated by the external inflows. On the contrary, should such a
constraint be violated, the network overload behavior is characterized. In
particular, it is established that there exists a cut with respect to which the
flow densities on every link grow linearly over time (resp. reach their
respective limits simultaneously) in the case where the buffer capacities are
infinite (resp. finite). The results employ an $l_1$-contraction principle for
monotone dynamical systems.
