  We study deterministic continuous-time lossy dynamical flow networks with
constant exogenous demands, fixed routing, and finite flow and buffer
capacities. In the considered model, when the total net flow in a cell
---consisting of the difference between the total flow directed towards it
minus the outflow from it--- exceeds a certain capacity constraint, then the
exceeding part of it leaks out of the system. The ensuing network flow dynamics
is a linear saturated system with compact state space that we analyze using
tools from monotone systems and contraction theory. Specifically, we prove that
there exists a set of equilibria that is globally asymptotically stable. Such
equilibrium set reduces to a single globally asymptotically stable equilibrium
for generic exogenous demand vectors. Moreover, we show that the critical
exogenous demand vectors giving rise to non-unique equilibria correspond to
phase transitions in the asymptotic behavior of the dynamical flow network.
