  We characterize the dynamics of all single-target networks under mass-action
kinetics: either the system is (i) globally stable for all choice of rate
constants (in fact, dynamically equivalent to a detailed-balanced system) or
(ii) has no positive steady states for any choice of rate constants and all
trajectories must converge to the boundary of the positive orthant or to
infinity. Moreover, global stability occurs if and only if the target vertex of
the network is in the relative interior of the convex hull of the source
vertices.
