Single-target networks

Reaction networks can be regarded as finite oriented graphs embedded in Euclidean space. Single-target networks are reaction networks with an arbitrarily set of source vertices, but only one sink vertex. We completely characterize the dynamics of all mass-action systems generated by single-target networks, as follows: either (i) the system is globally stable for all choice of rate constants (in fact, is dynamically equivalent to a detailed-balanced system with a single linkage class), or (ii) the system 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, we show that 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.