Single-Target Networks

TL;DR

This paper characterizes the stability and steady-state behavior of all single-target networks under mass-action kinetics, revealing conditions for global stability and the absence of positive steady states.

Contribution

It provides a complete classification of the dynamics of single-target networks, linking global stability to geometric conditions of the network structure.

Findings

Global stability occurs if the target vertex is in the relative interior of the convex hull of source vertices.

Networks are either globally stable or have no positive steady states, with no intermediate behavior.

The stability condition is characterized by a geometric property of the network vertices.

Abstract

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.