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# Single-target networks

• * Corresponding author: Jiaxin Jin

The authors are supported by NSF grant DMS-1816238

• 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.

Mathematics Subject Classification: Primary: 37N25, 92C42; Secondary: 80A30.

 Citation: • • Figure 1.  (a) A single-target network that is globally stable under mass-action kinetics. (b)–(c) Single-target networks with no positive steady states. (d) Not a single-target network

Figure 2.  Consider subnetworks of (a) under mass-action kinetics, whose associated dynamics is given by (9). If the coefficient of ${\boldsymbol{x}}^{ {\boldsymbol{y}}_1}$ in $\dot{x}$ is positive and the coefficient of ${\boldsymbol{x}}^{ {\boldsymbol{y}}_3}$ in $\dot{x}$ is negative, then the system (9) can be realized by a single-target network, determined by the sign of ${\boldsymbol{x}}^{ {\boldsymbol{y}}_2}$ in $\dot{x}$. If the net directions are as shown in (b), then (9) can be realized by the single-target network in (c). Similarly, if the net directions appear as in (d), then (9) can be realized by the network in (e)

Figure 3.  Reversible systems in (a) Example 3.12 and (b) Example 3.13 that are dynamically equivalent to detailed-balanced systems. Each undirected edge represents a pair of reversible edges

Figure 4.  Geometric argument for dynamically equivalence to single-target network in Example 3.12. Shown are the edges with $\mathrm{{X}}_1 + \mathrm{{X}}_2$ as their source. The centre $\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2} \right)^\top$ of the tetrahedron is marked in blue. With rate constants given in the example, the weighted sum of reaction vectors points from the source to the centre

Figure 5.  (a) The mass-action system with two target vertices from Example 4.2, which is dynamically equivalent to a complex-balanced system using a subnetwork of (b) if and only if $\frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25$

Figure 6.  The system in Figure 5(a) is dynamically equivalent to a complex-balanced system if and only if $\frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25$. The system is equivalent to (a) when $\kappa_2 \kappa_4 = 25 \kappa_1 \kappa_3$ and (b) when $25 \kappa_2 \kappa_4 = \kappa_1 \kappa_3$. For $\frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25$, the dynamically equivalent system is an appropriate convex combination of (a) and (b)

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