First-order endotactic reaction networks

Reaction networks are a general framework widely used in modeling diverse phenomena in different science disciplines. The dynamical process of a reaction network endowed with mass-action kinetics is a mass-action system which is an ODE defined by a directed graph, the so-called "reaction graph". Endotacticity is a graph property used to study persistence and permanence of mass-action systems. In this paper, we provide a detailed characterization of first-order endotactic reaction graphs. Besides, we provide a sufficient condition for endotacticity of reaction networks which are not necessarily of first-order. Such a characterization of a first-order endotactic reaction graph yields the spectral property of the adjacency matrix of the reaction graph. As a consequence, we prove that every first-order endotactic mass-action system as a linear ODE has a weakly reversible deficiency zero realization, and has a unique equilibrium which is exponentially globally asymptotically stable (and is positive) in each (positive) stoichiometric compatibility class. Using a stability result for asymptotically autonomous differential equations, examples are constructed to illustrate that the global stability results can be extended to mass-action systems of higher-order reaction networks modeled by nonlinear ODEs, which are not necessarily endotactic. Different from the classical approaches for proving global asymptotic stability, the proof does not rely on the construction of a Lyapunov function. This paper may serve as a starting point of characterizing higher-order endotactic reaction graphs and studying global stability of mass-action systems in general.