First order endotactic reaction networks

TL;DR

This paper characterizes first order endotactic reaction networks, linking their graph properties to stability and spectral features, and extends some stability results to higher order nonlinear systems without Lyapunov functions.

Contribution

It provides a detailed characterization of first order endotactic reaction graphs and establishes stability properties for associated mass-action systems, including a spectral analysis and stability extension.

Findings

First order endotactic reaction graphs have specific spectral properties.

Every first order endotactic mass-action system has a positive, globally stable equilibrium.

Stability results are extended to higher order nonlinear systems without Lyapunov functions.

Abstract

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 as 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 endotactic reaction graphs of higher orders and studying global stability of mass-action systems in general.