Global stability of first order endotactic reaction systems

Abstract

Reaction networks are a general framework widely used in modelling diverse phenomena in different science disciplines. The dynamical process of a reaction network endowed with mass-action kinetics is a mass-action system. In this paper we study dynamics of first order mass-action systems. We prove that every first order endotactic mass-action system 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. In particular, we prove that global attractivity conjecture holds for every linear complex balanced mass-action system. In this way, we exclude the possibility of first order endotactic mass-action systems to admit multistationarity or multistability. The result indicates that the importance of binding molecules in reactants is crucial for (endotactic) reaction networks to have complicated dynamics like limit cycles. The proof relies on the fact that $\mathcal{A}$-endotacticity of first order reaction networks implies endotacticity for a finite set $\mathcal{A}$, which is also proved in this paper. Out of independent interest, we provide a sufficient condition for endotacticity of reaction networks which are not necessarily of first order.