Stability Analysis for a Class of Non-Weakly Reversible Chemical Reaction Networks

Abstract

We consider ordinary differential equations (ODEs) that describe the time evolution of the concentrations of species in chemical reaction networks (CRNs). In order to analyze the convergence of solutions to the ODEs, the chemical reaction network theory has established an important theorem called Deficiency Zero Theorem (DZT). This theorem provides a sufficient condition for any solution to the ODEs to converge to an equilibrium point, based only on the graph structures of the CRNs and the algebraic properties of ODEs. In the present paper, we consider a class of non-weakly reversible chemical reaction networks, to which the DZT cannot be applied since one of the conditions, weak reversibility, is not satisfied. In order to make up for the failure of this important condition, by decomposing the network into weakly reversible sub-networks and applying the DZT to them, we show any solution to the ODEs for our class of networks with positive initial values converges to an equilibrium point on the boundary of the positive orthant.

Keywords

• chemical reaction networks
• weak reversibility
• ordinary differential equations
• stability
• equilibrium point