Revisiting Persistence of Chemical Reaction Networks through Lyapunov Function Partial Differential Equations

In this paper, we propose a novel technique, referred to as the Lyapunov Function PDEs technique, to diagnose persistence of chemical reaction networks with mass-action kinetics. The technique allows that every network is attached to a Lyapuonv function PDE and a boundary condition whose solutions are expected to be Lyapunov functions for the network. By means of solution of the PDEs, either in the forms of itself or its time derivative, some checkable criteria are proposed for persistence of network systems. These criteria show high validity in proving that neither non-semilocking boundary points nor semilocking boundary non-equilibrium points (with additional conditions included) in mass-action systems are $\omega$-$limit$ points. Further, we prove that a class of networks, called $\mathcal{W}_{I}$-endotactic networks that are regardless of values of the system parameters, a set larger than endotactic networks but smaller than $w_{I}$-endotactic networks, also give rise to persistent systems if the networks are $1$-dimensional. Although part of our results are covered by existing ones, part of them are still new. The proposed Lyapunov Function PDEs technique allows us to check persistence of mass action systems in an alternative corner, and is exhibiting large potential.