Lyapunov Function PDEs to the Stability of Some Complex Balancing Derivative and Compound Networks

This article contributes to extending the validity of Lyapunov function partial differential equations (PDEs) whose solution is conjectured to be able to behave as a Lyapunov function in stability analysis to more mass-action chemical reaction networks. First, we have proved that the Lyapunov function PDEs method is valid in capturing the asymptotic stability of the networks compounded of a complex balanced network and any species-dependent two-species autocatalytic network if some moderate conditions are included. Then, by defining a new class of networks, called complex balanced produced networks, we also show the asymptotic stability of this class of networks, and also to their compound with any species-independent one-dimensional network and with any species-dependent two-species autocatalytic network under some conditions by using the same method. A notable point is that these classes of networks are nonweakly reversible, of any dimension, and of any deficiency. Finally, we apply our results to some practical biochemical reaction networks, including birth-death processes, motifs related networks, etc., to illustrate validity.