Lyapunov Function PDEs Method to the Stability of Some Chemical Reaction Networks

TL;DR

This paper extends Lyapunov function PDEs to analyze the stability of complex-balanced produced chemical reaction networks, including non-weakly reversible and high-dimensional cases, with applications to biochemical networks.

Contribution

It introduces a new class of networks called complex-balanced produced networks and proves Lyapunov PDEs' validity for their stability analysis, even when combined with certain other networks.

Findings

Lyapunov PDEs effectively analyze stability of complex-balanced produced networks.

The method applies to non-weakly reversible, high-dimensional, and high-deficiency networks.

Practical biochemical networks confirm the theoretical results.

Abstract

This paper contributes to extending the validity of Lyapunov function PDEs (in-vented by Fang and Gao in [SIAM Journal on Applied Dynamical Systems, 18(2019), pp. 1163-1199]and whose solution is conjectured to be able to behave as a Lyapunov function) in stability analysisto more mass-action chemical reaction networks. By defining a new class of networks, called complexbalanced produced networks, we have proved that the Lyapunov function PDEs method is validin capturing the asymptotic stability of this class of networks, and also to their compound withany 1-dimensional independent network according to species and with any two-species autocatalyticnon-independent network if some moderate conditions are included. A notable point is that thesethree classes of networks are non-weakly reversible, any dimensional and of any deficiency. We applyour results to some practical biochemical reaction networks including birth-death processes, motifsrelated networks etc., to illustrate validity.