Stability Analysis of Biochemical Reaction Networks Linearly Conjugated to complex balanced Systems with Time Delays Added

TL;DR

This paper extends stability analysis of biochemical reaction networks linearly conjugated to complex balanced systems to include systems with time delays, using Lyapunov functionals and invariant set analysis.

Contribution

It introduces a novel framework for stability analysis of delayed linearly conjugated biochemical networks, incorporating Lyapunov functionals and invariant set decomposition.

Findings

Developed Lyapunov functionals for delayed systems

Identified invariant sets for trajectory analysis

Established local asymptotic stability conditions

Abstract

Linear conjugacy offers a new perspective to broaden the scope of stable biochemical reaction networks to the systems linearly conjugated to the well-established complex balanced mass action systems ($\ell$cCBMASs). This paper addresses the challenge posed by time delay, which can disrupt the linear conjugacy relationship and complicate stability analysis for delayed versions of $\ell$cCBMASs (D$\ell$cCBMAS). Firstly, we develop Lyapunov functionals tailored to some D$\ell$cCBMASs by using the persisted parameter relationships under time delays. Subsequently, we redivide the phase space as several invariant sets of trajectories and further investigate the existence and uniqueness of equilibriums in each newly defined invariant set. This enables us to determine the local asymptotic stability of some D$\ell$cCBMASs within an updated framework. Furthermore, illustrative examples are provided to demonstrate the practical implications of our approach.