Some dynamical properties of delayed weakly reversible mass-action systems

TL;DR

This paper investigates the dynamical behavior of delayed complex balanced mass-action systems, establishing conditions for positive equilibria and long-term stability, with extensions to weakly reversible networks and practical examples.

Contribution

It introduces new methods to analyze the existence of positive equilibria and long-term dynamics in delayed complex balanced systems, including weakly reversible networks.

Findings

Existence of positive equilibrium in each stoichiometric class

Long-term stability characterized by Lyapunov-Krasovskii functional

Extension of results to weakly reversible networks with constant delays

Abstract

This paper focuses on the dynamical properties of delayed complex balanced systems. We first study the relationship between the stoichiometric compatibility classes of delayed and non-delayed systems. Using this relation we give another way to derive the existence of positive equilibrium in each stoichiometric compatibility class for delayed complex balanced systems. And if time delays are constant, the result can be generalized to weakly reversible networks. Also, by utilizing the Lyapunov-Krasovskii functional, we can obtain a long-time dynamical property about $\omega$-limit set of the complex balanced system with constant time delays. An example is also provided to support our results.