# Semistability of complex balanced kinetic systems with arbitrary time delays

## TL;DR

This paper introduces a class of complex balanced kinetic systems with arbitrary time delays, proving their semistability and local asymptotic stability using Lyapunov-Krasovskii functionals.

## Contribution

It extends the theory of complex balanced kinetic systems to include arbitrary time delays and establishes their semistability and stability properties.

## Key findings

- Proves uniqueness of equilibrium solutions within delayed stoichiometric classes.
- Establishes semistability of equilibria using Lyapunov-Krasovskii functionals.
- Shows local asymptotic stability of positive equilibria.

## Abstract

In this letter we introduce a class of delayed kinetic systems derived from mass action type reaction network models. We define the time delayed positive stoichiometric compatibility classes and the notion of complex balanced time delayed kinetic systems. We prove the uniqueness of equilibrium solutions within the time delayed positive stoichiometric compatibility classes for such models. In our main result we prove the semistability of the equilibrium solutions for complex balanced systems with arbitrary time delays using an appropriate Lyapunov-Krasovskii functional and LaSalle's invariance principle. As a consequence, we obtain that every positive complex balanced equilibrium solution is locally asymptotically stable relative to its positive stoichiometric compatibility class.