Semistability of complex balanced kinetic systems with arbitrary time delays

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.