[Even] arbitrarily large time delays cannot impair the stability of complex balanced reaction networks

Presently, the class of kinetic systems has proven to be a useful representation of nonnegative system models not only in biochemistry, but also in other areas like population or disease dynamics, process systems, and even transportation networks. Naturally, kinetic systems are usually equipped with a network structure called the reaction graph, which is the abstraction of a set of chemical reactions, where the chemical complexes and reactions can be represented by vertices and directed edges, respectively. Generally, the complex balance condition requires that the signed sum of incoming and outgoing reaction rates at equilibrium to be zero for each complex in a chemical reaction network. Unfortunately, complex balance becomes a parameter dependent property when the deficiency of the network is higher than zero. Regardless, time-delays are often present in natural and technological processes, and the detailed mathematical treatment of such delays is sometimes necessary to model and understand important observed dynamical phenomena.

Recently, a team of researchers including Prof. Gábor Szederkényi, Prof. Katalin M. Hangos, Dr. György Lipták and Prof. Mihály Pituk from the Process Control Research Group, Systems and Control Laboratory, Institute for Computer Science and Control (MTA SZTAKI), from Pázmány Péter Catholic University, and from the University of Pannonia, Hungary proposed a study whose main objective was to incorporate the complex balance condition for kinetic systems with delayed reactions. Additionally, they purposed to investigate the stability properties of such systems using logarithmic Lyapunov–Krasovskii functionals and LaSalle’s invariance principle. Their work is currently published in the research journal, Systems & Control Letters.

The research method employed by the authors commenced with the introduction of a class of delayed kinetic systems derived from mass action type reaction network models. Next, the researchers defined the time delayed positive stoichiometric compatibility classes and the notion of complex balanced time delayed kinetic systems. Eventually, they proceeded to prove the uniqueness of equilibrium solutions within the time delayed positive stoichiometric compatibility classes for such models.

The authors observed that the equilibrium solutions of complex balanced kinetic systems could be directly obtained from the equilibria of the corresponding non-delayed kinetic system. Moreover, they noted that contrary to the classical mass action case, the classes were no longer linear manifolds in the state space. Furthermore, by introducing a logarithmic Lyapunov–Krasovskii functional and using LaSalle’s invariance principle, the semistability of equilibrium solutions in complex balanced systems with arbitrary time delays was also proven.

Professor Gábor Szederkényi and colleagues study presented the introduction of a class of delayed kinetic systems where different constant time-delays can be assigned to the individual reactions of the network. The authors of this work checked the complex balance property of a delayed network in an analogous way as in the non-delayed model. They also went ahead to 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. Altogether, it was established that every positive complex balanced equilibrium solution is locally asymptotically stable relative to its positive stoichiometric compatibility class.