Modelling and stability analysis of complex balanced kinetic systems with distributed time delays

Abstract

Kinetic systems are positive polynomial systems that can be equipped with discrete and/or distributed time delays for describing process models often appearing in complex reaction kinetic systems. Mathematically these models are in the form of delay-differential equations with discrete and/or distributed time delays.The origin of the delays may be an inherent phenomena or some approximation of the biochemical system model. It is shown in the paper that even the simplest transport mechanism, the spatially distributed convection combined with chemical reaction is resulted in a delayed chemical reaction network model, where the parameters of the transport are analytically related to the parameters of the induced delay – discrete for the plug flow and distributed for the laminar flow – in the model.We proved that – similarly to the case of kinetic systems without delay – every positive complex balanced equilibrium of a time delayed kinetic system with arbitrary nondecreasing cumulative delay distribution function is locally asymptotically stable relative to its positive stoichiometric compatibility class.The results and notations were illustrated on a simple example using simulation investigations.