ON THE EQUIVALENCE OF DELAYED ARGUMENTS AND TRANSFER EQUATIONS FOR MODELING DYNAMIC SYSTEMS

Development and improvement of mathematical methods used in modeling biological systems represents a topical issue of mathematical biology. In this paper, we considered a general form of a system of first-order delayed differential equations, traditionally used for describing the function of biological systems of different hierarchical levels. The main feature of this class of models is that some inherent processes (for example, elongation of DNA, RNA, and protein synthesis) are described in a subtle form and can be explicitly specified only through delayed arguments. In this paper, we propose an algorithm for rewriting systems with constant delayed arguments in an equivalent form that represents a system of partial differential equations with transfer equations. The algorithm is universal, since it does not impose any special conditions on the form of the right-hand parts of systems with delayed arguments. The proposed method is a multivariant algorithm. That is, based on one system of differential equations with delayed arguments, the algorithm allows writing out a number of special systems of partial differential equations, which are equivalent to the original system with delayed argument in the entire solution set. The results obtained indicate that delayed arguments and transfer equations are equivalent mathematical tools for describing all types of dynamic processes of energy and/or matter transfer in biological, chemical, and physical systems, indicating a deep-level similarity between properties of dynamic systems, regardless of their origin. At the same time, those processes that are subtle when retarded argument is used can be explicitly described in the form of transfer equations using systems of partial differential equations. This property is extremely important for the modeling of molecular genetic systems in which processes of DNA, RNA, and protein synthesis proceed at variable rates and need to be considered in certain problems, what can easily be done in models constructed using the mathematical tool of partial derivatives.

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