Nonlinear Differential and System Dynamics of the Interacting Companies

Based on ecological classification of interaction of populations, the mathematical models of production of interdependent branches of production were studied and constructed. When we speak about of interdependent branches of production, we refer to the system industries in which a prerequisite for the existence of one industry is the existence of another industry. A classic example of interdependent branches of production is coal and steel industry. The mathematical models are a nonlinear autonomous dynamical system (system of first order autonomous nonlinear differential equations). The dynamic variables of the system are yield producing by interdependent branches of production. Parameters of the system are production growth rate, resource constraints and restrictions of consumption of products. Autonomous systems with various functions of resource constraints and restrictions of consumption of products were studied. The system, having the closest solutions by empirical values of production output, was chosen from this set of systems. Using the capabilities of the WolframAlpha software the values of the equilibrium amounts of products were determined. Parametric, phase portraits as well as integral trajectories of the systems were constructed. Parametric conditions of stability and instability of the equilibrium points were defined. Behavior of the trajectories of the system depending on the initial conditions and system parameters were determined. These autonomous dynamical systems are used for the construction of a simulation model dynamics of output production of interdependent branches of production in modeling environment AnyLogic. And the possibility of using them in practice, as a decision support system, was considered.