the «predator-prey» model
the Levenberg–Marquardt algorithm in Fletcher modification
the parametric identification of overdetermined nonlinear systems of equations
identifying of tendencies in the socio-economic system
the regime with peaking
the socio-economic system of Russia
the motion equations
making of optimal management decisions
V.V. Andreev, M.I. Semenov
In our previous studies on the basis of mathematical models, created on the «predator-prey» principles, the dynamics of the socio-economic system of Russia and the United States was investigated at different time stages. The results of these studies adequately describe the real situation. The model «predator-prey» logically follows from the fact, that in the socio-economic system any separate its subsystem acts as a «predator» in relation to its other subsystem, and at the same time it can be a «victim» for the third subsystem.
In this paper we continue the study of the socio-economic system dynamics of Russia on the basis of mathematical models like the «predator-prey». In the mathematical model as the main interacting elements that characterize the socio-economic system, are selected: the consolidated budget revenues, the gross domestic product (GDP), the budget spending to science financing, the population incomes, the capital flight. The data of government statistics are used to determine the model parameters. Optimal parameter identification of the mathematical model is made on the basis of Levenberg-Marquardt algorithm in Fletcher modification. This algorithm gives the best results for the solution of overdetermined systems of nonlinear equations in comparison with other methods. In the study of mathematical model the methods of wavelet analysis are applied. Also the Poincaré sections are constructed.
It is shown that the model «predator-prey» follows from the general principles, which describe the interaction between the separate subsystems of the whole system. In fact, the equations of «predator-prey» model are the motion equations for the case when theirs right-hand sides expanded in a Taylor series up to second order. Thus, these equations are analogous to Newton's second law in the physical system.
In the paper also it was shown, that the «predator-prey» model allows us to define the major trends in the socio-economic system dynamics, prevailing to a specific point in time and allows us to determine its further evolution. The main advantage of used in this paper mathematical models and methods of their analysis is the ability to use them to control the state dynamics of socio-economic systems. By varying of the model parameters, which, in essence, means the changing of impacts on different levels of state administration at certain times, we can apply the model to achieve the most appropriate trends in the evolution of the socio-economic systems in the interest of time moments. For example, it is possible to obtain an estimate of which results in the future will lead one or another tendencies that have prevailed at present in the dynamics of the socio-economic system. In this case, if the expected in the future result is undesirable, on the basis of the proposed mathematical model it can be proposed such control actions on the socio-economic system, which will initiate the emergence of new tendencies in it and which conducive to long-term operation of the system in the desired mode.