Design and stability analysis of nondeterministic multidimensional populations dynamics models

The multidimensional model of the population dynamics is considered in the paper. This model is the generalization of the Lotka-Volterra model in case of interaction of the ultimate number of populations. The deterministic description of the model is given by the system of the ordinary nonlinear differential equations presented in the paper in the form of the multidimensional vector differential equation. The qualitative properties of the specified model are sufficiently well studied by means of Lyapunov methods. However, the probabilistic factors influencing on the behavior of model are not taken into account at the deterministic description of model. The new approaches to the modeling and stability analysis are of theoretical and applied interest in the nondeterministic case. In this paper, the methods for design of multidimensional nondeterministic models of interaction of populations are considered. The first method is connected with the transition from the vector nonlinear ordinary differential equation to the corresponding vector differential inclusions, fuzzy and stochastic differential equations. Using the principle of reduction, which allows us to study the stability problem of solving the differential inclusion to the stability problem of solving other types of equations, as a basis, the conditions of stability are obtained for the designed models. The second method is connected with the technique of design of the self-consistent stochastic models. The scheme of interaction is received on the basis of this technique. This scheme includes a symbolical record of possible interactions between the system elements. The structure of the multidimensional stochastic model is described, and the transition to the corresponding Fokker-Planck vector equation is carried out by means of the system state operators and the system state change operator. The rules for the transition to the multidimensional stochastic differential equation in the Langevin form are formulated. The execution of the numerical experiment with the application of the developed program complex for the solving the systems of the stochastic differential equations is possible for the models which are the concretization of the studied general model. The described approach to the modeling of the stochastic systems can find the application in the problems of comparing of the qualitative properties of the models in deterministic and stochastic cases. The obtained results are aimed at the developing methods for the analysis of nondeterministic nonlinear models. © Copyright 2017 for the individual papers by the papers' authors.