The multidimensional models of the population dynamics are considered in the paper. These models are the generalizations of the Lotka-Volterra model in case of interaction of the finite number of populations. The deterministic description of the models is given by the systems of the ordinary nonlinear differential equations presented in the paper in the form of the multidimensional vector differential equations. The qualitative properties of the specified models are sufficiently well studied by means of Lyapunov methods. However, the probabilistic factors influencing on the behavior of models are not taken into account at the deterministic description of models. 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. On the basis of the reduction principle, which makes it possible to reduce the problem of the stability of solutions of a differential inclusion to the problem of stability of solutions of other types of equations, stability conditions for the constructed models are obtained. 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 Lotka-Volterra models is described, and the transition to the corresponding Fokker-Planck vector equations 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 solving the systems of the stochastic differential equations is possible for the models which are the concretizations of the studied general models. The described approach to the modeling of the stochastic systems can be applied in the problems of comparing of the qualitative properties of the models in deterministic and stochastic cases. The obtained results are aimed at the development of the methods for the analysis of nondeterministic nonlinear models.
The problems of synthesis and analysis of multidimensional controlled models of population dynamics are of both theoretical and applied interest. The need to solve numerical optimization problems for such a class of models is associated with the expansion of ecosystem control requirements. The need to solve the problem of stochastization is associated with the emergence of new problems in the study of ecological systems properties under the influence of random factors. The aim of the work is to develop a new approach to studying the properties of population dynamics systems using methods of numerical optimization, stochastization and machine learning. The synthesis problems of nonlinear three-dimensional models of interconnected species number dynamics, taking into account trophic chains and competition in prey populations, are studied. Theorems on the asymptotic stability of equilibrium states are proved. A qualitative and numerical study of the models is carried out. Using computational experiments, the results of an analytical stability and permanent coexistence study are verified. The search for equilibrium states belonging to the stability and permanent coexistence region is made using the developed intelligent algorithm and evolutionary calculations. The transition is made from the model specified by the vector ordinary differential equation to the corresponding stochastic model. A comparative analysis of deterministic and stochastic models with competition and trophic chains is carried out. New effects are revealed that are characteristic of three-dimensional models, taking into account the competition in populations of prey. The formulation of the optimal control problem for a model with competition and trophic chains is proposed. To find optimal trajectories, new generalized algorithms for numerical optimization are developed. A methods for the synthesis of controllers based on the use of artificial neural networks and machine learning are developed. The results on the search for optimal trajectories and generation of control functions are presented.The obtained results can be used in modeling problems of ecological, demographic, socio-economic and chemical kinetics systems.
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