Design and Stability Analysis of NondeterministicMultidimensional Populations Dynamics Models

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The multidimensional models of the population dynamics are considered in the paper. Thesemodels are the generalizations of the Lotka-Volterra model in case of interaction of the finitenumber of populations. The deterministic description of the models is given by the systemsof the ordinary nonlinear differential equations presented in the paper in the form of themultidimensional vector differential equations. The qualitative properties of the specified modelsare sufficiently well studied by means of Lyapunov methods. However, the probabilistic factorsinfluencing on the behavior of models are not taken into account at the deterministic descriptionof models. The new approaches to the modeling and stability analysis are of theoretical andapplied interest in the nondeterministic case.In this paper, the methods for design of multidimensional nondeterministic models ofinteraction of populations are considered. The first method is connected with the transitionfrom the vector nonlinear ordinary differential equation to the corresponding vector differentialinclusions, 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 differentialinclusion to the problem of stability of solutions of other types of equations, stability conditionsfor the constructed models are obtained. The second method is connected with the technique ofdesign of the self-consistent stochastic models. The scheme of interaction is received on the basisof this technique. This scheme includes a symbolical record of possible interactions between thesystem elements. The structure of the multidimensional stochastic Lotka-Volterra models isdescribed, and the transition to the corresponding Fokker-Planck vector equations is carriedout by means of the system state operators and the system state change operator. The rules forthe transition to the multidimensional stochastic differential equation in the Langevin form areformulated. The execution of the numerical experiment with the application of the developedprogram complex for solving the systems of the stochastic differential equations is possible forthe models which are the concretizations of the studied general models. The described approachto the modeling of the stochastic systems can be applied in the problems of comparing of thequalitative properties of the models in deterministic and stochastic cases. The obtained resultsare aimed at the development of the methods for the analysis of nondeterministic nonlinearmodels.

A V Demidova

Principal contact for editorial correspondence.
Department of Applied Probability and Informatics Peoples’ Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

O V Druzhinina
Federal Research Center “Computer Science and Control” of RAS; V. A. Trapeznikov Institute of Control Sciences of RAS 40 Vavilov Str., Moscow, 119333, Russian Federation; 65 Profsoyuznaya St., Moscow, 117997, Russian Federation

O N Masina
Yelets State Ivan Bunin University 28 Communards St., Yelets, 399770, Russian Federation

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