Abstract
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.
Highlights
The stability research of the models of the population dynamics is an important problem
The questions of existence and stability of the solutions of the models described by the differential equations of various types were considered in [8,9,10,11,12] and in other works
The systemic approach is described in [2, 6,7,8,9] which allows us to consider properties of stability of the models described by the differential equations of various types from the unified point of view
Summary
The stability research of the models of the population dynamics is an important problem. The stability analysis is performed on the basis of the reduction principle in this work It is known [5, 13,14,15] that in the deterministic description of the model the probabilistic factors affecting the behavior of the model are not taken into account. In this connection, an important problem is construction and study of adequate stochastic models, as well as a comparative analysis of the properties of deterministic and corresponding stochastic models. It is shown that the used approach to construct multidimensional stochastic models can find application in problems of comparing the qualitative properties of the generalized Lotka–Volterra models
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
