Construction, stochastization and computer study of dynamic population models “two competitors - two migration areas”

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Abstract

When studying deterministic and stochastic population models, the actual problems are the formalization of processes, taking into account new effects caused by the interaction of species, and the development of computer research methods. Computer research methods make it possible to analyze the trajectories of multidimensional population systems. We consider the “two competitors - two migration areas” model, which takes into account intraspecific and interspecific competition in two populations, as well as bidirectional migration of both populations. For this model, we take into account the variability of the reproduction rates of species. A formalized description of the four-dimensional model “two competitors - two migration areas” and its modifications is proposed. Using the implementation of the evolutionary algorithm, a set of parameters is obtained that ensure the coexistence of populations under conditions of competition between two species in the main area, taking into account the migration of these species. Taking into account the obtained set of parameters, a positive stationary state is found. Two-dimensional and three-dimensional projections of phase portraits are constructed. Stochastization of the model “two competitors - two migration areas” is carried out based on the method of self-consistent one-step models constructing. The Fokker-Planck equations are used to describe the structure of the model. A transition to a four-dimensional stochastic differential equation in the Langevin form is performed. To carry out numerical experiments, a specialized software package is used to construct and study stochastic models, and a computer program based on differential evolution is developed. Algorithms for generating trajectories of the Wiener process and multipoint distributions and modifications of the Runge-Kutta method are used. In the deterministic and stochastic cases, the dynamics of the trajectories of populationmigration systems is studied. A comparative analysis of deterministic and stochastic models is carried out. The results can be used in modeling of different classes of dynamic systems.

About the authors

Irina I. Vasilyeva

Bunin Yelets State University

Email: irinavsl@yandex.ru
ORCID iD: 0000-0002-4120-2595

Assistant professor of Department of Mathematical Modeling, Computer Technologies and Information Security

28, Kommunarov St., Yelets, 399770, Russian Federation

Anastasia V. Demidova

Peoples’ Friendship University of Russia (RUDN University)

Email: demidova-av@rudn.ru
ORCID iD: 0000-0003-1000-9650

Candidate of Physical and Mathematical Sciences, Assistant professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Olga V. Druzhinina

Federal Research Center “Computer Science and Control” of RAS

Email: ovdruzh@mail.ru
ORCID iD: 0000-0002-9242-9730

Doctor of Physical and Mathematical Sciences, Chief Researche

44-2, Vavilov St., Moscow, 119333, Russian Federation

Olga N. Masina

Bunin Yelets State University

Author for correspondence.
Email: olga121@inbox.ru
ORCID iD: 0000-0002-0934-7217

Doctor of Physical and Mathematical Sciences, Deputy Head of Department of Mathematical Modeling, Computer Technologies and Information Security

28, Kommunarov St., Yelets, 399770, Russian Federation

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