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

**Authors:**Vasilyeva I.I.^{1}, Demidova A.V.^{2}, Druzhinina O.V.^{3}, Masina O.N.^{1}-
**Affiliations:**- Bunin Yelets State University
- Peoples’ Friendship University of Russia (RUDN University)
- Federal Research Center “Computer Science and Control” of RAS

**Issue:**Vol 31, No 1 (2023)**Pages:**27-45**Section:**Articles**URL:**https://journals.rudn.ru/miph/article/view/34460**DOI:**https://doi.org/10.22363/2658-4670-2023-31-1-27-45**EDN:**https://elibrary.ru/VFNJWV

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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.

## Full Text

1. Introduction The classical Lotka-Volterra models [1, 2] are further developed in numerous papers by researchers [3-8]. Significant progress is associated with the analysis of dynamic models of ecological systems using the methods of the theory of stability of solutions of differential equations and optimization theory [3-6, 9- 11]. It should be noted that when studying population models, the transition from the deterministic to the stochastic case is of great theoretical and applied interest [12-15]. Population dynamic models are characterized by the fact that when describing them, it is necessary to take into account various types of interaction in the population community, for example, intraspecific competition, interspecific competition, trophic interactions, migration, mutualism [16-18]. For example, research is being conducted related to the study of the properties of multidimensional ecological and demographic systems, taking into account competition and migration flows [19-22]. As the results show, the impact of migration can be significant, and the presence of migration flows leads to the emergence of new qualitative effects. The presence of migration flows in a population system is associated with an adaptive change in the behavior of an organism under changing environmental conditions, in particular, with a deterioration in the epidemiological situation or with an increase in population densities [23]. When constructing models, migration mechanisms are described using linear and nonlinear functions [15, 24-26]. The stability and qualitative behavior of population-migration models are considered in [18, 24, 26-29] and other papers. Despite a number of interesting results in the direction of studying systems with migration flows, there is a need to construct and research new models with migration. As is known, one-dimensional Fokker-Planck equations are used in the construction of Gaussian stochastic models of small dimension. For multidimensional models, the simplest linear models with additive noise are most often used, however, this approach does not fully take into account stochastic processes in the system. A promising direction is stochastic modeling of dynamic systems based on the method of constructing self-consistent one-step models [30-32]. Using this method, we can perform an algorithmic transition to a stochastic model and evaluate the influence of stochastics on the qualitative properties of the model. This assessment is performed through a comparative analysis of deterministic and stochastic models with selected sets of parameters. When studying high-dimensional models, the choice of parameters can be carried out by applying evolutionary algorithms [12, 33- 35]. Various systems of population dynamics (with competition, mutualism, migration) based on self-consistent models are considered in [12, 14, 28] and other papers. Researchers consider various generalizations and modifications of the classical Lotka-Volterra models in the direction of increasing the dimension and constructing non-deterministic models. When considering such models, there is a need for computer research, taking into account the capabilities of high-level languages and applied mathematical packages [36-39]. Numerical analysis of behavior and computer studies of the dynamics of trajectories are associated, among other things, with new problems in the study of nonlinear processes, taking into account the processing of large data arrays under uncertainty. A software package is developed for stochastic modeling of various dynamic systems based on the method of constructing self-consistent one-step models [30, 31]. For the controlled case, a set of programs is proposed that combines randomization, optimization and machine learning [12]. Modeling of population-migration systems is carried out using various software that have a fairly effective set of tools for constructing computer models and conducting computational experiments [38, 40]. The use of applied mathematical packages and high-level programming languages makes it possible to study multidimensional population systems taking into account different types of intraspecific and interspecific interactions, as well as taking into account the variation of parameters and variables. The three-dimensional model “predator-prey-one migration area” is considered in [29]. Four-dimensional population models with competition and one area of migration are studied in [20, 41]. This article is devoted to the study of such a four-dimensional population model of the type “two competitors - two migration areas”, which takes into account changes in the reproduction rates of populations. Section 2 of the paper considers the construction of the “two competitors - two migration areas” model with bidirectional migration (to two refuges) and its modifications. In particular, we offer a description of the model, in which the reproduction rate of population growth are different without varying the parameters of competition and migration. In Section 3, search for model parameters using an evolutionary algorithm is carried out. A study of a deterministic four-dimensional model is carried out, two-dimensional and three-dimensional projections of phase portraits are constructed. In Section 4, stochastic models “two competitors - two migration areas” are constructed using the method of constructing self-consistent stochastic models. In Section 5 the dynamics of trajectories for deterministic and stochastic models are studied. The results of computer experiments are presented and the interpretation of these results is given taking into account the comparison of stochastic and deterministic models. A software package developed in Python using the NumPy, SymPy, SсiPy libraries is used as a tool for studying models. 2. Description of the deterministic model “two competitors - two migration areas” and its modifications One of the basic population-migration models, taking into account competition and migration flows, is a three-dimensional model that describes the dynamics of two interrelated species. According to this model, the first species competes with the second species in the first area, taking into account the migration of the first species to the second area [19]. Four-dimensional generalizations of this population-migration model are studied in [19, 21, 22, 42] and in other papers. Next, we describe a four-dimensional model that takes into account the influence of interspecies competition in two populations with bidirectional migration of both populations. This model is given by a system of nonlinear differential equations of the form## 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## References

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