Computer research of deterministic and stochastic models “two competitors-two migration areas” taking into account the variability of parameters

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Abstract

Theanalysisoftrajectorydynamicsandthesolutionofoptimizationproblemsusingcomputermethods are relevant areas of research in dynamic population-migration models. In this paper, four-dimensional dynamic models describing the processes of competition and migration in ecosystems are studied. Firstly, we consider a modification of the “two competitors-two migration areas” model, which takes into account uniform intraspecific and interspecific competition in two populations as well as non-uniform bidirectional migration in both populations. Secondly, we consider a modification of the “two competitors-two migration areas” model, in which intraspecific competition is uniform and interspecific competition and bidirectional migration are non-uniform. For these two types of models, the study is carried out taking into account the variability of parameters. The problems of searching for model parameters based on the implementation of two optimality criteria are solved. The first criterion of optimality is associated with the fulfillment of such a condition for the coexistence of populations, which in mathematical form is the integral maximization of the functions product characterizing the populations densities. The second criterion of optimality involves checking the assumption of the such a four-dimensional positive vector existence, which will be a state of equilibrium. The algorithms developed on the basis of the first and second optimality criteria using the differential evolution method result in optimal sets of parameters for the studied population-migration models. The obtained sets of parameters are used to find positive equilibrium states and analyze trajectory dynamics. Using the method of constructing self-consistent one-step models and an automated stochastization procedure, the transition to the stochastic case is performed. The structural description and the possibility of analyzing two types of populationmigration stochastic models are provided by obtaining Fokker-Planck equations and Langevin equations with corresponding coefficients. Algorithms for generating trajectories of the Wiener process, multipoint distributions and modifications of the Runge-Kutta method are used. A series of computational experiments is carried out using a specialized software package whose capabilities allow for the construction and analysis of dynamic models of high dimension, taking into account the evaluation of the stochastics influence. The trajectory dynamics of two types of population-migration models are investigated, and a comparative analysis of the results is carried out both in the deterministic and stochastic cases. The results can be used in the modeling and optimization of dynamic models in natural science.

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1. Introduction The study of mathematical models of population systems began to develop actively in the twenties of the last century, thanks to the works of A. Lotka [1] and V. Volterra [2]. Currently, this direction includes the wide class study of models taking into account various interactions between populations © Vasilyeva I. I., Demidova A. V., Druzhinina O. V., Masina O. N., 2024 This work is licensed under a Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by-nc/4.0/legalcode (See, for example, [3-6]). Significant progress in the study is achieved due to the analysis of the dynamic stability models of ecological systems using the theory of differential equations, numerical methods and optimization methods [3, 4, 7-9]. A four-dimensional model of two competing species with migration between two ranges, taking into account the asymmetry coefficient, is considered in [10]. It is shown that the choice of the migration area is carried out depending on the value of this coefficient. The coefficient of asymmetry affects which of the habitats species migrate to first. Two-species Lotka-Volterra competition patch model is studied in [11]. It’s shown that in the long time, either the competition exclusion holds that one species becomes extinct, or the two species reach a coexistence equilibrium, and the outcome of the competition is determined by the strength of the inter-specific competition and the dispersal rates. The transition to the non-deterministic case based on the design stochastic self-consistent models (DSSM) method allows us to identify new qualitative properties of models and carry out a comparative analysis [12-16] and in the other works. For various types of population models, the DSSM method is used in [12, 17, 18]. In [18], a formalized description of the four-dimensional model “two competitors-two migration areas” and its modifications are proposed, taking into account the case when population growth coefficients are different (non-uniform reproduction of species). Using the implementation of the evolutionary algorithm, a set of parameters is obtained that ensure the coexistence of populations in the conditions of competition between two species in the general area, taking into account the migration of these species. Stochastization of the model “two competitors- two migration areas” (under conditions of non-uniform species reproduction) is carried out on the basis of the method of constructing self-consistent stochastic models. The dynamics of trajectories for deterministic and stochastic cases is studied, a comparative analysis is performed. This paper is a continuation of [18] and contains the construction and analysis of such modifications of the model “two competitors-two migration areas”, which allows us to study the influence of non-uniform migration flows and the influence of non-uniform interspecific competition on the trajectory dynamics both in the deterministic case and in the stochastic case. In section 2 of this paper, we consider the construction of two modifications of the model “two competitors-two migration areas” with bidirectional non-uniform migration (to two refuges), taking into account the uniformity and non-uniformness of the interspecific competition coefficients. In section 3, the search for model parameters is carried out using an evolutionary algorithm taking into account different optimality criteria. In section 4, a study of the obtained deterministic fourdimensional models is carried out, two-dimensional and three-dimensional projections of phase portraits are constructed. In section 5, the transition to stochastic models “two competitors-two migration areas” is made on the basis of the constructing self-consistent stochastic models method, the dynamics of trajectories in the stochastic case is 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. The developed in Python [19] software package [20] is used to study the models. Section 7 discusses the results. 2. Description of the model modifications “two competitors-two migration areas” taking into account non-uniform migration Ref. [18] describes a general four-dimensional deterministic model, which takes into account the influence of interspecific and intraspecific competition in two populations with bidirectional migration of both populations, and the non-uniform growth of population reproduction. We describe further such a model “two competitors-two migration areas”, for which the growth of population reproduction, interspecific and intraspecific competition are uniform, and migration is non-uniform. The specified model is given by a system of equations of the form
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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

RUDN University

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

Candidate of Physical and Mathematical Sciences, Assistant professor of Department of Probability Theory and Cyber Security

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

Olga V. Druzhinina

Federal Research Center “Computer Science and Control” of Russian Academy of Sciences

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

Doctor of Physical and Mathematical Sciences, Chief Researche

44 Vavilov St, bldg 2, 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, Professor of Department of Mathematical Modeling, Computer Technologies and Information Security

28 Kommunarov St, Yelets, 399770, Russian Federation

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Copyright (c) 2024 Vasilyeva I.I., Demidova A.V., Druzhinina O.V., Masina O.N.

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