Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)3446010.22363/2658-4670-2023-31-1-27-45Research ArticleConstruction, stochastization and computer study of dynamic population models “two competitors - two migration areas”VasilyevaIrina I.<p>Assistant professor of Department of Mathematical Modeling, Computer Technologies and Information Security</p>irinavsl@yandex.ruhttps://orcid.org/0000-0002-4120-2595DemidovaAnastasia V.<p>Candidate of Physical and Mathematical Sciences, Assistant professor of Department of Applied Probability and Informatics</p>demidova-av@rudn.ruhttps://orcid.org/0000-0003-1000-9650DruzhininaOlga V.<p>Doctor of Physical and Mathematical Sciences, Chief Researche</p>ovdruzh@mail.ruhttps://orcid.org/0000-0002-9242-9730MasinaOlga N.<p>Doctor of Physical and Mathematical Sciences, Deputy Head of Department of Mathematical Modeling, Computer Technologies and Information Security</p>olga121@inbox.ruhttps://orcid.org/0000-0002-0934-7217Bunin Yelets State UniversityPeoples’ Friendship University of Russia (RUDN University)Federal Research Center “Computer Science and Control” of RAS30032023311274520042023Copyright © 2023, Vasilyeva I.I., Demidova A.V., Druzhinina O.V., Masina O.N.2023<p style="text-align: justify;">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.</p>population dynamics modelsstochastic differential equationsone-step processesstochastizationcompetitionmigrationtrajectory dynamicsprojections of phase portraitscomputer modelingsoftware packageмодели динамики популяцийстохастические дифференциальные уравненияодношаговые процессыстохастизацияконкуренциямиграциятраекторная динамикапроекции фазовых портретовкомпьютерное моделированиепрограммный комплекс1. 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[V. Volterra, “Fluctuations in the abundance of a species considered mathematically,” Nature, no. 118, pp. 558-560, 1926. DOI: 10.1038/118558a0.][A. J. Lotka, Elements of physical biology. Baltimore, MD, USA: Williams and Wilkins Company, 1925.][A. D. Bazykin, Nonlinear dynamics of interacting populations [Nelineynaya dinamika vzaimodeystvuyushchikh populyatsiy]. Moscow-Izhevsk: Institute of Computer Research, 2003, in Russian.][A. Y. Aleksandrov, A. V. Platonov, V. N. Starkov, and N. A. Stepenko, Study of mathematical modeling and sustainability of biological societies [Matematicheskoye modelirovaniye i issledovaniye ustoychivykh biologicheskikh soobshchestv]. St. Petersburg: Lan, 2017, in Russian.][P. Turchin, Complex population dynamics. Princeton: Princeton University Press, 2013.][Y. A. Pykh, Generalized Lotka-Volterra systems: theory and applications [Obobshchennyye sistemy Lotki-Vol’terra: teoriya i prilozheniya]. St. Petersburg: SPbGIPSR, 2017, in Russian.][L. Stucchi, J. Pastor, J. Garcia-Algarra, and J. Galeano, “A general model of population dynamics accounting for multiple kinds of interaction,” Complexity, vol. 2020, p. 7961327, 2020. DOI: 10.1155/2020/7961327.][J. S. Link, F. Pranovi, and S. Libralato, “Simulations and interpretations of cumulative trophic theory,” Ecological Modelling, vol. 463, p. 109800, 2022. DOI: 10.1016/j.ecolmodel.2021.109800.][A. A. Shestakov, Generalized direct method of Lyapunova for systems with distributed parameters [Obobshchennyy pryamoy metod Lyapunova dlya sistem s raspredelennymi parametrami]. Moscow: URSS, 2007, in Russian.][A. I. Moskalenko, Methods of nonlinear mappings in optimal control. Theory and applications to models of natural systems [Metody nelineynykh otobrazheniy v optimal’nom upravlenii (teoriya i prilozheniya k modelyam prirodnykh sistem)]. Novosibirsk: Nauka, 1983, in Russian.][O. V. Druzhinina and O. N. Masina, Methods for analyzing the stability of dynamic intelligent control systems [Metody analiza ustoychivosti dinamicheskikh sistem intellektnogo upravleniya]. Moscow: URSS, 2016, in Russian.][A. V. Demidova, O. V. Druzhinina, O. N. Masina, and A. A. Petrov, “Synthesis and computer study of population dynamics controlled models using methods of numerical optimization, stochastization and machine learning,” Mathematics, vol. 9, no. 24, p. 3303, 2021. DOI: 10.3390/math9243303.][A. V. Demidova, “Equations of population dynamics in the form of stochastic differential equations,” RUDN Journal of Mathematics, Information Sciences and Physics, vol. 1, pp. 67-76, 2013, in Russian.][A. V. Demidova, O. V. Druzhinina, and O. N. Masina, “Design and stability analysis of nondeterministic multidimensional populations dynamics models,” RUDN Journal of Mathematics, Information Sciences and Physics, vol. 25, no. 4, pp. 363-372, 2017.][I. N. Sinitsyn, O. V. Druzhinina, and O. N. Masina, “Analytical modeling and stability analysis of nonlinear broadband migration flows,” Nonlinear World, vol. 16, no. 3, pp. 3-16, 2018, in Russian.][Y. M. Svirezhev and D. O. Logofet, Stability of biological communities. Moscow: Nauka, 1978, in Russian.][H. I. Freedman and B. Rai, “Can Mutualism alter Competitive Outcome: a Mathematical Analysis,” The Rocky Mountain Journal of Mathematics, vol. 25, no. 1, pp. 217-230, 1995.][Z. Lu and Y. Takeuchi, “Global asymptotic behavior in single-species discrete diffusion systems,” Journal of Mathematical Biology, vol. 32, pp. 67-77, 1993. DOI: 10.1007/BF00160375.][X.-a. Zhang and L. Chen, “The linear and nonlinear diffusion of the competitive Lotka-Volterra model,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 12, pp. 2767-2776, 2007. DOI: 10.1016/j.na.2006.04.006.][A. V. Demidova, O. V. Druzhinina, O. N. Masina, and E. D. Tarova, “Computer research of nonlinear stochastic models with migration flows,” in Proceedings of the Selected Papers of the 10th International Conference “Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems” (ITTMM-2019). CEUR Workshop Proceedings, vol. 2407, 2019, pp. 26-37.][A. A. Petrov, O. V. Druzhinina, O. N. Masina, and I. I. Vasilyeva, “The construction and analysis of four-dimensional models of population dynamics taking into account migration flows,” Uchenye zapiski UlGU. Series: Mathematics and Information Technology, no. 1, pp. 43-55, 2022, in Russian.][I. I. Vasilyeva, “Computer modeling of the system of population dynamics taking into account the variation of migration parameters,” Uchenye zapiski UlGU. Series: Mathematics and Information Technology, no. 2, pp. 21-30, 2022, in Russian.][S. Cui and M. Bai, “Mathematical analysis of population migration and its effects to spread of epidemics,” Discrete and Continuous Dynamical Systems - B, vol. 20, no. 9, pp. 2819-2858, 2015. DOI: 10.3934/dcdsb. 2015.20.2819.][H. C. Tuckwell, “A study of some diffusion models of population growth,” Theoretical Population Biology, vol. 5, no. 3, pp. 345-357, 1974. DOI: 10.1016/0040-5809(74)90057-4.][H. I. Freedman and P. Waltman, “Mathematical models of population interactions with dispersal. I: Stability of two habitats with and without a predator,” SIAM Journal on Applied Mathematics, vol. 32, no. 3, pp. 631-648, 1977. DOI: 10.1137/0132052.][L. J. S. Allen, “Persistence and extinction in single-species reactiondiffusion models,” Bulletin of Mathematical Biology, vol. 45, no. 2, pp. 209-227, 1983. DOI: 10.1016/S0092-8240(83)80052-4.][Y. Takeuchi, Global dynamical properties of Lotka-Volterra systems. Singapore: World Scientific, 1996.][A. V. Demidova, O. V. Druzhinina, M. Jacimovic, O. N. Masina, and N. Mijajlovic, “Synthesis and analysis of multidimensional mathematical models of population dynamics,” in 2018 10th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), vol. 2018-November, 2019, pp. 361-366. DOI: 10.1109/ICUMT.2018.8631252.][A. Demidova, O. Druzhinina, M. Jacimovic, O. Masina, N. Mijajlovic, N. Olenev, and A. Petrov, “The Generalized algorithms of global parametric optimization and stochastization for dynamical models of interconnected populations,” in Optimization and Applications, Cham: Springer International Publishing, 2020, pp. 40-54. DOI: 10.1007/978-3-030-628673_4.][M. N. Gevorkyan, T. R. Velieva, A. V. Korolkova, D. S. Kulyabov, and L. A. Sevastyanov, “Stochastic Runge-Kutta software package for stochastic differential equations,” in Dependability Engineering and Complex Systems, Cham: Springer International Publishing, 2016, pp. 169-179. DOI: 10.1007/978-3-319-39639-2_15.][M. Gevorkyan, A. Demidova, T. Velieva, A. Korolkova, D. Kulyabov, and L. Sevastyanov, “Implementing a method for stochastization of one-step processes in a computer algebra system,” Programming and Computer Software, vol. 44, pp. 86-93, Mar. 2018. DOI: 10.1134/S0361768818020044.][A. Korolkova and D. Kulyabov, “One-step stochastization methods for open systems,” EPJ Web of Conferences, vol. 226, p. 02014, 2020. DOI: 10.1051/epjconf/202022602014.][A. P. Karpenko, Modern search engine optimization algorithms. Algorithms inspired by nature [Sovremennyye algoritmy poiskovoy optimizatsii. Algoritmy vdokhnovlennyye prirodoy], 2nd ed. Moscow: N.E. Bauman MSTU, 2016, in Russian.][D. Simon, Algorithms for evolutionary optimization [Algoritmy evolyutsionnoy optimizatsii]. Moscow: DMK Press, 2020, in Russian.][A. A. Petrov, O. V. Druzhinina, and O. N. Masina, “Application of the computational intelligence method to modeling the dynamics of multidimensional population system,” in Data Science and Algorithms in Systems, vol. 597, Cham: Springer International Publishing, 2023, pp. 565-575. DOI: 10.1007/978-3-031-21438-7_45.][R. Lamy, Instant SymPy Starter. Packt Publishing, 2013.][T. E. Oliphant, “Python for scientific computing,” Computing in Science Engineering, vol. 9, no. 3, pp. 10-20, 2007. DOI: 10.1109/MCSE.2007.58.][C. Fuhrer, J. Solem, and O. Verdier, Scientific computing with Python. Second edition. Packt Publishing, 2021.][C. Hill, Learning scientific programming with Python, Second Edition. Cambridge: Cambridge University Press, 2020.][N. Sillero, J. C. Campos, S. Arenas-Castro, and A. M. Barbosa, “A curated list of R packages for ecological niche modelling,” Ecological Modelling, vol. 476, p. 110242, 2023. DOI: 10.1016/j.ecolmodel.2022.110242.][O. V. Druzhinina, O. N. Masina, and E. D. Tarova, “Synthesis, computer research and stability analysis for the multidimensional models of the dynamics of the interconnected population,” Nonlinear World, vol. 17, no. 2, pp. 48-58, 2019, in Russian. DOI: 10.18127/j20700970-20190206.][I. I. Vasilyeva, O. V. Druzhinina, and O. N. Masina, “Design and research of population dynamic model “two competitors - two migration areas”,” Nonlinear World, vol. 20, no. 4, pp. 60-68, 2022, in Russian. DOI: 10.18127/j20700970-202204-06.][C. W. Gardiner, Handbook of stochastic methods: for Physics, Chemistry and the Natural Sciences. Heidelberg: Springer, 1985.][N. G. Van Kampen, Stochastic processes in Physics and Chemistry. Amsterdam: Elsevier, 1992.]