Discrete and Continuous Models and Applied Computational ScienceDiscrete 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)3446010.22363/2658-4670-2023-31-1-27-45VFNJWVArticlesСтатьиResearch ArticleConstruction, stochastization and computer study of dynamic population models “two competitors - two migration areas”Построение, стохастизация и компьютерное исследование динамических популяционных моделей «два конкурента - два ареала миграции»https://orcid.org/0000-0002-4120-2595VasilyevaIrina I.ВасильеваИ. И.

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

irinavsl@yandex.ruhttps://orcid.org/0000-0003-1000-9650DemidovaAnastasia V.ДемидоваА. В.

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

demidova-av@rudn.ruhttps://orcid.org/0000-0002-9242-9730DruzhininaOlga V.ДружининаО. В.

Doctor of Physical and Mathematical Sciences, Chief Researche

ovdruzh@mail.ruhttps://orcid.org/0000-0002-0934-7217MasinaOlga N.МасинаО. Н.

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

olga121@inbox.ruBunin Yelets State UniversityЕлецкий государственный университет им. И.А. БунинаPeoples’ Friendship University of Russia (RUDN University)Российский университет дружбы народовFederal Research Center “Computer Science and Control” of RASФедеральный исследовательский центр «Информатика и управление» РАН30032023311VOL 31, NO1 (2023)ТОМ 31, №1 (2023)274520042023Copyright ©; 2023, Vasilyeva I.I., Demidova A.V., Druzhinina O.V., Masina O.N.Copyright ©; 2023, Васильева И.И., Демидова А.В., Дружинина О.В., Масина О.Н.2023Vasilyeva I.I., Demidova A.V., Druzhinina O.V., Masina O.N.Васильева И.И., Демидова А.В., Дружинина О.В., Масина О.Н.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/34460

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.

При изучении детерминированных и стохастических популяционных моделей актуальными задачами являются формализация процессов с учётом новых эффектов, обусловленных взаимодействием видов, и развитие компьютерных методов исследования. Компьютерные методы исследования позволяют выполнить анализ траекторий многомерных популяционных систем. Мы рассматриваем модель «два конкурента - два ареала миграции», в которой учитывается внутривидовая и межвидовая конкуренция в двух популяциях, а также двунаправленная миграция обеих популяций. Для указанной модели мы учитываем вариативность параметров естественного воспроизводства видов. Предложено формализованное описание четырёхмерной модели «два конкурента - два ареала миграции» и её модификаций. С помощью реализации эволюционного алгоритма получен набор параметров, обеспечивающих сосуществование популяций в условиях конкуренции двух видов в основном ареале с учётом миграции этих видов. Исходя из полученного набора параметров найдено положительное состояние равновесия. Построены двумерные и трёхмерные проекции фазовых портретов. Осуществлена стохастизация модели «два конкурента - два ареала миграции» на основе метода построения самосогласованных одношаговых моделей. Для описания структуры модели использованы уравнения Фоккера-Планка. Выполнен переход к четырёхмерному стохастическому дифференциальному уравнению в форме Ланжевена. Для проведения численных экспериментов использован специализированный программный комплекс, предназначенный для построения и изучения стохастических моделей, а также разработана компьютерная программа на основе дифференциальной эволюции. Использованы алгоритмы генерирования траекторий винеровского процесса и многоточечных распределений и модификации метода Рунге-Кутты. В детерминированном и стохастическом случаях изучена динамика траекторий популяционно-миграционных систем. Проведён сравнительный анализ детерминированных и стохастических моделей. Результаты могут найти применение в задачах моделирования популяционных, экономических, демографических и химических систем.

population dynamics modelsstochastic differential equationsone-step processesstochastizationcompetitionmigrationtrajectory dynamicsprojections of phase portraitscomputer modelingsoftware packageмодели динамики популяцийстохастические дифференциальные уравненияодношаговые процессыстохастизацияконкуренциямиграциятраекторная динамикапроекции фазовых портретовкомпьютерное моделированиепрограммный комплексV. 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.