Computer research of deterministic and stochastic models “two competitors-two migration areas” taking into account the variability of parameters
- Authors: Vasilyeva I.I.1, Demidova A.V.2, Druzhinina O.V.3, Masina O.N.1
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Affiliations:
- Bunin Yelets State University
- RUDN University
- Federal Research Center “Computer Science and Control” of Russian Academy of Sciences
- Issue: Vol 32, No 1 (2024)
- Pages: 61-73
- Section: Articles
- URL: https://journal-vniispk.ru/2658-4670/article/view/316839
- DOI: https://doi.org/10.22363/2658-4670-2024-32-1-61-73
- EDN: https://elibrary.ru/GZIFWL
- ID: 316839
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Full Text
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.
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 FederationAnastasia 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 FederationOlga 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 FederationOlga 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 FederationReferences
- Lotka, A. J. Elements of Physical Biology (Williams and Wilkins Company, Baltimore, MD, USA, 1925).
- Volterra, V. Fluctuations in the abundance of a species considered mathematically. Nature, 558-560. doi: 10.1038/118558a0 (1926).
- Bazykin, A. D. Nonlinear Dynamics of Interacting Populations Russian. [in Russian] (Institute of Computer Research, Moscow-Izhevsk, 2003).
- Pykh, Y. A. Generalized Lotka-Volterra Systems: Theory and Applications Russian. [in Russian] (SPbGIPSR, St. Petersburg, 2017).
- Li, A. Mathematical Modelling of Ecological Systems in Patchy Environments in Electronic Thesis and Dissertation Repository (2021), 8059.
- Hsu, S. B., Ruan, S. & Yang, T. H. Analysis of three species Lotka-Volterra food web models with omnivory. Journal of Mathematical Analysis and Applications 426, 659-687. doi: 10.1016/j.jmaa.2015.01.035 (2015).
- Shestakov, A. A. Generalized Direct Method for Systems with Distributed Parameters Russian. [in Russian] (URSS, Moscow, 2007).
- Moskalenko, A. I. Methods of Nonlinear Mappings in Optimal Control. Theory and Applications to Models of Natural Systems Russian. [in Russian] (Nauka, Novosibirsk, 1983).
- Zhou, Q. Modelling Walleye Population and Its Cannibalism Effect in Electronic Thesis and Dissertation Repository (2017), 4809.
- Sadykov, A. & Farnsworth, K. D. Model of two competing populations in two habitats with migration: Application to optimal marine protected area size. Theoretical Population Biology 142, 114-122. doi: 10.1016/j.tpb.2021.10.002 (2021).
- Chen, S., Shi, J., Shuai, Z. & Wu, Y. Global dynamics of a Lotka-Volterra competition patch model. Nonlinearity 35, 817. doi: 10.1088/1361-6544/ac3c2e (2021).
- Demidova, A. V. Equations of Population Dynamics in the Form ofStochastic Differential Equations. Russian. RUDN Journal of Mathematics, Information Sciences and Physics 1. [in Russian], 67-76 (2013).
- Gevorkyan, M. N., Velieva, T. R., Korolkova, A. V., Kulyabov, D. S. & Sevastyanov, L. A. Stochastic Runge-Kutta Software Package for Stochastic Differential Equations in Dependability Engineering and Complex Systems (eds Zamojski, W., Mazurkiewicz, J., Sugier, J., Walkowiak, T. & Kacprzyk, J.) (Springer International Publishing, Cham, 2016), 169-179. doi: 10.1007/978-3-319-39639-2_15.
- Korolkova, A. & Kulyabov, D. One-step Stochastization Methods for Open Systems. EPJ Web of Conferences 226, 02014. doi: 10.1051/epjconf/202022602014 (2020).
- Gardiner, C. W. Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences (Springer, Heidelberg, 1985).
- Van Kampen, N. G. Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 1992).
- Demidova, A. V., Druzhinina, O. V., Masina, O. N. & Petrov, A. A. Synthesis and Computer Study of Population Dynamics Controlled Models Using Methods of Numerical Optimization, Stochastization and Machine Learning. Mathematics 9, 3303. doi: 10.3390/math9243303 (2021).
- Vasilyeva, I. I., Demidova, A.V., Druzhinina, O.V. & Masina, O. N. Construction, stochastization and computer study of dynamic population models “two competitors - two migration areas”. Discrete and Continuous Models and Applied Computational Science 31, 27-45. doi: 10.22363/2658-4670-2023-31-1-27-45 (2023).
- Sullivan, E. Numerical Methods - An Inquiry-Based Approach with Python 2021.
- Demidova, A. V., Druzhinina, O. V., Masina, O. N. & Petrov, A. A. Development of Algorithms and Software for Modeling Controlled Dynamic Systems Using Symbolic Computations and Stochastic Methods. Programming and Computer Software 49, 108-121. doi: 10.1134/S036176882302007X (2023).
- Storn, R. & Price, K. Differential Evolution: A Simple and Efficient Adaptive Scheme for Global Optimization Over Continuous Spaces. Journal of Global Optimization 23 (1995).
- Das, S. & Suganthan, P. N. Differential Evolution: A Survey of the State-of-the-Art. IEEE Transactions on Evolutionary Computation 15, 4-31. doi: 10.1109/TEVC.2010.2059031 (2011).
- Eltaeib, T. & Mahmood, A. Differential Evolution: A Survey and Analysis. Applied Sciences 8, 1945. doi: 10.3390/app8101945 (2018).
- Petrov, A. A., Druzhinina, O. V., Masina, O. N. & Vasilyeva, I. I. The construction and analysis of four-dimensional models of population dynamics taking into account migration flows. Russian. Uchenye zapiski UlGU. Series: Mathematics and Information Technology”. [in Russian], 43-55 (2022).
- Vucetic, D. Fuzzy Differential Evolution Algorithm in Electronic Thesis and Dissertation Repository (2012), 503.
- Karpenko, A. P. Modern Search Engine Optimization Algorithms. Algorithms Inspired by Nature 2nd ed. Russian. [in Russian] (N.E. Bauman MSTU, Moscow, 2016).
- Simon, D. Evolutionary Optimization Algorithms. Biologically-Inspired and Population-Based Approaches to Computer Intelligence (John Wiley & Sons, Inc., New York, 2013).
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