Анализ динамики решений гибридной разностной системы типа Лотки—Вольтерры

Анализ динамики решений гибридной разностной системы типа Лотки—Вольтерры

Платонов А. В.
Сибирский журнал индустриальной математики, 27, 4(100), 99-112 (2024)
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УДК 517.962 
DOI: 10.33048/SIBJIM.2024.27.407

Аннотация:

Рассматривается дискретная система типа Лотки—Вольтерры. Предполагается, что эта система может функционировать как в некотором плановом, так и в возмущённом режимах. Исследуются ограничения на время пребывания системы в этих режимах, обеспечивающие желаемое динамическое поведение. В частности, определяются условия предельной ограниченности решений и перманентности системы. Используется прямой метод Ляпунова, причём в разных частях фазового пространства строятся разные функции Ляпунова. Оцениваются размеры области допустимых начальных значений решений и области предельного пребывания решений, соответствующих требуемой динамике системы. Устанавливаются ограничения на величину шага дискретизации системы.

Литература:
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  13. Hu H., Wang K., Wu D. Permanence and global stability for nonautonomous N-species Lotka—Volterra competitive system with impulses and infinite delays // J. Math. Anal. Appl. 2011. V. 377, N 1. P. 145– 160; DOI: 10.1016/j.jmaa.2010.10.031
     
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  19. Wang S., Wu W., Lu J., She Zh. Inner-approximating domains of attraction for discrete-time switched systems via multi-step multiple Lyapunov-like functions // Nonlinear Anal. Hybrid Syst. 2021. V. 40. Article 100993; DOI: 10.1016/j.nahs.2020.100993
     
  20. Platonov A. V. Analysis of the dynamical behavior of solutions for a class of hybrid generalized Lotka—Volterra models // Commun. Nonlinear Sci. Numer. Simul. 2023. V. 119. Article 107068; DOI: 10.1016/j.cnsns.2022.107068

Данная работа финансировалась за счёт средств бюджета Санкт-Петербургского государственного университета. Других источников финансирования проведения или руководства данным конкретным исследованием не было.

А. В. Платонов
  1. Санкт-Петербургский государственный университет, 
    Университетская наб. 7/9, г. Санкт-Петербург 199034, Россия

E-mail: a.platonov@spbu.ru

Статья поступила 28.06.2023 г.
После доработки — 04.09.2024 г.
Принята к публикации 06.11.2024 г.

Abstract:

A difference system of the Lotka—Volterra type is considered. It is assumed that this system can operate both in some program and perturbed modes. The restrictions on the time of the system’s stay in these modes, providing the desired dynamical behavior, are investigated. In particular, the conditions of the ultimate boundedness of solutions and the permanence of the system are obtained. The direct Lyapunov method is used, and different Lyapunov functions are constructed in different parts of the phase space. The sizes of the domain of permissible initial values of solutions and the domain of the ultimate bound of solutions corresponding to the required dynamics of the system are estimated. Constraints are set on the size of the digitization step of the system.

References:
  1. J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics (Cambridge Univ. Press, Cambridge, 1998).
     
  2. E. Kazkurewicz and A. Bhaya, Matrix Diagonal Stability in Systems and Computation (Birkhäuser, Boston, 1999). 
     
  3. Yu. A. Pykh, Equilibrium and Stability in Population Dynamics Models (Nauka, Moscow, 1983) [in Russian].
     
  4. J. Hofbauer, V. Hutson, and W. Jansen, “Coexistence for systems governed by difference equations of Lotka–Volterra type,” J. Math. Biol. 25 (5), 553–570 (1987). https://doi.org/10.1007/BF00276199
     
  5. F. D. Chen, “Permanence and global attractivity of a discrete multispecies Lotka– Volterra competition predator-prey systems,” Appl. Math. Comput. 182 (1), 3–12 (2006). https://doi.org/10.1016/j.amc.2006.03.026 
     
  6. Z. Lu and W.Wang, “Permanence and global attractivity for Lotka—Volterra difference systems,” J. Math. Biol. 39 (3), 269–282 (1999). https://doi.org/10.1007/s002850050171
     
  7. N. V. Pertsev, B. Yu. Pichugin, and K. K. Loginov, “Statistical modeling of population dynamics developing under the influence of toxic substances,” Sib. Zh. Ind. Mat. 14 (2), 84–94 (2011) [in Russian].
     
  8. F. Capone, R. De Luca, and S. Rionero, “On the stability of non-autonomous perturbed Lotka—Volterra models,” Appl. Math. Comput. 219 (12), 6868–6881 (2013). https://doi.org/10.1016/j.amc.2013.01.003
     
  9. L. Li and Zh.-J. Wang, “Global stability of periodic solutions for a discrete predator—prey system with functional response,” Nonlinear Dyn. 72 (3), 507–516 (2013). https://doi.org/10.1007/s11071-012-0730-6
     
  10. K. Chakraborty, S. Haldar, and T. K. Kar, “Global stability and bifurcation analysis of a delay induced prey-predator system with stage structure,” Nonlinear Dyn. 73 (3), 1307–1325 (2013). https://doi.org/10.1007/s11071-013-0864-1
     
  11. J. Balbus, “Permanence in nonautonomous competitive systems with nonlocal dispersal,” J. Math. Anal. Appl. 447 (1), 564–578 (2017). https://doi.org/10.1016/j.jmaa.2016.10.030 
     
  12. J. Bao, X. Mao, G. Yin, and C. Yuan, “Competitive Lotka—Volterra population dynamics with jumps,” Nonlinear Anal. 74 (17), 6601–6616 (2011). https://doi.org/10.1016/j.na.2011.06.043
     
  13. H. Hu, K. Wang, and D. Wu, “Permanence and global stability for nonautonomous N-species Lotka— Volterra competitive system with impulses and infinite delays,” J. Math. Anal. Appl. 377 (1), 145–160 (2011). https://doi.org/10.1016/j.jmaa.2010.10.031
     
  14. D. Liberzon, Switching in Systems and Control (Birkhäuser, Boston, 2003).
     
  15. G. Zhai, B. Hu, K. Yasuda, and A. N. Michel, “Disturbance attention properties of timecontrolled switched systems,” J. Franklin Inst. 338 (7), 765–779 (2001). https://doi.org/10.1016/S0016- 0032(01)00030-8
     
  16. L. Zu, D. Jiang, and D. O’Regan, “Conditions for persistence and ergodicity of a stochastic Lotka— Volterra predator-prey model with regime switching,” Commun. Nonlinear Sci. Numer. Simul. 29 (1–3), 1–11 (2015). https://doi.org/10.1016/j.cnsns.2015.04.008
     
  17. A. Yu. Aleksandrov, Y. Chen, A. V. Platonov, and L. Zhang, “Stability analysis and uniform ultimate boundedness control synthesis for a class of nonlinear switched difference systems,” J. Differ. Equ. Appl. 18 (9), 1545–1561 (2012). https://doi.org/10.1080/10236198.2011.581665
     
  18. A. V. Platonov, “On the global asymptotic stability and ultimate boundedness for a class of nonlinear switched systems,” Nonlinear Dyn. 92 (4), 1555–1565 (2018). https://doi.org/10.1007/s11071-018-4146-9
     
  19. S. Wang, W. Wu, J. Lu, and Zh. She, “Inner-approximating domains of attraction for discrete-time switched systems via multi-step multiple Lyapunov-like functions,” Nonlinear Anal. Hybrid Syst. 40, 100993 (2021). https://doi.org/10.1016/j.nahs.2020.100993
     
  20. A. V. Platonov, “Analysis of the dynamical behavior of solutions for a class of hybrid generalized Lotka—Volterra models,” Commun. Nonlinear Sci. Numer. Simul. 119, 107068 (2023). https://doi.org/10.1016/j.cnsns.2022.107068