Instability in dynamic balance of volterra-lotka systems with perturbations in the right side

Authors

  • Ракан Абед Алнаби Альджаафрех Мохаммад Kharkiv National University of Radio Electronics Lenina 14, Kharkov, Ukraine, 61166, Ukraine https://orcid.org/0000-0002-8326-2764

DOI:

https://doi.org/10.15587/1729-4061.2014.23138

Keywords:

Lotka-Volterra model, stability problem, phase space, attractor, chaos, non-resonant torus

Abstract

The basic effects and patterns that characterize the model of coexistence of two species with weak sinusoidal external effect on the reproduction rate is considered. Solving Lotka-Volterra differential equations describes the behavior of the elementary ecosystem. Numerical solutions for exposure frequencies, close to the frequency of the unperturbed system cycle are found. The stability of such a non-autonomous system is investigated.

It is determined that the sinusoidal effect on the population, e.g., by changing the reproduction rate of one or both species because of seasonal changes in nutrition or hunting leads to a nonperiodic system dynamics, having the type of degenerate 2-dimensional non-resonant torus. Various forms of irregular behavior of “predators” and “victims” appear in the phase portraits for similar perturbations. All this confirms that even relatively simple models of ecosystems reveal their instability, i. e., sensitivity to small external perturbations

Author Biography

Ракан Абед Алнаби Альджаафрех Мохаммад, Kharkiv National University of Radio Electronics Lenina 14, Kharkov, Ukraine, 61166

Graduate student
Department of Applied Mathematics

References

  1. Вольтерра, В. Математическая теория борьбы за существование [Текст] / В. Вольтерра. – М.-Ижевск: Институт компьютерных исследований, 2004. – 288 с.
  2. Jost, C. The wolves of Isle Royale display scale-invariant satiation and density dependent predation on moose [Теxt] / С. Jost, G. Devulder, J. A. Vucetich, R. Peterson, R. Arditi // J. Anim. Ecol. – 2005. – № 74(5). – С. 809–816.
  3. Мартынюк, А. А. Хаотическая потеря предельного цикла в задаче Вольтерра [Текст] / А. А. Мартынюк, Н. В. Никитина // Докл. АН Украины. – 1996. – № 4. – С. 1–7.
  4. Hayashi, С. Bifurcations and the Generation of Chaotic States in the Solutions of Nonlinear Differential Еquations [Теxt]: Докл. Кн. 1/ С. Hayashi, H. Kawakami // Теорегическая и прикладная механика. – Варна, София, 1981 – С. 537–542.
  5. Hoppensteadt, F. Predator-prey model [Теxt] / F. Hoppensteadt // Scholarpedia. – 2006. – № 1(10). – 1563 с.
  6. Brauer, F. Mathematical Models in Population Biology and Epidemiology [Теxt] / F. Brauer, C. Castillo-Chavez, Springer-Verlag, 2000. – 201 p.
  7. Сорокин, П. А. Моделирование биологических популяций с использованием комплексных моделей, включающих в себя индивидуум-ориентированные и аналитические компоненты [Текст] : дис. ... канд. физ.-мат. наук / П. А. Сорокин. – Долгопрудный, 2004.– 153 c.
  8. Arditi, R. How Species Interact: Altering the Standard View on Trophic Ecology [Теxt] / R. Arditi, L. R. Ginzburg. – Oxford University Press, 2012. – 112 р.
  9. Гусятников, П. П. Качественные и численные методы в задачах оптимального управления в моделях хищник-жертва и популяции леммингов [Текст] : дис. ... канд. физ.-мат. наук [Текст] / П. П. Гусятников. – Москва, 2006.– 101 с.
  10. Nasritdinov, G. Limit cycle, trophic function and the dynamics of intersectoral interaction [Теxt] / G. Nasritdinov, R. T. Dalimov // Current Research J. of Economic Theory. – 2010. – № 2(2). – С. 32–40.
  11. Эрроусмит, Д. К. Обыкновенные дифференциальные уравнения. Качественная теория с приложениями [Текст] / Д. К. Эрроусмит, К. М. Плейнс. – М.: Мир, 1986. – 243 с.
  12. Арнольд, В. И. Дополнительные главы теории обыкновенных дифференциальных уравнений [Текст] / В. И. Арнольд. – М.: Наука, 1987. – 304 с.
  13. Vol`terra, V. (2004). Matematicheskaia teoriia bor`by` za sushchestvovanie. Moscow-Izhevsk, Russia. Institut komp`iuterny`kh issledovanii`, 288.
  14. Jost, C., Devulder, G., Vucetich, J. A., Peterson, R., Arditi, R. (2005). The wolves of Isle Royale display scale-invariant satiation and density dependent predation on moose. J. Anim. Ecol., 74 (5), 809–816.
  15. Marty`niuk, A. A., Nikitina, N. V. (1996). Haoticheskaia poteria predel`nogo tcicla v zadache Vol`terra Docl. AN Ukrainy, 4, 1–7.
  16. Hayashi, C., Kawakami, H. (1981). Bifurcations and the Generation of Chaotic States in the Solutions of Nonlinear Differential Еquations. Teoregicheskaia i pricladnaia mehanika, 537–542.
  17. Hoppensteadt, F. (2006). Predator-prey model. Scholarpedia, 1 (10), 1563.
  18. Brauer, F., Castillo-Chavez, C. (2000). Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, 201.
  19. Sorokin, P. A. (2004). Modelirovanie biologicheskikh populiatcii` s ispol`zovaniem kompleksny`kh modelei`, vcliuchaiushchikh v sebia individuum-orientirovanny`e i analiticheskie komponenty. Dolgoprudny`i`, Russia, 153.
  20. Arditi, R., Ginzburg, L. R. (2012). How Species Interact: Altering the Standard View on Trophic Ecology. Oxford University Press, 112.
  21. Gusiatneykov, P. P. (2006). Kachestvenny`e i chislenny`e metody` v zadachakh optimal`nogo upravleniia v modeliakh hishchnik-zhertva i populiatcii lemmingov, 101.
  22. Nasritdinov, G., Dalimov, R. T. (2010). Limit cycle, trophic function and the dynamics of intersectoral interaction. Current Research J. of Economic Theory, 2 (2), 32–40.
  23. E`rrousmit, D. K., Plei`s, K. M. (1986). Oby`knovenny`e differentcial`ny`e uravneniia. Kachestvennaia teoriia s prilozheniiami. Мoscow, Мir, 243.
  24. Arnold, V. I. (1987). Dopolnitel`ny`e glavy` teorii oby`knovenny`kh differentcial`ny`kh uravnenii`. Мoscow, Nauka. 304.

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Published

2014-04-09

How to Cite

Мохаммад, Р. А. А. А. (2014). Instability in dynamic balance of volterra-lotka systems with perturbations in the right side. Eastern-European Journal of Enterprise Technologies, 2(4(68), 47–50. https://doi.org/10.15587/1729-4061.2014.23138

Issue

Vol. 2 No. 4(68) (2014): Mathematics and Cybernetics - applied aspects

Section

Mathematics and Cybernetics - applied aspects

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Copyright (c) 2014 Ракан Абед Алнаби Альджаафрех Мохаммад

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