Developing methods for investigating stable motions in lotka-volterra systems with periodic perturbations

Authors

  • Валид Ахмед Альрефаи Kharkov State University of Radio-electronics 14, Lenin av., Kharkov, Ukraine, 61166, Ukraine
  • 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/10.15587/1729-4061.2015.37800

Keywords:

Lotka-Volterra model, model perturbations, stability problems, periodic solutions, attractor, limit cycle

Abstract

Destabilization effects of trophic coexistence of two populations, described by the Lotka-Volterra differential equation system at weak sinusoidal external influences on the reproduction rate were investigated. The stability of such a non-autonomous system was examined. Numerical solutions at frequencies of exposure close to the frequency of the cycle of the unperturbed system were found.

Such systems are soft classical models of many real objects in the ecology, economy and other areas, therefore their studies are relevant.

It is known that such systems of nonlinear equations with the perturbed right side generally can not be solved. Numerical experiment has allowed to reveal bifurcations when changing the amplitude n, and the perturbation period W. Trophic parameters of the unperturbed system, as it is known for the classical Lotka-Volterra system do not lead to bifurcations.

As a result of the research, it was found that the amplitude variations (within 1±0.05) lead to a transition of the system from periodic motions to sustainable growth, and then to chaotic oscillations. At the same time, Lyapunov exponents may have opposite signs. So bifurcation introduces an asymmetry and instability in the structure of the characteristic exponents, and trajectory "goes" to infinity. Herewith, both monotonous and chaotic types are possible.

Author Biographies

Валид Ахмед Альрефаи, Kharkov State University of Radio-electronics 14, Lenin av., Kharkov, Ukraine, 61166

postgraduate student

Department of Applied Mathematics

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

postgraduate Student
Department of Applied Mathematics

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Published

2015-02-27

How to Cite

Альрефаи, В. А., & Мохаммад, Р. А. А. А. (2015). Developing methods for investigating stable motions in lotka-volterra systems with periodic perturbations. Eastern-European Journal of Enterprise Technologies, 1(4(73), 58–61. https://doi.org/10.15587/1729-4061.2015.37800

Issue

Section

Mathematics and Cybernetics - applied aspects