DOI: https://doi.org/10.15587/1729-4061.2015.37800

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

Валид Ахмед Альрефаи, Ракан Абед Алнаби Альджаафрех Мохаммад

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.


Keywords


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

References


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ISSN (print) 1729-3774, ISSN (on-line) 1729-4061