Abstract
In this paper, we analyse qualitatively a cubic Kolmogorov system:\(\left\{ \begin{gathered} \frac{{dx}}{{dt}} = x[a_0 + a_1 x - a_3 x^2 - a_4 y + a_5 xy], \hfill \\ \frac{{dy}}{{dt}} = y(x - 1)(1 + by), \hfill \\ \end{gathered} \right.\) which is one of the mathematical models in ecology describing the interaction between Predator-Prey of two populations; and give the conditions of nonexistence, existence and uniqueness of limit cycles for three different cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen Lansun, Mathematical Ecologic and Method of Research, Science Press, Beijing, 1988 (in Chinese).
Zhang Jinyan, On the Problems of the Branch and the Geometrical Theory of Ordinary Differential Equation, Beijing University Press, 1981, 46–51; 189–190 (in Chinese).
Zhang Zhifen, The Uniqueness of Cycle for Nonlinear Oscillatory Equation,ДАН СССР,119: 4 (1958), 659–662 (in Russian).
Ye Yanqian and Ye Weiyin, Cubic Kolmogorov Differential System with Two Limit Cycles Surrounding the Same Focus,Ann. of Diff. Eqs.,1 (2) (1985), 201–207.
Author information
Authors and Affiliations
Additional information
Fulfilled during engagement in advanced studies at the Institute of Mathematics, Academia Sinica.
Rights and permissions
About this article
Cite this article
Gan, W. Existence and uniqueness of limit cycles on a cubic kolmogorov differential system in the Predator-Prey relation. Acta Mathematicae Applicatae Sinica 4, 245–256 (1988). https://doi.org/10.1007/BF02006221
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02006221