Explicit limit cycles of a class of Kolmogorov system

Year 2018, Volume: 1 Issue: 3, 148 - 154, 30.09.2018

Salah Benyoucef , Ahmed Bendjeddou

https://doi.org/10.32323/ujma.425056

Abstract

A class of Kolmogorov differential system is introduced. It is shown that under suitable assumptions on degrees and parameters, algebraic limit cycles can occur. we propose an easy algorithm to test the existence of limit cycles and we give them explicit expressions.

Keywords

Kolmogorov differential system, Invariant curve, Periodic solution, Hyperbolic limit cycle

References

• [1] A. Bendjeddou and R. Cheurfa, On the exact limit cycle for some class of planar differential systems, Nonlinear differ. equ. appl. 14 (2007), 491-498.
• [2] A. Bendjeddou and R. Cheurfa, Cubic and quartic planar differential systems with exact algebraic limit cycles, Elect. J. of Diff. Equ., no15 (2011), 1-12.
• [3] S. Benyoucef, A. Barbach, and A. Bendjeddou, A class of Differential system with at most four limit cycles, Annals of applied mathematics, 31, no 4, 2015, 1-9.
• [4] S. Benyoucef and A. Bendjeddou, A class of Kolmogorov system with exact algebraic limit cycles, Int.J.of Pure and Applied Mathematics, V103 no 3, 2015, 439-451.
• [5] Shen Boqian and Liu Demeng. Existence of limit cycles for a cubic Kolmogorov system with a hyperbolic solution. Northwest Math.16(1), 2000, 91–95.
• [6] Cheng K.S, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal, 12 (4) (1981),541-548.
• [7] SI Chengbin, Shen Boqian. The existence of limit cycles for the Kolmogorov cubic system with a quartic curve solution.J.Sys. Sci.& Math. Scis.28(3) (2008), 334–339.
• [8] H. Giacomini, M. Grau, On the stability of limit cycles for planar differential systems, J. of Diff. Equ, v 213 issue 2, 2005, 368-388.
• [9] Xun C. Huang and Lemin Zhu, Limit cycles in a general Kolmogorov model, Nonlin. Anal. Theo. Meth. and Appl. 60 (2005), 1393-1414.
• [10] Huang X.C, Limit cycle in a Kolmogorov-type model, Internat. J. Math. & Math Sci.vol 13 no 3 (1990) 555-566.
• [11] X. Huang, Y. Wang, A. Cheng, Limit cycles in a cubic predator–prey differential system, J. Korean Math. Soc. 43 no 4 (2006) 829–843.
• [12] Y. Kuang and H.I Freedman, Uniqueness of limit cycles in Gause-type models of Predator-prey systems, Math. Biosci.. 88 (1988), 67-84.
• [13] N. G. Lloyd, J. M. Pearson, E S´aez, I. Sz´ant´o, Limit cycles of a Cubic Kolmogorov System, Appl. Math. Lett. vol 9 no1, (1996) pp 15-18.
• [14] N. G. Lloyd, J. M. Pearson, E. S´aez, and I. Sz´ant´o, A cubic Kolmogorov system with six limit cycles, International Journal Computers and Mathematics with Applications 44 (2002), 445-455.
• [15] R.M May , Limit cycles in predator-prey communities, Science 177 (1972), 900-902.
• [16] L. Perko, Differential equations and dynamical systems, Third edition. Texts in Applied Mathematics, 7. Springer-Verlag, New York, 2001.
• [17] Peng Yue-hui. Limit Cycles in a Class of Kolmogorov Model with Two Positive equilibrium Points. Natural Science journal of Xiangtan University,Vol. 32 No.4 Dec.2010, 10-15.