Geodesic
Kolmogorov system with explicit hyperbolic limit cycle
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 10 (2017) no. 2, pp. 216-222.

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A class of Kolmogorov differential system is introduced.We show that under suitable assumptions on parameters, an algebraic hyprbolic limit cycle can occur, the explicit expression of this limit cycle is given.
Keywords: Kolmogorov differential system, invariant curve, singular point, periodic solution, algebraic limit cycle. 
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Salah Benyoucef; Ahmed Bendjeddou. Kolmogorov system with explicit hyperbolic limit cycle. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 10 (2017) no. 2, pp. 216-222. http://geodesic.mathdoc.fr/item/JSFU_2017_10_2_a8/

[1] A. Bendjeddou, R. Cheurfa, “On the exact limit cycle for some class of planar differential systems”, Nonlinear differ. equ. appl., 14 (2007), 491–498 | DOI | MR | Zbl

[2] A. Bendjeddou, R. Cheurfa, “Cubic and quartic planar differential systems with exact algebraic limit cycles”, Elect. J. of Diff. Equ., 15 (2011), 1–12 | MR

[3] S. Benyoucef, A. Bendjeddou, “A class of Kolmogorov system with exact limit cycles”, International journal of pure and applied mathematics, 103:4 (2015), 439–451 | MR

[4] Shen Boqian, Liu Demeng, “Existence of limit cycles for a cubic Kolmogorov system with an hyperbolic solution”, Northwest Math., 16:1 (2000), 91–95 | MR | Zbl

[5] K. S. Cheng, “Uniqueness of a limit cycle for a predator-prey system”, SIAM J. Math. Anal., 12:4 (1981), 541–548 | DOI | MR | Zbl

[6] S. I. 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 | MR | Zbl

[7] H. Giacomini, M. Grau, “On the stability of limit cycles for planar differential systems”, J. of Diff. Equ., 213:2 (2005), 368–388 | DOI | MR | Zbl

[8] X. C. Huang, “Limit cycle in a Kolmogorov-type model”, Internat. J. Math. Math Sci., 13:3 (1990), 555–566 | DOI | MR | Zbl

[9] X. Huang, Y. Wang, A. Cheng, “Limit cycles in a cubic predator–prey differential system”, J. Korean Math. Soc., 43:4 (2006), 829–843 | DOI | MR | Zbl

[10] X. C. Huang, Lemin Zhu, “Limit cycles in a general Kolmogorov model”, Nonlin. Anal. Theo. Meth. and Appl., 60 (2005), 1393–1414 | DOI | MR | Zbl

[11] N. G. Lloyd, J. M. Pearson, E. Sáez, I. Szántó, “Limit cycles of a Cubic Kolmogorov System”, Appl. Math. Lett., 9:1 (1996), 15–18 | DOI | MR | Zbl

[12] N. G. Lloyd, J. M. Pearson, E. Sáez, I. Szántó, “A cubic Kolmogorov system with six limit cycles”, International Journal Computers and Mathematics with Applications, 44 (2002), 445–455 | DOI | MR | Zbl

[13] R. M. May, “Limit cycles in predator-prey communities”, Science, 177 (1972), 900–902 | DOI

[14] L. Perko, Differential equations and dynamical systems, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 2001 | DOI | MR | Zbl

[15] Peng Yue-hui, “Limit Cycles in a Class of Kolmogorov Model with Two Positive equilibrium Points”, Natural Science journal of Xiangtan University, 32:4 (2010), 10–15