Limit cycles in a cubic Kolmogorov system with harvest and two positive equilibrium points. (English) Zbl 1474.92136

Zhang, Qi-Ming; Li, Feng; Zhao, Yulin

Abstr. Appl. Anal. 2014, Article ID 786962, 6 p. (2014).

Summary: A class of planar cubic Kolmogorov systems with harvest and two positive equilibrium points is investigated. With the help of computer algebra system MATHEMATICA, we prove that five limit cycles can be bifurcated simultaneously from the two critical points (1, 1) and (2, 2), respectively, in the first quadrant. Moreover, the necessary conditions of centers are obtained.

MSC:

92D40	Ecology
92-08	Computational methods for problems pertaining to biology
34C23	Bifurcation theory for ordinary differential equations
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations

Software:

Mathematica

References:

[1]	Liu, X. S.; Zhuo, Z. H., Existence and uniqueness of limit cycles of a cubic Kolmogorov differential system, Biomathematics, 3, 266-271, (2001)
[2]	Huang, X. C.; Zhu, L., Limit cycles in a general Kolmogorov model, Nonlinear Analysis: Theory, Methods and Applications, 60, 8, 1393-1414, (2005) · Zbl 1074.34035
[3]	Ye, Y.; Ye, W., Cubic Kolmogorov differential systems with two limit cycles surrounding the same focus, Annals of Differential Equations, 2, 201-207, (1985) · Zbl 0597.34020
[4]	Saez, E.; Szanto, I., Limit cycle s of a cubic Kolmogorov system, Applied Mathematics Letters, 9, 15-18, (1996) · Zbl 0858.34023
[5]	F. Li, Integrability and bifurcations of limit cycles in a cubic Kolmogorov system, International Journal of Bifurcation and Chaos, 23, 4, (2013) · Zbl 1270.34043
[6]	Du, C.; Huang, W., Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model, Nonlinear Dynamics, 72, 1-2, 197-206, (2013) · Zbl 1269.92065
[7]	Du, C.; Wang, Q.; Huang, W., Three-dimensional Hopf bifurcation for a class of cubic Kolmogorov model, International Journal of Bifurcation and Chaos, 24, 3, (2014) · Zbl 1296.34096
[8]	Du, C.; Liu, Y.; Huang, W., Limit cycles bifurcations for a class of Kolmogorov model in symmetrical vector field, International Journal of Bifurcation and Chaos, 24, 3, (2014) · Zbl 1296.34095
[9]	Liu, Y.; Li, J., Theory of values of singular point in complex autonomous differential system, Science China A, 3, 245-255, (1989)