One and three limit cycles in a cubic predator-prey system

Mathematical Methods in the Applied Sciences

Volumn 30, Issue 5, 2007, Pages 501-511

  Huang, Xuncheng ^a   Wang, Yuanming ^a   Zhu, Lemin ^a  

^a Yangzhou Polytechnic College   (China)

Author keywords

Cubic differential system; Predator prey models; Three limit cycles

Indexed keywords

MATHEMATICAL MODELS; OPTIMIZATION;

CUBIC DIFFERENTIAL SYSTEMS; LIMIT CYCLES; PREDATOR-PREY SYSTEM; THREE LIMIT CYCLES;

DIFFERENTIAL EQUATIONS;

EID: 33847746696     PISSN: 01704214     EISSN: 10991476     Source Type: Journal    
DOI: 10.1002/mma.791     Document Type: Article

References (18)

1. Ye YQ, Wang SL, Yang XA. Theory of Limit Cycles. AMS: Providence, RI, 1986.
2. Iliashenko IS, Yakovenko S, Iliashenko Y, Iakovenko SB (eds). Concerning the Hilbert 16th problem. AMS Translations Series 2, Providence, RI, 1995.
3. Roussaire R. Bifurcations of Planar Vector Fields and Hilbert's Sixteen Problem. Springer: New York, 1998.
4. Gavrilov L. The infinitesimal 16th Hilbert problem in the quadratic case. Inventiones Mathematicae 2001; 143:449-497.
5. Li BY, Zhang ZF. A note on a result of G. S. Petro about the weakened 16th Hilbert problem. Journal of Mathematical Analysis and Applications 2001; 190:266-271.
6. Kuno E. Mathematical models for predator-prey interaction. Advances in Ecological Research 1987; 16:252-265.
7. Zheng JB, Yu ZX, Sun JT. Existence and uniqueness of limit cycle about prey-predator system with sparssing effect. Journal of Biomathematics 2001; 16(2): 156-161.
8. Shen C, Shen BQ. A necessary and sufficient condition of the existence and uniqueness of the limit cycles for a class of prey-predator system with sparssing effect. Journal of Biomathematics 2003; 18(2):207-210.
9. May RM. Limit cycles in predator-prey communities. Science 1972; 177:900-902.
10. May RM. Stability and Complexity in Model Ecosystems. Princeton University Press: Princeton, NJ, 1974.
11. Cronin J. Differential Equations: Introduction and Qualitative Theory. Marcel Dekker: New York, 1994.
12. Cherkas LA, Zhilevich LI. Some criteria for the absence of limit cycles and for the existence of a single limit cycle. Differential Equations 1970; 6:891-897.
13. Huang X, Zhu L. Limit cycles in a general Kolmogorov model. Nonlinear Analysis: Theory, Methods and Applications 2005; 60(8):1393-1414.
14. Huang X, Zhu L. A study of a general Kolmogorov model. Journal of Yangzhou Polytechnic University 2004; 8(1):20-32.
15. Huang XC. Existence of more limit cycles in general predator-prey models. Journal of Physics A - Mathematical and General 1989; 22:L61-L66.
16. Huang XC, Merrill SJ. Conditions for uniqueness of limit cycles in general predator-prey system. Mathematical Biosciences 1989; 96:47-60.
17. Huang XC. Stability of a general predator-prey model. Journal of Franklin Institute 1990; 327(5):751-769.
18. Huang XC, Zhu LM. A three-dimensional chemostat with quadratic yields. Journal of Mathematical Chemistry 2005; 38(3):399-412.