Uniqueness of limit cycles in a predator-prey system with Holling-type functional response

Author:
Jitsuro Sugie

Journal:
Quart. Appl. Math. **58** (2000), 577-590

MSC:
Primary 92D25; Secondary 34C05, 34C60

DOI:
https://doi.org/10.1090/qam/1770656

MathSciNet review:
MR1770656

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the problem of uniqueness of limit cycles in a predator-prey system with Holling's functional response , where and are positive parameters. The problem has not yet been settled only in the case . This paper gives a sufficient condition under which the predator-prey system with has exactly one limit cycle by using a result of Zhang and Gao. Finally, the fact that our condition is also necessary is mentioned.

**[1]**Jun-Ping Chen and Hong-De Zhang,*The qualitative analysis of two species predator-prey model with Holling's type*III*functional response*, Appl. Math. Mech.**7**, 77-86 (1986) MR**857154****[2]**Kuo-Shung Cheng,*Uniqueness of a limit cycle for a predator-prey system*, SIAM J. Math. Anal.**12**, 541-548 (1981) MR**617713****[3]**Sun-Hong Ding,*On a kind of predator-prey system*, SIAM J. Math. Anal.**20**, 1426-1435 (1989) MR**1019308****[4]**Xun-Cheng Huang,*Uniqueness of limit cycles of generalised Liénard systems and predator-prey systems*, J. Phys. A: Math. Gen.**21**, L685-L691 (1988) MR**953455****[5]**R. E. Kooij and A. Zegeling,*Qualitative properties of two-dimensional predator-prey systems*, Nonlinear Anal.**29**, 693-715 (1997) MR**1452753****[6]**Yang Kuang and H. I. Freedman,*Uniqueness of limit cycles in Gause-type models of predator-prey system*, Math. Biosci.**88**, 67-84 (1988) MR**930003****[7]**H. N. Moreira,*On Liénard's equation and the uniqueness of limit cycles in predator-prey systems*, J. Math. Biol.**28**, 341-354 (1990) MR**1047169****[8]**J. Sugie and M. Katayama,*Global asymptotic stability of a predator-prey system of Holling type*, Nonlinear Anal.**38**, 105-121 (1999) MR**1693000****[9]**J. Sugie, R. Kohno, and R. Miyazaki,*On a predator-prey system of Holling type*, Proc. Amer. Math. Soc.**125**, 2041-2050 (1997) MR**1396998****[10]**J. Sugie, K. Miyamoto, and K. Morino,*Absence of limit cycles of a predator-prey system with a sigmoid functional response*, Appl. Math. Lett.**9**, 85-90 (1996) MR**1415457****[11]**Xian-Wu Zeng, Zhi-Fen Zhang, and Su-Zhi Gao,*On the uniqueness of the limit cycle of the generalized Liénard equation*, Bull. London Math. Soc.**26**, 213-247 (1994) MR**1289041****[12]**Zhi-Fen Zhang,*Proof of the uniqueness theorem of limit cycles of generalized Liénard equation*, Appl. Anal.**23**, 63-74 (1986) MR**865184****[13]**Zhi-Fen Zhang and Su-Zhi Gao,*On the uniqueness of the limit cycle of Liénard equation*, Acta Math. Peking Univ.**22**, 1-13 (1986) MR**865031**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
92D25,
34C05,
34C60

Retrieve articles in all journals with MSC: 92D25, 34C05, 34C60

Additional Information

DOI:
https://doi.org/10.1090/qam/1770656

Article copyright:
© Copyright 2000
American Mathematical Society