Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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
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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 $ {x^p}/\left( a + {x^p} \right)$, where $ a$ and $ p$ are positive parameters. The problem has not yet been settled only in the case $ 1 < p < 2$. This paper gives a sufficient condition under which the predator-prey system with $ 1 < \\ p < 2$ has exactly one limit cycle by using a result of Zhang and Gao. Finally, the fact that our condition is also necessary is mentioned.


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  • [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

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Additional Information

DOI: https://doi.org/10.1090/qam/1770656
Article copyright: © Copyright 2000 American Mathematical Society

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