On a predator-prey system of Holling type

Authors:
Jitsuro Sugie, Rie Kohno and Rinko Miyazaki

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2041-2050

MSC (1991):
Primary 34C05, 92D25; Secondary 58F21, 70K10

DOI:
https://doi.org/10.1090/S0002-9939-97-03901-4

MathSciNet review:
1396998

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the predator-prey system with a fairly general functional response of Holling type and give a necessary and sufficient condition under which this system has exactly one stable limit cycle. Our result extends previous results and is an answer to a conjecture which was recently presented by Sugie, Miyamoto and Morino.

**1.**K.-S. Cheng,*Uniqueness of a limit cycle for a predator-prey system*, SIAM J. Math. Anal.**12**(1981), 541- 548. MR**82h:34035****2.**S.-H. Ding,*On a kind of predator-prey system*, SIAM J. Math. Anal.**20**(1989), 1426-1435. MR**91f:92018****3.**H. I. Freedman,*Deterministic Mathematical Models in Population Ecology*, Marcel Dekker, New York, 1980. MR**83h:92043****4.**A. Gasull and A. Guillamon,*Non-existence of limit cycles for some predator-prey systems*, Proceedings of Equadiff' 91, pp. 538-543, World Scientific, Singapore, 1993. CMP**94:02****5.**C. S. Holling,*The functional response of predators to prey density and its role in mimicry and population regulation*, Mem. Ent. Soc. Can.**45**(1965), 1-60.**6.**X.-C. Huang,*Uniqueness of limit cycles of generalised Liénard systems and predator-prey systems*, J. Phys. A: Math. Gen.**21**(1988), L685-L691.MR**89i:92043****7.**Y. Kuang,*Global stability of Gause-type predator-prey systems*, J. Math. Biol.**28**(1990), 463-474. MR**91g:92017****8.**Y. Kuang and H. I. Freedman,*Uniqueness of limit cycles in Gause-type models of predator-prey system*, Math. Biosci.**88**(1988), 67-84. MR**89g:92045****9.**R. May,*Stability and Complexity in Model Ecosystems*, 2nd ed., Princeton Univ. Press, Princeton, 1974.**10.**L. A. Real,*Ecological determinants of functional response*, Ecology**60**(1979), 481-485.**11.**J. Sugie and T. Hara,*Non-existence of periodic solutions of the Liénard system*, J. Math. Anal. Appl.**159**(1991), 224-236. MR**92m:34095****12.**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**(1996), 85-90. CMP**97:03**

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Additional Information

**Jitsuro Sugie**

Affiliation:
Department of Mathematics, Faculty of Science, Shinshu University, Matsumoto 390, Japan

Address at time of publication:
Department of Mathematics and Computer Science, Shimane University Matsue 690, Japan

Email:
jsugie@riko.shimane-u.ac.jp

**Rie Kohno**

Affiliation:
Department of Mathematics, Faculty of Science, Shinshu University, Matsumoto 390, Japan

**Rinko Miyazaki**

Affiliation:
Department of Mathematical Sciences, Osaka Prefecture University, Sakai 593, Japan

Email:
rinko@ms.osakafu-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-97-03901-4

Keywords:
Limit cycles,
global asymptotic stability,
predator-prey system,
functional response

Received by editor(s):
January 25, 1996

Additional Notes:
The first author was supported in part by Grant-in-Aid for Scientific Research 06804008.

Communicated by:
Hal L. Smith

Article copyright:
© Copyright 1997
American Mathematical Society