On a predatorprey system of Holling type
Authors:
Jitsuro Sugie, Rie Kohno and Rinko Miyazaki
Journal:
Proc. Amer. Math. Soc. 125 (1997), 20412050
MSC (1991):
Primary 34C05, 92D25; Secondary 58F21, 70K10
MathSciNet review:
1396998
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: We consider the predatorprey 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.
Kuo
Shung Cheng, Uniqueness of a limit cycle for a predatorprey
system, SIAM J. Math. Anal. 12 (1981), no. 4,
541–548. MR
617713 (82h:34035), http://dx.doi.org/10.1137/0512047
 2.
Sun
Hong Ding, On a kind of predatorprey system, SIAM J. Math.
Anal. 20 (1989), no. 6, 1426–1435. MR 1019308
(91f:92018), http://dx.doi.org/10.1137/0520092
 3.
Herbert
I. Freedman, Deterministic mathematical models in population
ecology, Monographs and Textbooks in Pure and Applied Mathematics,
vol. 57, Marcel Dekker, Inc., New York, 1980. MR 586941
(83h:92043)
 4.
A. Gasull and A. Guillamon, Nonexistence of limit cycles for some predatorprey systems, Proceedings of Equadiff' 91, pp. 538543, 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), 160.
 6.
Xun
Cheng Huang, Uniqueness of limit cycles of generalised
Liénard systems and predatorprey systems, J. Phys. A
21 (1988), no. 13, L685–L691. MR 953455
(89i:92043)
 7.
Y.
Kuang, Global stability of Gausetype predatorprey systems,
J. Math. Biol. 28 (1990), no. 4, 463–474. MR 1057049
(91g:92017), http://dx.doi.org/10.1007/BF00178329
 8.
Yang
Kuang and H.
I. Freedman, Uniqueness of limit cycles in Gausetype models of
predatorprey systems, Math. Biosci. 88 (1988),
no. 1, 67–84. MR 930003
(89g:92045), http://dx.doi.org/10.1016/00255564(88)900491
 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), 481485.
 11.
Jitsuro
Sugie and Tadayuki
Hara, Nonexistence of periodic solutions of the Liénard
system, J. Math. Anal. Appl. 159 (1991), no. 1,
224–236. MR 1119432
(92m:34095), http://dx.doi.org/10.1016/0022247X(91)90232O
 12.
J. Sugie, K. Miyamoto and K. Morino, Absence of limit cycles of a predatorprey system with a sigmoid functional response, Appl. Math. Lett. 9 (1996), 8590. CMP 97:03
 1.
 K.S. Cheng, Uniqueness of a limit cycle for a predatorprey system, SIAM J. Math. Anal. 12 (1981), 541 548. MR 82h:34035
 2.
 S.H. Ding, On a kind of predatorprey system, SIAM J. Math. Anal. 20 (1989), 14261435. 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, Nonexistence of limit cycles for some predatorprey systems, Proceedings of Equadiff' 91, pp. 538543, 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), 160.
 6.
 X.C. Huang, Uniqueness of limit cycles of generalised Liénard systems and predatorprey systems, J. Phys. A: Math. Gen. 21 (1988), L685L691.MR 89i:92043
 7.
 Y. Kuang, Global stability of Gausetype predatorprey systems, J. Math. Biol. 28 (1990), 463474. MR 91g:92017
 8.
 Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gausetype models of predatorprey system, Math. Biosci. 88 (1988), 6784. 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), 481485.
 11.
 J. Sugie and T. Hara, Nonexistence of periodic solutions of the Liénard system, J. Math. Anal. Appl. 159 (1991), 224236. MR 92m:34095
 12.
 J. Sugie, K. Miyamoto and K. Morino, Absence of limit cycles of a predatorprey system with a sigmoid functional response, Appl. Math. Lett. 9 (1996), 8590. 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.shimaneu.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.osakafuu.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002993997039014
PII:
S 00029939(97)039014
Keywords:
Limit cycles,
global asymptotic stability,
predatorprey system,
functional response
Received by editor(s):
January 25, 1996
Additional Notes:
The first author was supported in part by GrantinAid for Scientific Research 06804008.
Communicated by:
Hal L. Smith
Article copyright:
© Copyright 1997
American Mathematical Society
