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 ${x^p}/\left ( a + {x^p} \right )$, 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.
- Jun Ping Chen and Hong De Zhang, The qualitative analysis of two species predator-prey model with Holling’s type ${\rm III}$ functional response, Appl. Math. Mech. 7 (1986), no. 1, 73–80 (Chinese, with English summary); English transl., Appl. Math. Mech. (English Ed.) 7 (1986), no. 1, 77–86. MR 857154, DOI https://doi.org/10.1007/BF01896254
- Kuo Shung Cheng, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal. 12 (1981), no. 4, 541–548. MR 617713, DOI https://doi.org/10.1137/0512047
- Sun Hong Ding, On a kind of predator-prey system, SIAM J. Math. Anal. 20 (1989), no. 6, 1426–1435. MR 1019308, DOI https://doi.org/10.1137/0520092
- Xun Cheng Huang, Uniqueness of limit cycles of generalised Liénard systems and predator-prey systems, J. Phys. A 21 (1988), no. 13, L685–L691. MR 953455
- R. E. Kooij and A. Zegeling, Qualitative properties of two-dimensional predator-prey systems, Nonlinear Anal. 29 (1997), no. 6, 693–715. MR 1452753, DOI https://doi.org/10.1016/S0362-546X%2896%2900068-5
- Yang Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, Math. Biosci. 88 (1988), no. 1, 67–84. MR 930003, DOI https://doi.org/10.1016/0025-5564%2888%2990049-1
- Helmar Nunes Moreira, On Liénard’s equation and the uniqueness of limit cycles in predator-prey systems, J. Math. Biol. 28 (1990), no. 3, 341–354. MR 1047169, DOI https://doi.org/10.1007/BF00178782
- Jitsuro Sugie and Masaki Katayama, Global asymptotic stability of a predator-prey system of Holling type, Nonlinear Anal. 38 (1999), no. 1, Ser. B: Real World Appl., 105–121. MR 1693000, DOI https://doi.org/10.1016/S0362-546X%2899%2900099-1
- Jitsuro Sugie, Rie Kohno, and Rinko Miyazaki, On a predator-prey system of Holling type, Proc. Amer. Math. Soc. 125 (1997), no. 7, 2041–2050. MR 1396998, DOI https://doi.org/10.1090/S0002-9939-97-03901-4
- 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), no. 4, 85–90. MR 1415457, DOI https://doi.org/10.1016/0893-9659%2896%2900056-0
- 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 (1994), no. 3, 213–247. MR 1289041, DOI https://doi.org/10.1112/blms/26.3.213
- Zhi Fen Zhang, Proof of the uniqueness theorem of limit cycles of generalized Liénard equations, Appl. Anal. 23 (1986), no. 1-2, 63–76. MR 865184, DOI https://doi.org/10.1080/00036818608839631
- Zhi Fen Zhang and Su Zhi Gao, The problem of uniqueness of limit cycles for a class of nonlinear equations, Beijing Daxue Xuebao 1 (1986), 1–13 (Chinese, with English summary). MR 865031
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)
Kuo-Shung Cheng, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal. 12, 541–548 (1981)
Sun-Hong Ding, On a kind of predator-prey system, SIAM J. Math. Anal. 20, 1426–1435 (1989)
Xun-Cheng Huang, Uniqueness of limit cycles of generalised Liénard systems and predator-prey systems, J. Phys. A: Math. Gen. 21, L685–L691 (1988)
R. E. Kooij and A. Zegeling, Qualitative properties of two-dimensional predator-prey systems, Nonlinear Anal. 29, 693–715 (1997)
Yang Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey system, Math. Biosci. 88, 67–84 (1988)
H. N. Moreira, On Liénard’s equation and the uniqueness of limit cycles in predator-prey systems, J. Math. Biol. 28, 341–354 (1990)
J. Sugie and M. Katayama, Global asymptotic stability of a predator-prey system of Holling type, Nonlinear Anal. 38, 105–121 (1999)
J. Sugie, R. Kohno, and R. Miyazaki, On a predator-prey system of Holling type, Proc. Amer. Math. Soc. 125, 2041–2050 (1997)
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)
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)
Zhi-Fen Zhang, Proof of the uniqueness theorem of limit cycles of generalized Liénard equation, Appl. Anal. 23, 63–74 (1986)
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)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
92D25,
34C05,
34C60
Retrieve articles in all journals
with MSC:
92D25,
34C05,
34C60
Additional Information
Article copyright:
© Copyright 2000
American Mathematical Society