Absence of limit cycles of a predator-prey system with a sigmoid functional response. (English) Zbl 0865.34032

Summary: A large number of studies have been made on the predator-prey system with Holling’s functional response, namely, \(\phi(x)=x^n/(a+x^n)\) \((n=1,2)\). This paper presents a sufficient condition under which the predator-prey system has no limit cycles for \(n=3\). The argument used here is based on a result of Liénard dynamics. The relation between previous results \((n=1,2)\) and our result \((n=3)\) is cleared. Some phase portraits of trajectories of the predator-prey system are also given as an example of our result.


34C25 Periodic solutions to ordinary differential equations
92D25 Population dynamics (general)
Full Text: DOI


[1] Holling, C. S., The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 45, 1-60 (1965)
[2] Kazarinoff, N. D.; van den Driessche, P., A model pedator-prey system with functional response, Math. Biosci., 39, 125-134 (1978) · Zbl 0382.92007
[3] May, R., Stability and Complexity in Model Ecosystems (1974), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ
[4] Real, L. A., The kinetics of functional response, Amer. Natur., 111, 289-300 (1977)
[5] Real, L. A., Ecological determinants of functional response, Ecology, 60, 481-485 (1979)
[6] Cheng, K.-S., Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 12, 541-548 (1981) · Zbl 0471.92021
[7] Ding, S.-H., On a kind of pedator-prey system, SIAM J. Math. Anal., 20, 1426-1435 (1989) · Zbl 0678.92013
[8] Gasull, A.; Guillamon, A., Non-existence of limit cycles for some predator-prey systems, (Proceedings of Equadiff’91 (1993), World Scientific: World Scientific Singapore), 538-543 · Zbl 0938.34515
[10] Huang, X.-C., Uniqueness of limit cycles of generalised Liénard systems and predator-prey systems, J. Phys. A: Math. Gen., 21, L685-L691 (1988) · Zbl 0661.34028
[11] Kuang, Y.; Freedman, H. I., Uniqueness of limit cycles in Guase-type models of predator-prey systems, Math. Biosci, 88, 67-84 (1988) · Zbl 0642.92016
[12] Sugie, J.; Hara, T., Non-existence of periodic solutions of the Liénard system, J. Math Anal. Appl., 159, 224-236 (1991) · Zbl 0731.34042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.