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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, 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, (), 538-543 · Zbl 0938.34515
[9] A. Gasull and A. Guillamon, On the periodic solutions of a generalized Lienard equation with applications to predator-prey systems (preprint). · Zbl 0876.34035
[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
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