zbMATH — the first resource for mathematics

Extinction and permanence of the predator-prey system with stocking of prey and harvesting of predator impulsively. (English) Zbl 1086.92051
Summary: A predator-prey system with stocking of prey and harvesting of predator impulsively is studied. Here, the prey population is stocked with a constant quantity and the predator population is harvested at a rate proportional to the species itself at fixed moments. Under some conditions, the existence and global asymptotic stability of the boundary periodic solution are proved, which implies that the system will be extinct; and given some different restrictions, ultimate positive upper and lower bounds of all solutions are obtained, showing the system being permanent. At last, two examples are given to illustrate our results.

92D40 Ecology
34C25 Periodic solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
34A37 Ordinary differential equations with impulses
34D05 Asymptotic properties of solutions to ordinary differential equations
Full Text: DOI
[1] Goh, Journal of Mathematical Biology 3 pp 313– (1976) · Zbl 0362.92013
[2] Agur, Proceedings of the National Academy of Sciences, U.S.A. 90 pp 11689– (1993)
[3] Shulgin, Bulletin of Mathematical Biology 60 pp 1– (1998)
[4] Panetta, Bulletin of Mathematical Biology 58 pp 425– (1996)
[5] Lakmeche, Dynamics of Continuous, Discrete and Impulsive Systems 7 pp 265– (2000)
[6] Lakmeche, Nonlinear Analysis (RWA) 2 pp 455– (2001)
[7] . System with Impulse Effect. Ellis Horwood: Chichester, U.K., 1982.
[8] . Impulsive Differential Equations: Periodic Solutions and Applications. Longman: England, 1993.
[9] et al. Theory of Impulsive Differential Equations. World Scientific: Singapore, 1989.
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.