Journal of Applied Mathematics

Extinction and Permanence of a General Predator-Prey System with Impulsive Perturbations

Xianning Liu and Lansun Chen

Full-text: Open access

Abstract

A general predator-prey system is studied in a scheme where there is periodic impulsive perturbations. This scheme has the potential to protect the predator from extinction but under some conditions may also serve to lead to extinction of the prey. Conditions for extinction and permanence are obtained via the comparison methods involving monotone theory of impulsive systems and multiple Liapunov functions, which establish explicit bounds on solutions. The existence of a positive periodic solution is also studied by the bifurcation theory. Application is given to a Lotka-Volterra predator-prey system with periodic impulsive immigration of the predator. It is shown that the results are quite different from the corresponding system without impulsive immigration, where extinction of the prey can never be achieved. The prey will be extinct or permanent independent of whether the system without impulsive effect immigration is permanent or not. The model and its results suggest an approach of pest control which proves more effective than the classical one.

Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 521729, 19 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.jam/1355495250

Digital Object Identifier
doi:10.1155/2012/521729

Mathematical Reviews number (MathSciNet)
MR2959991

Zentralblatt MATH identifier
1255.34049

Citation

Liu, Xianning; Chen, Lansun. Extinction and Permanence of a General Predator-Prey System with Impulsive Perturbations. J. Appl. Math. 2012 (2012), Article ID 521729, 19 pages. doi:10.1155/2012/521729. https://projecteuclid.org/euclid.jam/1355495250


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