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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 521729, 19 pages
http://dx.doi.org/10.1155/2012/521729
Research Article

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

1Key Laboratory of Eco-Environments in Three Gorges Reservoir Region (Ministry of Education), School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2Institute of Mathematics, Academy of Mathematics and System Sciences, Beijing 100080, China

Received 23 February 2012; Accepted 18 May 2012

Academic Editor: A. A. Soliman

Copyright © 2012 Xianning Liu and Lansun Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.