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Discrete Dynamics in Nature and Society
Volume 2013, Article ID 412409, 8 pages
Research Article

Modeling the Dynamics of a Single-Species Model with Pollution Treatment in a Polluted Environment

1College of Mathematics and Information Science, Anshan Normal University, Anshan, Liaoning 114007, China
2Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
3Department of Mathematics, Beihang University, Beijing 100083, China

Received 19 November 2012; Accepted 25 December 2012

Academic Editor: Zhen Jin

Copyright © 2013 Bing Liu et al. 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.


Without any question, environmental pollution is the main cause for the species extinction in recent times. In this paper, based on impulsive differential equation, the dynamics of a single-species model with impulsive pollution treatment at fixed time in a polluted environment is considered, in which we assume that the species is directly affected by the pollutants. Sufficient conditions for permanence and extinction of the species are given. The results show that the species is permanent when the impulsive period is less than some critical value, otherwise the species will be extinct. Although shortening the impulsive period can protect the species from extinction, it is expensive. To see how pollution treatment applications could be economical, we also establish a hybrid impulsive model involving periodic pollution treatment at fixed time with state-dependent pollution treatment applied when the pollution concentration reaches the given Environment Threshold (ET). It indicates that the hybrid method is the most effective method to protect the species from extinction. Numerical simulations confirm our theoretical results.