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Journal of Applied Mathematics
Volume 2014, Article ID 978758, 10 pages
http://dx.doi.org/10.1155/2014/978758
Research Article

Stability Analysis of a Ratio-Dependent Predator-Prey Model Incorporating a Prey Refuge

1School of Mathematics and Statistics, Hebei University of Economics & Business, Shijiazhuang 050061, China
2Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China

Received 24 November 2013; Revised 11 February 2014; Accepted 12 February 2014; Published 23 March 2014

Academic Editor: Shan Zhao

Copyright © 2014 Lingshu Wang and Guanghui Feng. 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 ratio-dependent predator-prey model incorporating a prey refuge with disease in the prey population is formulated and analyzed. The effects of time delay due to the gestation of the predator and stage structure for the predator on the dynamics of the system are concerned. By analyzing the corresponding characteristic equations, the local stability of a predator-extinction equilibrium and a coexistence equilibrium of the system is discussed, respectively. Further, it is proved that the system undergoes a Hopf bifurcation at the coexistence equilibrium, when . By comparison arguments, sufficient conditions are obtained for the global stability of the predator-extinction equilibrium. By using an iteration technique, sufficient conditions are derived for the global attractivity of the coexistence equilibrium of the proposed system.