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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 152459, 19 pages
http://dx.doi.org/10.1155/2013/152459
Research Article

Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional Response

1Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China
2Department of Mathematics, Tongji University, Shanghai 200092, China

Received 21 December 2012; Accepted 7 April 2013

Academic Editor: Kunquan Lan

Copyright © 2013 Jiao Jiang and Yongli Song. 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 delayed Leslie-Gower predator-prey model with nonmonotonic functional response is studied. The existence and local stability of the positive equilibrium of the system with or without delay are completely determined in the parameter plane. Using the method of upper and lower solutions and monotone iterative scheme, a sufficient condition independent of delay for the global stability of the positive equilibrium is obtained. Hopf bifurcations induced by the ratio of the intrinsic growth rates of the predator and prey and by delay, respectively, are found. Employing the normal form theory, the direction and stability of Hopf bifurcations can be explicitly determined by the parameters of the system. Some numerical simulations are given to support and extend our theoretical results. Two limit cycles enclosing an equilibrium, one limit cycle enclosing three equilibria and different types of heteroclinic orbits such as connecting two equilibria and connecting a limit cycle and an equilibrium are also found by using analytic and numerical methods.