Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 509871, 19 pages
doi:10.1155/2011/509871
Research Article

The Dynamic Complexity of a Holling Type-IV Predator-Prey System with Stage Structure and Double Delays

Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China

Received 31 October 2010; Revised 17 January 2011; Accepted 28 February 2011

Academic Editor: Mingshu Peng

Copyright © 2011 Yakui Xue and Xiafeng Duan. 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

We invest a predator-prey model of Holling type-IV functional response with stage structure and double delays due to maturation time for both prey and predator. The dynamical behavior of the system is investigated from the point of view of stability switches aspects. We assume that the immature and mature individuals of each species are divided by a fixed age, and the mature predator only attacks the mature prey. Based on some comparison arguments, sharp threshold conditions which are both necessary and sufficient for the global stability of the equilibrium point of predator extinction are obtained. The most important outcome of this paper is that the variation of predator stage structure can affect the existence of the interior equilibrium point and drive the predator into extinction by changing the maturation (through-stage) time delay. Our linear stability work and numerical results show that if the resource is dynamic, as in nature, there is a window in maturation time delay parameters that generate sustainable oscillatory dynamics.