Bifurcation Analysis for a Delayed Predator-Prey System with Stage Structure
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Research Article Bifurcation Analysis for a Delayed Predator-Prey System with Stage Structure Zhichao Jiang and Guangtao Cheng Fundamental Science Department, North China Institute of Astronautic Engineering, Langfang Hebei 065000, China Correspondence should be addressed to Zhichao Jiang, [email protected] Received 9 August 2010; Revised 10 October 2010; Accepted 14 October 2010 Academic Editor: Massimo Furi Copyright q 2010 Z. Jiang and G. Cheng. 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. A delayed predator-prey system with stage structure is investigated. The existence and stability of equilibria are obtained. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the normal form and the center manifold theory. Finally, a numerical example supporting the theoretical analysis is given.
1. Introduction The age factors are important for the dynamics and evolution of many mammals. The rates of survival, growth, and reproduction almost always depend heavily on age or developmental stage, and it has been noticed that the life history of many species is composed of at least two stages, immature and mature, with significantly different morphological and behavioral characteristics. The study of stage-structured predator-prey systems has attracted considerable attention in recent years see 1–6 and the reference therein. In 4, Wang considered the following predator-prey model with stage structure for predator, in which the immature predators can neither hunt nor reproduce. xt ˙ xt r − axt − y˙ 1 t
by2 t , 1 mxt
kbxty2 t − D v1 y1 t, 1 mxt
y˙ 2 t Dy1 t − v2 y2 t,
1.1
2
Fixed Point Theory and Applications
where xt denotes the density of prey at time t, y1 t denotes the density of immature predator at time t, y2 t denotes the density of mature predator at time t, b is the search rate, m is the search rate multiplied by the handling time, and r is the intrinsic growth rate. It is assumed that the reproduction rate of the mature predator depends on the quality of prey considered, the efficiency of conversion of prey into newborn immature predators being denoted by k. D denotes the rate at which immature predators become mature predators. v1 and v2 denote the mortality rates of immature and mature predators, respectively. All coefficients are positive constants. In 4, he concluded that the system under some conditions has a unique positive equilibrium, which is globally asymptotically stable. Georgescu and Moros¸anu 7 generalized the system 1.1 as xt ˙ nxt − fxty2 t, 1.2
y˙ 1 t kfxty2 t − D v1 y1 t, y˙ 2 t Dy1 t − v2 y2 t, satisfying the following hypotheses: H1 a fx is the predator functional response and satisfies that
f ∈ C1 0, ∞, 0, ∞,
f0
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