tailieunhanh - báo cáo hóa học:" Research Article Bifurcation Analysis for a Delayed Predator-Prey System with Stage Structure"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Bifurcation Analysis for a Delayed Predator-Prey System with Stage Structure | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 527864 14 pages doi 2010 527864 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 jzhsuper@ Received 9 August 2010 Revised 10 October 2010 Accepted 14 October 2010 Academic Editor Massimo Furi Copyright 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. x f x f r - ax f - ky2 f w 1 mx f J . _ kbx f y2 f y1 f 1 mx D D V1W f mx 1 2 f D 1 f - v2y2 f 2 Fixed Point Theory and .