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báo cáo hóa học:" Research Article The Permanence and Extinction of a Discrete Predator-Prey System with Time Delay and Feedback Controls"

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 The Permanence and Extinction of a Discrete Predator-Prey System with Time Delay and Feedback Controls | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 738306 20 pages doi 10.1155 2010 738306 Research Article The Permanence and Extinction of a Discrete Predator-Prey System with Time Delay and Feedback Controls Qiuying Li Hanwu Liu and Fengqin Zhang Department of Mathematics Yuncheng University Yuncheng 044000 China Correspondence should be addressed to Qiuying Li liqy-123@163.com Received 23 May 2010 Revised 4 August 2010 Accepted 7 September 2010 Academic Editor Yongwimon Lenbury Copyright 2010 Qiuying Li et al. 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 discrete predator-prey system with time delay and feedback controls is studied. Sufficient conditions which guarantee the predator and the prey to be permanent are obtained. Moreover under some suitable conditions we show that the predator species y will be driven to extinction. The results indicate that one can choose suitable controls to make the species coexistence in a long term. 1. Introduction The dynamic relationship between predator and its prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. The traditional predator-prey models have been studied extensively e.g. see 1-10 and references cited therein but they are questioned by several biologists. Thus the Lotka-Volterra type predator-prey model with the Beddington-DeAngelis functional response has been proposed and has been well studied. The model can be expressed as follows W X1 I b -a11 xw - i pSmArn un ạ 1 1 An d - 22V 1.1 The functional response in system 1.1 was introduced by Beddington 11 and DeAngelis 2 Advances in Difference Equations et al. 12 . It is similar to the well-known Holling type II functional response but has an extra term Yy in .