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 Permanence of a Discrete n-Species Schoener Competition System with Time Delays and Feedback Controls | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 515706 10 pages doi 2009 515706 Research Article Permanence of a Discrete n-Species Schoener Competition System with Time Delays and Feedback Controls Xuepeng Li and Wensheng Yang School of Mathematics and Computer Science Fujian Normal University Fuzhou 350007 China Correspondence should be addressed to Wensheng Yang ywensheng@ Received 4 March 2009 Revised 25 July 2009 Accepted 3 September 2009 Recommended by John Graef A discrete n-species Schoener competition system with time delays and feedback controls is proposed. By applying the comparison theorem of difference equation sufficient conditions are obtained for the permanence of the system. Copyright 2009 X. Li and W. Yang. 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. 1. Introduction In 1974 Schoener 1 proposed the following competition model cộ I1 X r1x - X e1 - T11X - my - I2 y r2y y e2 - r21x - r22y - cộ where ri Ii ei rij ci i 1 2 j 1 2 are all positive constants. May 2 suggested the following set of equations to describe a pair of mutualists 1 u u r1u a1 b1v c1u v r2v a2 b2u c2v 1 v 2 Advances in Difference Equations where u v are the densities of the species U V at time t respectively. ri a-i bi Ci i 1 2 are positive constants. He showed that system has a globally asymptotically stable equilibrium point in the region u 0 v 0. Both of the above-mentioned works are considered the continuous cases. However many authors 3-5 have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Bai et al. 6 argued that the discrete case of cooperative system is more appropriate and they proposed the following system x1 k 1 X1 k exp
