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: ASYMPTOTIC BEHAVIOR OF A COMPETITIVE SYSTEM OF LINEAR FRACTIONAL DIFFERENCE EQUATIONS | ASYMPTOTIC BEHAVIOR OF A COMPETITIVE SYSTEM OF LINEAR FRACTIONAL DIFFERENCE EQUATIONS M. R. S. KULENOVIC AND M. NURKANOVIC Received 18 July 2005 Revised 3 April 2006 Accepted 5 April 2006 We investigate the global asymptotic behavior of solutions of the system of difference equations xn 1 a xn b yn yn 1 d yn e xn n 0 1 . where the parameters a b d and e are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. In certain range of parameters we prove the existence of the global stable manifold of the unique positive equilibrium of this system which is the graph of an increasing curve. We show that the stable manifold of this system separates the positive quadrant of initial conditions into basins of attraction of two types of asymptotic behavior. In the case where a d and b e we find an explicit equation for the stable manifold to be y x. Copyright 2006 M. R. S. Kulenovic and M. Nurkanovic. 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 and preliminaries The following system of difference equations was considered in 12 a xn . d yn . n xn 1 1. . yn 1 . . n 0 1 b yn e xn where the parameters a b d and e are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. Ithasbeen shown in 12 that has the unique positive equilibrium which is globally asymptotically stable in the following three cases 1 b 1 e 1 2 b 1 e 1 a d 3 b 1 e 1 a d. It has been also shown in 12 that has the unique positive equilibrium E x y which is a saddle point in the following three cases Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 19756 Pages 1-13 DOI ADE 2006 19756 2 Competitive system of rational difference equations 4 b 1 e 1 5 b 1 e 1 a d 6 b 1 e 1 d a. We also proved that all solutions of
