Advances in Difference Equations
Volume 2006 (2006), Article ID 19756, 13 pages
doi:10.1155/ADE/2006/19756
Asymptotic behavior of a competitive system of linear fractional difference equations
1Department of Mathematics, University of Rhode Island, Kingston RI 02881-0816, USA
2Department of Mathematics, University of Tuzla, Tuzla 75000, Bosnia and Herzegovina
Received 18 July 2005; Revised 3 April 2006; Accepted 5 April 2006
Copyright © 2006 M. R. S. Kulenović and M. Nurkanović. 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.
Abstract
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.