Advances in Difference Equations
Volume 2007 (2007), Article ID 41541, 22 pages
doi:10.1155/2007/41541
Research Article
Global Asymptotic Behavior of yn+1=(pyn+yn−1)/(r+qyn+yn−1)
Department of Mathematics, University of Rhode Island, Kingston 02881-0816, RI, USA
Received 9 July 2007; Accepted 19 November 2007
Academic Editor: Elena Braverman
Copyright © 2007 A. Brett and M. R. S. Kulenović. 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 stability character of the equilibrium points and the period-two solutions of
yn+1=(pyn+yn−1)/(r+qyn+yn−1),n=0,1,…, with positive parameters and nonnegative initial conditions. We show that every solution of the equation in the title converges to either the zero equilibrium, the positive equilibrium, or the period-two solution, for all values of parameters outside of a specific set defined in the paper. In the case when the equilibrium points and period-two solution coexist, we give a precise description of the basins of attraction of all points. Our results give an affirmative answer to Conjecture 9.5.6 and the complete answer to Open Problem 9.5.7 of Kulenović and Ladas, 2002.