Advances in Difference Equations
Volume 2007 (2007), Article ID 41541, 22 pages
doi:10.1155/2007/41541

Global Asymptotic Behavior of yn+1=(pyn+yn−1)/(r+qyn+yn−1)

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.