Dynamics of a Higher Order Nonlinear Difference Equation

Abstract

We consider the higher-order nonlinear difference equation x n 1 α x n / A Bx n x n−k , n 0, 1, . . ., where parameters are positive real numbers and initial conditions x −k , . . . , x 0 are nonnegative real numbers, k ≥ 2. We investigate the periodic character, the invariant intervals, and the global asymptotic stability of all positive solutions of the abovementioned equation. We show that the unique equilibrium of the equation is globally asymptotically stable under certain conditions.