Abstract

We investigate the dynamical behavior of the following fourth-order rational difference equation xn+1 = xnxb n−1xn−3 + xn + xb n−1 + xn−3 + a xnxb n−1 + xb n−1xn−3 + xnxn−3 +1+ a , n = 0, 1, 2, ... where a, b ∈ [0,∞) and the initial values x−3, x−2, x−1, x0 ∈ (0,∞). We find that the successive lengths of positive and negative semicycles of nontrivial solutions of the above equation occur periodically. We also show that the positive equilibrium of the equation is globally asymptotically stable.