Dynamics of xn+1=xn-2k+1xn-2k+1+αxn-2l

We investigate the global stability of all positive solutions of the difference equation xn+1=xn-2k+1xn-2k+1+αxn-2l, where α is a positive real number, k ∈ {1, 2 , …}, l ∈ {0, 1 , …} and the initial conditions are positive real numbers. When α < 1 we show the positive equilibrium x¯=1α+1 is globally asymptotically stable. Also, when α > 1 we show that this equation possesses solutions that converge to the two-cycle …, 0, 1, 0, 1 , …