global dynamics of certain mix monotone difference equation

We investigate global dynamics of the following second order rational difference equation x n + 1 = x n x n − 1 + α x n + β x n − 1 a x n x n − 1 + b x n − 1 , where the parameters α , β , a , b are positive real numbers and initial conditions x − 1 and x 0 are arbitrary positive real numbers. The map associated to the right-hand side of this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the corresponding parametric space. In most cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability.