Basins of attraction of equilibrium and boundary points of second-order difference equations

We investigate the global behaviour of the difference equation of the form x(n+1) = Bx(n)x(n-1)/bx(n)x(n-1) + dx(n) + ex(n-1) + f, n = 0, 1, ... with non-negative parameters and initial conditions such that B > 0, b + d + e + f > 0. We give a precise description of the basins of attraction of different equilibrium points, and show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium points or non-hyperbolic equilibrium points are in fact the global stable manifolds of neighbouring saddle or non-hyperbolic equilibrium points. Different types of bifurcations when one or more parameters b, d, e, f are 0 are explained.