We investigate the global character of the difference equation of the form $$x_{n+1} = f(x_{n}, x_{n-1}),\quad n=0,1, \ldots $$ with several period-two solutions, where f is increasing in all its variables. We show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium solutions or period-two solutions are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium solutions or period-two solutions. As an application of our results we give the global dynamics of three feasible models in population dynamics which includes the nonlinearity of Beverton-Holt and sigmoid Beverton-Holt types.