We investigate the global stability character of the equilibrium points and the period-two solutions of , with positive parameters and nonnegative initial conditions. We show that every solution of the equation in the title converges to either the zero equilibrium, the positive equilibrium, or the period-two solution, for all values of parameters outside of a specific set defined in the paper. In the case when the equilibrium points and period-two solution coexist, we give a precise description of the basins of attraction of all points. Our results give an affirmative answer to Conjecture 9.5.6 and the complete answer to Open Problem 9.5.7 of Kulenović and Ladas, 2002.

We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = (a+xn)/(b+yn), yn+1 = (d+yn)/(e+xn), n = 0,1,..., where the parameters a, b, d, and e are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. In certain range of parameters, we prove the existence of the global stable manifold of the unique positive equilibrium of this system which is the graph of an increasing curve. We show that the stable manifold of this system separates the positive quadrant of initial conditions into basins of attraction of two types of asymptotic behavior. In the case where a = d and b = e, we find an explicit equation for the stable manifold to be y = x.

We investigate global behavior of $x_{n+1} = T(x_{n}),\quad n=0,1,2,...$ (E) where $T:\mathcal{ R}\rightarrow \mathcal{ R}$ is a competitive (monotone with respect to the south-east ordering) map on a set $\mathcal{R}\subset \mathbb{R}^2$ with nonempty interior. We assume the existence of a unique fixed point $\overline{e}$ in the interior of $\mathcal{ R}$. We give very general conditions which are easily verifiable for (E) to exhibit either competitive-exclusion or competitive-coexistence. More specifically, we obtain sufficient conditions for the interior fixed point $\overline{ e}$ to be a global attractor when $\mathcal{ R}$ is a rectangular region. We also show that when $T$ is strongly monotone in $\mathcal{ R}^{\circ}$ (interior of $\mathcal{ R}$), $\mathcal{ R}$ is convex, the unique interior equilibrium $\overline{ e}$ is a saddle, and a technical condition is satisfied, the corresponding global stable and unstable manifolds are the graphs of monotonic functions, and the global stable manifold splits the domain into two connected regions, which under additional conditions on $\mathcal{R}$ and on $T$ are shown to be basins of attraction of fixed points on the boundary of $\mathcal{R}$. Applications of the main results to specific difference equations are given.

We investigate the global asymptotic behavior of solutions of the system of difference equations,,, where the parameters,,, and are positive numbers and the initial conditions and are arbitrary nonnegative numbers. We obtain some asymptotic results for the positive equilibrium of this system.