We first investigate the Lyapunov stability of the period-three solution of Todd's equation with a period-three coefficient: where α,β, and γ positive. Then for k = 2,3,… we extend our stability result to the k-order equation, where pn is a periodic coefficient of period k with positive real values and x-k+1,…,x-1, x0 ∈ (0, ∞).

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 the unbounded solutions of the second order difference equation where all parameters and C and initial conditions are nonnegative and such that for all n. We give a characterization of unbounded solutions for this equation showing that whenever an unbounded solution exists the subsequence of even indexed (resp. odd) terms tends to and the subsequence of odd indexed (resp. even) terms tends to a nonnegative number. We also show that two sets in the plane of initial conditions corresponding to the two cases are separated by the global stable manifold of the unique positive equilibrium. Our result answers two open problems posed by Kulenović and Ladas (2001, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Boca Raton/London: Chapman and Hall/CRC).

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 attractivity of the equilibrium of second-order difference equation where the parameters p, q, q < p and initial conditions x − 1, x 0 are nonnegative for all n. We prove that the unique equilibrium of this equation is global attractor which gives the affirmative answer to a conjecture of Kulenović and Ladas. The method of proof is innovative, and it has the potential to be used in the proof of global attractivity of equilibria of many similar equations.

We present a global attractivity result for maps generated by systems of autonomous difference equations. It is assumed that the map of the system leaves invariant a box, is monotone in a coordinate-wise sense (but not necessarily monotone with respect to a standard cone), and satisfies certain algebraic condition. It is shown that there exists a unique equilibrium, and that it is a global attractor. As an application, it is shown that a discretized version of the Lotka-Volterra system of differential equations of order $k$ has a global attractor in the positive orthant for certain range of parameters.

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.

We investigate the global character of solutions of the system of difference equations with positive parameters and non-negative initial conditions.

Our aim here is to present a summary of our recent work and a large number of open problems and conjectures on third order rational difference equations of the form with non-negative parameters and non-negative initial conditions.

The role of compensatory and overcompensatory dynamics in generating multiple attractors in density-dependent Leslie models with or without the Allee effects are studied. In the presence of the Allee effect the models support multiple attractors. However, in the absence of the Allee effect single attractors are supported when the dynamics are compensatory while multiple attractors are supported under overcompensatory dynamics. The existence of multiple attractors in density-dependent Leslie models implies that the qualitative population dynamics depend on initial conditions.

We investigate the periodic nature, the boundedness character and the global asymptotic stability of solutions of the difference equation where the parameter p n is a period-two sequence with positive values and the initial conditions are positive.