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.

We investigate the global character of solutions of the equation in the title with positive parameters and non-negative initial conditions.

We investigate the rate of convergence of solutions of some special cases of the equation , with positive parameters and nonnegative initial conditions. We give precise results about the rate of convergence of the solutions that converge to the equilibrium or period-two solution by using Poincaré's theorem and an improvement of Perron's theorem.

In this section, we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relevant information to G. Ladas.

We investigate the global character of solutions of the equation in the title with positive parameters and positive initial conditions. We obtain results about the global attractivity of the equilibrium, the existence and attractivity of the period-two solution and the semicycles.

Dedicated to Allan Peterson on the Occasion of His 60th Birthday. We investigate the global asymptotic behavior of solutions of the system of difference equations where the parameters A and B are in (0, ∞) and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers. We show that the stable manifold of this system separates the positive quadrant into the basins of attraction of two types of asymptotic behavior.