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Dedicated to Professor Ronald E. Mickens on the occasion of his 60th Birthday. May his outstanding contributions continue for many years.We investigate the global stability, the periodic character, and the boundedness nature of the solutions of the difference equationwhere the parameters a, g, A, B and the initial conditions x 22 , x 21 , x 0 are non-negative real numbers. We show that the solutions of the equation exhibit a trichotomy character which depends upon whether g is less than A, equal to A, or greater than A.

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Our aim is to investigate the global asymptotic behavior, the existence of invariant intervals, oscillatory behavior, structure of semicycles, and periodicity of a nonlinear discrete population model of the form xn+1= F(xn); for n = 0,1,...,where x0> 0; and the function F is a positive piecewise continuous function with two jump discontinuities satisfying some additional conditions. The motivation for study of this general model was inspired by the classical Williamson's discontinuous population model, some recent results about the dynamics of the discontinuous Beverton-Holt model, and applications of discontinuous maps to the West Nile epidemic model. In the first section we introduce the population model which is a focal point of this paper. We provide background information including a summary of related results, a comparison between characteristics of continuous and discontinuous population models (with and without the Allee-type effect), and a justification of hypotheses introduced in the model. In addition we review some basic concepts and formulate known results which will be used later in the paper. The second and third sections are dedicated to the study of the dynamics and the qualitative analysis of solutions of the model in two distinct cases. An example, illustrating the obtained results, together with some computer experiments that provide deeper insight into the dynamics of the model are presented in the fourth section. Finally, in the last section we formulate three open problems and provide some concluding remarks.