Abstract
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.
Highlights
In recent years there has been increased interest in studying the dynamics of discontinuous maps
In several papers [6, 7, 8, 10, 12, 14, 19, 20, 22, 23, 24] the study of the dynamics was directed toward a detailed understanding of bifurcations and the structure of periodic solutions
Other exotic types of bifurcations such as period adding bifurcations, period incriminating bifurcations, and Farey tree type bifurcations are very common in this equation
Summary
Where x0> 0, and the function F is a positive piecewise continuous function with two jump discontinuities satisfying some additional conditions. 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. In the last section we formulate three open problems and provide some concluding remarks
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