INVESTIGATION OF THE GLOBAL DYNAMICS FOR NONLINEAR DIFFERENCE EQUATION MODELS INCLUDING NEGATIVE EXPONENTIAL TERMS

Abstract

It is well known that the subject of difference equations and systems of difference equations containing exponential terms are of great importance and play a vital role in the field of mathematics as well as in other sciences. The applications of the difference equations appear as discrete mathematical models of many real life phenomena such as in biology, economics, ecology, control theory, physics, social sciences, engineering, and other fields. The dynamic of any situation refers to how the situation changes over the course of time. Many population models in biology are governed by nonlinear difference equations containing negative exponential terms, and scientists published lot of papers recently in this matter. Researchers need to study difference equations that contain parameters, so it is important to study the behaviour of these equations as the value of certain parameter varies The usefulness of population dynamics to predictability and resource management depends on the underlying assumptions of the mathematical models The main purpose of this study is to investigate the global dynamics including existence, uniqueness, boundedness, periodic nature, local and global asymptotic behavior of the positive solutions to a difference equation models with negative exponential terms ,where are positive real numbers and the initial conditions  are non-negative real numbers where  is an even number. This equation could be also viewed as a model in mathematical biology, in which case we consider  denote the movement rate, the growth rate and the carrying capacity of the one species, respectively.

Finally, we present Numerical simulations, examples and figures to show the effectiveness and the applicability of our proposed results.