On the global character of some systems of rational difference equations in the plane

Manuscript 1: We investigate the global character of the solutions of the rational system in the title, with nonnegative parameters and nonnegative initial terms such that the denominators are always positive. We also pose several open problems related to systems of rational difference equations. Manuscript 2: In this paper, we analyze the global character of the solutions of an anticompetitive system of rational difference equations. We prove that the solutions of the system can have three different types of global behavior, corresponding to different regions of the parameter space. Our analysis utilizes a global convergence theorem from Camouzis and Ladas, and two theorems from Kulenovi¢ and Merino that apply to competitive systems. Manuscript 3: In this paper, we analyze the global character of the solutions of an anticompetitive system of rational difference equations. We prove that the solutions of the system can have four types of global behavior, corresponding to different regions of the parameter space. We also show that, in a range of the parameter space, there is a curve of period-2 points. These period-2 points are fixed points of the 2nd iterate of the map corresponding to the system; and each fixed point has a linear global stable manifold. Given an initial point (x0; y0), we use the equations of the invariant manifolds to calculate the limit points of the solution of the system of difference equations. Manuscript 4: There are 49 linear fractional, 1st-order, difference equations with two variables. Thus, there are 49 × 49 = 2,401 systems of two equations of this form. In this paper, we analyze ten of the 49 linear fractional equations and demonstrate that they can be separated into three families. We prove three theorems (one for each family) which provide the boundedness characterization of the solutions of each of the 880 linear fractional systems that contains one or more of these ten equations. Manuscript 5: In this paper, we analyze each of the 686 linear fractional systems that have the form given in the title, where the parameters and the initial terms are nonnegative numbers such that the denominators are always positive. We demonstrate that we can separate these systems into six families and we present six theorems (one for each family) which provide the boundedness characterization of the solutions of each of the 686 systems. This paper summarizes and extends the results from Manuscript 1 and Manuscript 4