Qualitative behavior of solutions of a two-dimensional rational system of difference equations

In this study, the rational system \begin{equation*} x_{n+1}=\frac{\alpha _{1}+\beta _{1}y_{n-1}}{a_{1}+b_{1}y_{n}}, \quad y_{n+1}=\frac{\alpha _{2}+\beta_{2}x_{n-1}}{a_{2}+b_{2}x_{n}}, \quad n\in\mathbb{N}_{0}, \end{equation*} where $\alpha_{i}$, $\beta_{i}$, $a_{i}$, $b_{i}$, $(i=1,2)$, and $x_{-j}$, $y_{-j}$, $(j=0,1)$, are positive real numbers, is defined and its qualitative behavior is discussed. The system in question is a two-dimensional extension of an old difference equation in the literature. The results obtained generalize the results in the literature on the equation in question.