Global dynamics of some system of second-order difference equations

In this paper, we study the boundedness and persistence of positive solution, existence of invariant rectangle, local and global behavior, and rate of convergence of positive solutions of the following systems of exponential difference equations \begin{document}$ \begin{align*} x_{n+1} = \dfrac{\alpha_1+\beta_1e^{-x_{n-1}}}{\gamma_1+y_n},\ y_{n+1} = \dfrac{\alpha_2+\beta_2e^{-y_{n-1}}}{\gamma_2+x_n},\\ x_{n+1} = \dfrac{\alpha_1+\beta_1e^{-y_{n-1}}}{\gamma_1+x_n},\ y_{n+1} = \dfrac{\alpha_2+\beta_2e^{-x_{n-1}}}{\gamma_2+y_n}, \end{align*} $\end{document} where the parameters \begin{document}$ \alpha_i,\ \beta_i,\ \gamma_i $\end{document} for \begin{document}$ i \in \{1,2\} $\end{document} and the initial conditions \begin{document}$ x_{-1}, x_0, y_{-1}, y_0 $\end{document} are positive real numbers. Some numerical example are given to illustrate our theoretical results.