Global dynamics of some system of second-order difference equations (Q2055219)

The authors investigate the qualitative behavior of the systems of exponential difference equations \[ x_{n+1}=\frac{\alpha_1+\beta_1e^{-x_{n-1}}}{\gamma_1+y_n}, \quad y_{n+1}=\frac{\alpha_2+\beta_2e^{-y_{n-1}}}{\gamma_2+x_n}, \] \[ x_{n+1}=\frac{\alpha_1+\beta_1e^{-y_{n-1}}}{\gamma_1+x_n}, \quad y_{n+1}=\frac{\alpha_2+\beta_2e^{-x_{n-1}}}{\gamma_2+y_n}, \] where the parameters \(\alpha_i, \beta_i, \) and \( \gamma_i \) for \( i\in \{1, 2\} \) and the initial conditions \( x_{-1}, x_0, y_{-1} \) and \( y_0 \) are positive real numbers. They prove the boundedness and the persistence of positive solutions of this system. Moreover, it is shown that the unique positive equilibrium point of the system is globally asymptotically stable under certain conditions on the parameters. Furthermore, the rate of convergence of the positive solutions which converge to their unique positive equilibrium point is computed. Finally, some illustrative numerical examples are provided.