Behavior of an exponential system of difference equations (Q2321452)

Summary: We study the qualitative behavior of the following exponential system of rational difference equations: \(x_{n + 1}=\left(\alpha e^{- y_n} + \beta e^{- y_{n - 1}}\right)/\left(\gamma + \alpha x_n + \beta x_{n - 1}\right)\), \(y_{n + 1}=\left(\alpha_1 e^{- x_n} + \beta_1 e^{- x_{n - 1}}\right)/\left(\gamma_1 + \alpha_1 y_n + \beta_1 y_{n - 1}\right)\), \(n=0,1, \ldots\), where \(\alpha\), \(\beta\), \(\gamma\), \(\alpha_1\), \(\beta_1\), and \(\gamma_1\) and initial conditions \(x_0\), \(x_{- 1}\), \(y_o\), and \(y_{- 1}\) are positive real numbers. More precisely, we investigate the boundedness character and persistence, existence and uniqueness of positive equilibrium, local and global behavior, and rate of convergence of positive solutions that converges to unique positive equilibrium point of the system. Some numerical examples are given to verify our theoretical results.