Long-Term Behavior of Positive Solutions of a Certain Nonlinear System of Difference Equations

In this paper, we study the boundedness, stability, and the rate of convergence of positive solutions of the following system of difference equations:

$$\begin{aligned} x_{n+1}=\alpha +\left( \dfrac{y_{n-1}}{x_{n }}\right) ^p,\ y_{n+1}=\alpha + \left( \dfrac{x_{n-1} }{y_{n} }\right) ^q, \ n=0, 1, 2, ... \end{aligned}$$

where parameters \(\alpha \), p, \(q \in (0, +\infty )\), and the initial values \(x_{i}\), \(y_{i}\) are arbitrary positive numbers for \( i= -1, 0\).