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: \(\begin{aligned} x_{n+1}=\alpha +\left( \dfrac{y_{n-1}}{x_{n }}\right) ^p,\ y_{n+1}=\alpha + \left( \dfrac{x_{n-1} }{y_{n} }\right) ^q, \ n=0, 1, 2, ... \end{aligned}\) $$\begin{array}{c}\hfill x_{n+1}=\alpha +{\left({\displaystyle \frac{y_{n-1}}{x_n}}\right)}^p,y_{n+1}=\alpha +{\left({\displaystyle \frac{x_{n-1}}{y_n}}\right)}^q,n=0,1,2,...\end{array}$$ where parameters \(\alpha \) $\alpha$ , p, \(q \in (0, +\infty )\) $q\in (0,+\infty )$ , and the initial values \(x_{i}\) $x_i$ , \(y_{i}\) $y_i$ are arbitrary positive numbers for \( i= -1, 0\) $i=-1,0$ .