Dynamic behavior of a third-order system of difference equations

In this paper, we investigate the global behavior of positive solutions of the system of difference equations \(\begin{aligned} x_{n+1}=\gamma + \dfrac{x^p_{n-2}}{y^p_{n}},\ y_{n+1}=\gamma + \dfrac{y^q_{n-2}}{x^q_{n}}, \ n=0, 1, 2,\ldots \end{aligned}\) $$\begin{array}{c}\hfill x_{n+1}=\gamma +{\displaystyle \frac{x_{n-2}^p}{y_n^p}},y_{n+1}=\gamma +{\displaystyle \frac{y_{n-2}^q}{x_n^q}},n=0,1,2,\dots \end{array}$$ where parameters \(\gamma >1\) $\gamma >1$ , \(0< p, q \le 1\) $0<p,q\le 1$ and the initial values \(x_{i}\) $x_i$ , \(y_{i}\) $y_i$ are arbitrary positive numbers for \( i= -2,-1, 0\) $i=-2,-1,0$ . Precisely, we study the boundedness, persistence, and the stability of positive solutions of above system. We also give some numerical examples to demonstrate theoretical results.