Global behavior of positive solutions of a third order difference equations system

\begin{abstract} In this paper, we consider the following system of difference equations \begin{equation*} x_{n+1}=\alpha+\dfrac{y_{n}^p}{y_{n-2}^p},\ y_{n+1}=\alpha+ \dfrac{x_{n}^q}{x_{n-2}^q}, \ n=0, 1, 2, ... \end{equation*} where parameters $\alpha, p, q \in (0, \infty)$ and the initial values $x_{-i}$, $y_{-i}$ are arbitrary positive numbers for $ i=-2,-1, 0$. Our main aim is to investigate semi-cycle analysis of solutions of above system. Also, we study the boundedness of the positive solutions and the global asymptotic stability of the equilibrium point in case $\alpha>1$, $ 0<p,\ q\leq 1$. Moreover, the rate of convergence of the solutions is established. Finally, some numerical examples are given to illustrate our theoretical results.