This paper is concerned with the dynamics of positive solutions for a system of rational difference equations of the following form \begin{equation*} u_{n+1}=\frac{lpha u_{n-1}^{2}}{\beta +\gamma v_{n-2}},ext{ }v_{n+1}=% \frac{lpha _{1}v_{n-1}^{2}}{\beta _{1}+\gamma _{1}u_{n-2}},\quad n=0,1,\dots, \end{equation*}% where the parameters $\alpha ,\beta ,\gamma ,\alpha _{1},\beta _{1},\gamma_{1}$ and the initial values $u_{-i},v_{-i}\in (0,\infty )$, $i=0,1,2$. Moreover, the rate of convergence of a solution that converges to the zero equilibrium of the system is discussed. Finally, some numerical examples are given to demonstrate the effectiveness of the results obtained.