We study kth order systems of two rational difference equations$$x_n=rac{lpha+sum^{k}_{i=1}eta_{i}x_{n-i} +sum^{k}_{i=1}gamma_{i}y_{n-i}}{A+sum^{k}_{j=1}B_{j}x_{n-j} +sum^{k}_{j=1}C_{j}y_{n-j}},quad ninmathbb{N},$$$$y_n=rac{p+sum^{k}_{i=1}delta_{i}x_{n-i} +sum^{k}_{i=1}epsilon_{i}y_{n-i}}{q+sum^{k}_{j=1}D_{j}x_{n-j} +sum^{k}_{j=1}E_{j}y_{n-j}},quad ninmathbb{N}.$$ In particular we assumenon-negative parameters and non-negative initial conditions. We develop severalapproaches which allow us to extend well known boundedness results on the kthorder rational difference equation to the setting of systems in certain cases.
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