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We study k th order systems of two rational difference equations $x_{n} = \frac{α + \sum_{i = 1}^{k} β_{i} x_{n - 1} + \sum_{i = 1}^{k} γ_{i} y_{n - 1}}{A + \sum_{j = 1}^{k} B_{j} x_{n - j} + \sum_{j = 1}^{k} C_{j} y_{n - j}}, y_{n} = \frac{p + \sum_{i = 1}^{k} δ_{i} x_{n - i} + \sum_{i = 1}^{k} ε_{i} y_{n - i}}{q + \sum_{j = 1}^{k} D_{j} x_{n - j} + \sum_{j = 1}^{k} E_{j} y_{n - j}} n \in ℕ$ . In particular, we assume non-negative parameters and non-negative initial conditions, such that the denominators are nonzero. We develop several approaches which allow us to extend well known boundedness results on the k th order rational difference equation to the setting of systems in certain cases.

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