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We consider a wireless cognitive radio network in which a set of secondary users (SUs) opportunistically utilize the wireless spectrum licensed to the primary users (PUs) to transmit packets to the secondary base station (SBS). It is challenging to maximize the spectrum utilization while limiting the interference imposed to PUs due to SUs. To achieve the optimal tradeoff between the spectrum utilization and the interference caused by SUs, we propose the adaptive spectrum sharing schemes for code division multiple access (CDMA) based cognitive medium access control (MAC) in the uplink communications over the cognitive radio networks. Our proposed schemes address the joint problems of channel sensing, data transmission, and power and rate allocations. Under our proposed schemes, the SUs can adaptively select between the intrusive spectrum sharing and the non-intrusive spectrum sharing operations to transmit data to SBS based on the channel utilization, traffic load, and interference constraints. Our proposed schemes enable the SUs to efficiently utilize the available frequency spectrum which is licensed to the PUs while stringently limiting the interference to the PUs. Also conducted are extensive simulations to validate and evaluate our proposed schemes, which show the superiority of our proposed schemes as compared with the other schemes.