The dual-branch selection-and-stay combining (SES) is analyzed for the diversity reception on correlated Nakagami fading channels, where the conventional selection combining is employed only at the switching instance, and the receiver will stay at the selected branch till its estimated signal-to-noise ratio (SNR) becomes lower than a preset threshold. In this combining scheme, the receiver only needs to continuously observe and estimate the SNR of a single selected branch. To analyze the performance of the SES scheme, we derive the switching rate and the average bit error rate (BER) of the binary DPSK (BDPSK). Numerical results are illustrated for performance examination.