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Most research on replay detection has focused on developing a stand-alone countermeasure that runs independently of a speaker verification system by training a single spoofed model and a single genuine model for all speakers. In this paper, we explore the potential benefits of adapting the back-end of a spoofing detection system towards the claimed target speaker. Specifically, we characterize and quantify speaker variability by comparing speaker-dependent and speaker-independent (SI) models of feature distributions for both genuine and spoofed speech. Following this, we develop an approach for implementing speaker-dependent spoofing detection using a Gaussian mixture model (GMM) back-end, where both the genuine and spoofed models are adapted to the claimed speaker. Finally, we also develop and evaluate a speaker-specific neural network-based spoofing detection system in addition to the GMM based back-end. Evaluations of the proposed approaches on replay corpora BTAS2016 and ASVspoof2017 v2.0 reveal that the proposed speaker-dependent spoofing detection outperforms equivalent SI replay detection baselines on both datasets. Our experimental results show that the use of speaker-specific genuine models leads to a significant improvement (around 4% in terms of equal error rate (EER)) as previously shown and the addition of speaker-specific spoofed models adds a small improvement on top (less than 1% in terms of EER).