Sensorineural hearing loss typically results in elevated thresholds and steepened loudness growth significantly conditioned by a damage of outer hair cells (OHC). In hearing aids, amplification and dynamic compression aim at widening the limited available dynamic range. However, speech perception particularly in complex acoustic scenes often remains difficult. Here, a physiologically motivated, fast acting, model-based dynamic compression algorithm (MDC) is introduced which aims at restoring the behaviorally estimated basilar membrane input-output (BM I/O) function in normal-hearing listeners. A system-specific gain prescription rule is suggested, based on the same model BM I/O function and a behavioral estimate of the individual OHC loss. Cochlear off-frequency component suppression is mimicked using an instantaneous frequency (IF) estimate. Increased loudness as a consequence of widened filters in the impaired system is considered in a further compensation stage. In an extended version, a subband estimate of the speech presence probability (MDC+SPP) additionally provides speech-selective amplification in stationary noise. Instrumental evaluation revealed that the IF control enhances the spectral contrast of vowels and benefits in quality predictions at higher signal-to-noise ratios (SNRs) were observed. Compared with a conventional multiband dynamic compressor, MDC achieved objective quality and intelligibility benefits for a competing talker at lower SNRs. MDC+SPP outperformed the conventional compressor in the quality predictions and reached comparable instrumental speech intelligibility as achieved with linear amplification. The proposed algorithm provides a first promising basis for auditory model-based compression with signal-type- and bandwidth-dependent gains.