We study the oscillatory dynamics of an enzyme-diffusion system with product inhibition of substrate flux. A lumped model is considered in which the diffusion medium is a thin membrane whose permeabilities to substrate and product undergo discrete transitions in response to product concentration, with hysteresis. Assuming piecewise constant permeabilities and linear reaction kinetics, we derive analytical criteria for single steady state, double steady state, and oscillatory behavior as a function of system parameters. System behavior is pulsatile in the oscillatory regime. Analytical equations for pulse widths and interpulse intervals are presented, along with numerical calculations that confirm the validity of these equations. This system is similar but not completely analogous to chemical oscillators based on bistability/feedback mechanisms.