An age-structured model for erythropoiesis is extended to include the active destruction of the oldest mature cells and possible control by apoptosis. The former condition, which is applicable to other population models where the predator satiates, becomes a constant flux boundary condition and results in a moving boundary condition. The method of characteristics reduces the age-structured model to a system of threshold type differential delay equations. Under certain assumptions, this model can be reduced to a system of delay differential equations with a state dependent delay in an uncoupled differential equation for the moving boundary condition. Analysis of the characteristic equation for the linearized model demonstrates the existence of a Hopf bifurcation when the destruction rate of erythrocytes is modified. The parameters in the system are estimated from experimental data, and the model is simulated for a normal human subject following a loss of blood typical of a blood donation. Numerical studies for a rabbit with an induced auto-immune hemolytic anemia are performed and compared with experimental data.