Global analysis of a model pulsing drug delivery oscillator based on chemomechanical feedback with hysteresis

A simple model for an autonomous pulsing drug delivery system was previously introduced. This model involves negative feedback action, with hysteresis, of an enzyme on the permeability of a membrane through which substrate, at constant external concentration, must diffuse to reach the enzyme. The qualitative dynamics of this model permit, depending on system parameters and external driving substrate concentration, two separate single steady state, double steady state, and permanently alternating (oscillatory) behaviors. The present contribution is concerned with rigorous proofs regarding the global stability of steady states when permanent alternation is precluded, and the existence and globally asymptotic stability of a limit cycle in the permanently alternating case. Also, we prove that more restrictive but often realistic conditions on the system parameters imply limitations on the number of alternations the system can undergo before reaching steady state.