Modelling the regulatory system of a chemostat model with a threshold window

The chemostat model involving a control is either on or off which can be described by piecewise (or non-smooth) system. To improve the regulatory system for microorganism culture, piecewise chemostat models involving control strategy with threshold window are proposed and analysed. A special case is investigated first, i.e., chemostat models with a single threshold, the existence and stability of regular, virtual, pseudo-equilibria and tangent points are addressed, and it is shown that the regular equilibria and the pseudo-equilibrium cannot coexist. Then the global stabilities of the regular equilibria and pseudo-equilibrium have been studied by using qualitative analysis techniques of non-smooth Filippov dynamic systems. Furthermore, sliding bifurcations related to boundary node bifurcations are investigated with theoretical and numerical techniques, and the corresponding biological implications are discussed. For chemostat models with a threshold window, the effects of the relations between the threshold window and equilibria on the global dynamics of the proposed models are indicated, and subsequently we elaborate how the widths of the threshold windows affect the durations of No control state and Control state. All results suggest that the microorganism concentration can be maintained within a prescribed range using the proposed piecewise chemostat models. Moreover, it is concluded that the initial microorganism concentrations play an important role in the control process.