Effects of impulsive control on permanence and extinction of lactic acid fermentation

• A lactic acid fermentation model with fixed time impulsive effects is investigated.
• The extinction and permanence of this model are obtained.
• A lactic acid fermentation model with state dependent impulsive effects is proposed.
• Dynamical behaviors of the model are analyzed.
• Numerical results are carried out to illustrate the feasibility of our main results.

With the consideration of mechanism of control for lactic acid fermentation in membrane bioreactor, we propose, in this paper, two kinds of different control strategies for a lactic acid fermentation model. Firstly, by introducing a new study method, some sufficient conditions on the permanence and extinction of the microorganisms with periodic impulsive control strategy are obtained. Secondly, we state and prove some general criteria for the existence and stability of positive order-k periodic solution of the microorganisms with state dependent impulsive control strategy by using the Poincaré map, differential inequality and method of qualitative analysis. Finally, numerical simulations are carried out to illustrate the main results and compare numerically periodic impulsive control strategy and state dependent control strategy.