Dynamic behaviors of a turbidostat model with Tissiet functional response and discrete delay

In this paper, dynamic behaviors of a turbidostat model with Tissiet functional response, linear variable yield and time delay are investigated. The existence and boundedness of solutions, the local asymptotic stability of its equilibria and the phenomenon of Hopf bifurcation for this system are considered. Using the Liapunov–LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Furthermore, based on some knowledge of limit set, we show the necessary and sufficient conditions of permanent of the turbidostat model. Finally, numerical simulations are offered to support our results.