LAUSR

In this paper, a chemostat model with general monotone response functions and two discrete time delays is proposed to describe the dynamical behavior about predator–prey system. First, by analyzing the characteristic equation associated with the model, we obtain the conditions of the existence and stability of extinction equilibrium and positive equilibrium. Choosing delays as bifurcation parameters, the existence of Hopf bifurcations is investigated in detail. Second, by virtue of the Poincaré normal form method and center manifold theorem, explicit formulas are derived to determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Finally, some numerical simulations are carried out to illustrate the theoretical results and the biological significance.