Hopf bifurcation of a chemostat model

A chemostat model with discrete time delay is proposed to describe oscillatory behavior in microbial continuous culture. Stability and Hopf bifurcation are discussed, including the changing regularity of bifurcation value. Explicit algorithm for determining the direction of Hopf bifurcation and the stability of periodic solution is derived, using the theory of normal form and center manifold. Finally, some numerical simulations are performed to illustrate the analytical results found.