Dynamics of a delay turbidostat system with contois growth rate

In this contribution, the dynamic behaviors of a turbidostat model with Contois growth rate and delay are investigated. The qualitative properties of the system are carried out including the stability of the equilibria and the bifurcations. More concretely, we exhibit the transcritical bifurcation by reducing the system without delay to a 1-dimensional system on a center manifold and find that Hopf bifurcation occurs by choosing the delay as bifurcation parameter. Also, using the normal form theory and the center manifold theorem we determine the direction and stability of the bifurcating periodic solutions induced by the Hopf bifurcation. Finally, numerical simulations are presented to support our theoretical results.