Hopf bifurcation direction of a delayed bacteria-immunity system with quorum sensing mechanism

Considering the mechanism of quorum sensing, we formulate a bacteria-immunity model to describe the competition between bacteria and immune cells on the basis of Zhang’s model (see [9] for more details). A time delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. In the sequel, the length of delay which preserves the stability of the positive equilibrium is estimated, and the existence of Hopf bifurcation when the delay crosses through a critical value is investigated. Further, by using the normal form theory and center manifold theory, the explicit formulae are calculated which determine the stability, the direction and the period of bifurcating periodical solutions. Finally, numerical simulations are employed to verify the mathematical conclusions.