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

On the basis of Zhang’s model (see [P. Fergola, J. Zhang, M. Cerasuolo, Z.E. Ma, On the influence of quorum sensing in the competition between bacteria and immune system of invertebrates, in: Collective Dynamic: Topics on Competition and Cooperation in the Biosciences: A Selection of Papers in the Proceedings of the BIOCOMP2007 International Conference, AIP Conference Proceedings, vol. 1028, 2008, pp. 215–232] for more details), we formulate a bacteria-immunity model to describe the competition between bacteria and immune cells with bacterial quorum sensing mechanism. A time delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. Subsequently, the length of delay which preserves the stability of the positive equilibrium is estimated and Hopf bifurcation occurs when time delay crosses through a critical value are researched. Further, by using the normal form theory and center manifold theory, the explicit formulaes 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.