Analysis of a periodic bacteria-immunity model with delayed quorum sensing

Based on [J. Zhang, M. Cerasuolo, P. Fergola, Z.E. Ma, On the influence of quorum sensing in the competition between bacteria and immune system (in press)], a bacteria-immunity model with periodically bacterial quorum sensing is formulated which describes the competition between bacteria and immune cells. A discrete delay is introduced to characterize the time in which bacteria receive signal molecules and combat with immune cells. In this paper, we focus on a subsystem of the bacteria-immunity model, investigate the existence and stability of the positively periodic solution, and discuss the existence of subharmonic bifurcation by the perturbation method.