Bactria and Lytic Phage Coexistence in a Chemostat with Periodic Nutrient Supply

The dynamics of bacteria and bacteriophage coexistence was examined in a chemostat in which the externally driven supply of nutrient for bacteria, and washout rate oscillates periodically. The proposed mathematical model for three interacting variables, bacteria, phage, and nutrient, consists of 3 differential equations with time delay, due to the phage latent period of lysing. The study was carried out in an interval of physical parameters where an equivalent model with constant supply of nutrient and washout rate is mathematically unstable, running in limit cycle regimes, with known self-frequencies. It addresses mainly the asymptotically persistent dynamics of the system. Bifurcation maps in terms of two externally controlled parameters, the amplitude and frequency of the controlled nutrient supply were constructed for various latent lysis periods, in order to determine the frequency entrainment, i.e., the resulting main operating frequency of the system, relative to the known external and self-frequencies. Also presented are bifurcation maps for the rich variety of dynamical types observed in the study. Bifurcation diagrams in terms of the lysing time delay were also included for completion. A new type of entrainment, combining in a simple way the external and self-periods (reciprocal frequencies), is shown to exist for a range of parameters.