Dynamical analysis on a bacteria-phages model with delay and diffusion

Abstract

Delay and diffusion have important significance in enriching the dynamic behavior of nonlinear dynamical systems. In this paper, we study the effects of delay and diffusion on the dynamics of a bacteria-phages model. By analyzing the stability of equilibria and the existence of Hopf bifurcation, we obtain the following results: (i) Under certain condition, only diffusion cannot contribute to the Turing instability, which is different from the general results that the diffusion can destabilize the model and lead model to produce the Turing instability; (ii) Delay or the combination of diffusion and delay can destabilize the model and lead model to generate the Hopf bifurcation and patterns. Moreover, the formulae for determining the properties of Hopf bifurcation are derived. Numerical simulations are carried out to reveal the effects of delay or the combination of diffusion and delay on the stability and Hopf bifurcation for the model. The results obtained are of significance to predict the coexistence of bacteria and phages and to find the appropriate time to implement the phage therapy.