Stability and Hopf bifurcation for a delayed predator–prey model with disease in the prey

This paper is concerned with a mathematical model dealing with a predator–prey system with disease in the prey. Mathematical analysis of the model regarding stability has been performed. The effect of delay on the above system is studied. By regarding the time delay as the bifurcation parameter, the stability of the positive equilibrium and Hopf bifurcations are investigated. Furthermore, the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, to verify our theoretical predictions, some numerical simulations are also included.

► We proposed a new predator–prey model with time delay and disease in the prey.
► We can predict the development of epidemiology by the stability of the different equilibria.
► The stability of equilibria and periodic solutions correspond to the transmission or the end of the epidemic.
► Some numerical simulations be used to verify our theoretical prediction.