Hopf bifurcation and stability of periodic solutions in a delayed eco-epidemiological system

In this paper, a delayed predator–prey epidemiological system with disease spreading in predator population is considered. By regarding the delay as the bifurcation parameter and analyzing the characteristic equation of the linearized system of the original system at the positive equilibrium, the local asymptotic stability of the positive equilibrium and the existence of local Hopf bifurcation of periodic solutions are investigated. Moreover, we also study the direction of Hopf bifurcations and the stability of bifurcated periodic solutions, an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.