Dynamics of a predator-prey model with discrete and distributed delay

This paper considers a predator-prey model with discrete time delay representing prey handling time and assumed equal to the predator maturation period, and a distributed time delay describing intra-species interactions. We show that due to the delayed logistic growth of the prey, it is impossible for the species to become extinct through predation. Conditions for existence and local stability of the co-existence equilibrium are derived in terms of system parameters. Using techniques of centre manifold reduction and the normal form theory, we establish the direction of Hopf bifurcation of the co-existence equilibrium, as well as the stability of the bifurcating period solution. Numerical bifurcation analysis and simulations are performed to illustrate regions of stability of the co-existence equilibrium, to investigate how the amplitude and the period of bifurcating periodic solutions depend on parameters, and to demonstrate different types of dynamics of the system.