Complexity in a predator-prey-parasite model with nonlinear incidence rate and incubation delay

This paper is concerned with a predator-prey-parasite model with nonlinear infection rate and incubation delay. To explore the system dynamics, we study the distribution of roots of the characteristic equation of the Jacobian matrix of the system which has delay-dependent coefficients. The dynamics displayed by the system can exhibit some of the key features observed in the natural systems, such as appearance and disappearance of cycles in succession. It is shown that the switching phenomenon between stable coexistence and oscillatory coexistence of interior equilibrium as well as of predator-free equilibrium is an interplay of three system parameters, viz. the crowding coefficient, the prey consumption rate and the length of incubation delay. We also discuss the stability of the delayed-system when the non-delayed system is assumed to be unstable.