Switching from simple to complex dynamics in a predator-prey-parasite model: An interplay between infection rate and incubation delay

- Delay-induced Leslie-Gower type predator-prey-parasite model is proposed.
- Stability and instability of the endemic equilibrium point is studied with respect to the force of infection and length of delay.
- Populations coexist in stable state for all values of delay if the force of infection is low.
- Conditional stability is observed when the force of infection is intermediate and the system is stable in absence of delay.
- Complicated dynamical behavior is observed when the force of infection is high and the system is unstable in absence of delay.

Parasites play a significant role in trophic interactions and can regulate both predator and prey populations. Mathematical models might be of great use in predicting different system dynamics because models have the potential to predict the system response due to different changes in system parameters. In this paper, we study a predator-prey-parasite (PPP) system where prey population is infected by some micro parasites and predator-prey interaction occurs following Leslie-Gower model with type II response function. Infection spreads following SI type epidemic model with standard incidence rate. The infection process is not instantaneous but mediated by a fixed incubation delay. We study the stability and instability of the endemic equilibrium point of the delay-induced PPP system with respect to two parameters, viz., the force of infection and the length of incubation delay under two cases: (i) the corresponding non-delayed system is stable and (ii) the corresponding non-delayed system is unstable. In the first case, the system populations coexist in stable state for all values of delay if the force of infection is low; or show oscillatory behavior when the force of infection is intermediate and the length of delay crosses some critical value. The system, however, exhibits very complicated dynamics if the force of infection is high, where the system is unstable in absence of delay. In this last case, the system shows oscillatory, stable or chaotic behavior depending on the length of delay.