Mathematical Biosciences 277 (2016) 1–14 Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs Switching from simple to complex dynamics in a predator–prey–parasite model: An interplay between infection rate and incubation delay N. Bairagi ∗ , D. Adak Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India a r t i c l e i n f o Article history: Received 14 October 2015 Revised 29 March 2016 Accepted 31 March 2016 Available online 14 April 2016 Keywords: Leslie–Gower predator–prey model Standard incidence Hopf bifurcation Period doubling Chaos a b s t r a c t Parasites play a significant role in trophic interactions and can regulate both predator and prey pop- ulations. 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 insta- bility 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. © 2016 Elsevier Inc. All rights reserved. 1. Introduction Theoreticians have used different mathematical models to un- derstand, explain and predict complex dynamics of predator–prey interactions. Classical Lotka–Volterra or Rosenzweig–MacArthur predator–prey models and its variants assume that prey popula- tion only grows logistically to its carrying capacity but the preda- tor has no such limitation. Leslie [1], for the first time, considered logistic growth of predator population with prey density as its up- per limit. Thus, the predator’s growth equation contains a nega- tive term which has a reciprocal relationship with per capita avail- ability of its preferred prey [2,3]. If X(t) and Y(t) are, respectively, the prey and predator densities at time t then Leslie–Gower model with type II predator’s response function is represented by dX dt = r 1 X − b 1 X 2 − a 1 XY k 1 + X , dY dt = Y  r 2 − a 2 Y X  . (1) ∗ Corresponding author. Tel.: +91 9433110781. E-mail address: nbairagi@math.jdvu.ac.in (N. Bairagi). It says, in absence of predator, the prey population grows exponen- tially with intrinsic per capita growth rate r 1 when prey is rare. However, prey population follows density-dependent birth rate, with b 1 as the strength of density dependency, when its size in- creases. Note that this model does not state a carrying capacity for the prey population in an explicit way, but models in an im- plicit way by means of intraspecific competition coefficient. This is known as emergent carrying capacity [4] since it is an emergent property based on actual life-history traits of prey, rather being a predetermined number (say K) as in popular logistic model. How- ever, as an special case, one can easily obtain the explicit carrying capacity (K = r 1 /b 1 ) from the emerging carrying capacity. Preda- tor regulates prey population following Type II response function a 1 X k 1 +X , where a 1 is the maximal per capita prey consumption rate and k 1 is the half saturation constant. Predators also grow logisti- cally to its carrying capacity r 2 a 2 X with maximum per capita growth rate r 2 by consuming its prey. One can observe that predator’s car- rying capacity is not a constant, but a function of prey density, X. The proportionality constant a 2 represents the number of prey re- quired to support one predator at equilibrium when the maximum per capita growth rate of predator is unity. All parameters are http://dx.doi.org/10.1016/j.mbs.2016.03.014 0025-5564/© 2016 Elsevier Inc. All rights reserved.