| Summary: | Abstract The dynamical behavior of the predator–prey system is influenced effectively due to the mutual interaction of parasites. Regulations are imposed on biodiversity due to such type of interaction. With implementation of nonlinear saturated incidence rate and piecewise constant argument method of differential equations, a three-dimensional discrete-time model of prey–predator–parasite type is studied. The existence of equilibria and the local asymptotic stability of these steady states are investigated. Moreover, explicit criteria for a Neimark–Sacker bifurcation and a period-doubling bifurcation are implemented at positive equilibrium point of the discrete-time model. Chaos control is discussed through implementation of a hybrid control technique based on both parameter perturbation and a state feedback strategy. At the end, some numerical simulations are provided to illustrate our theoretical discussion.
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