In the present paper we have investigated the impact of predator harvesting in a two-dimensional prey–predator model with Holling type III functional response. The main objective of this paper is to study the change of dynamical behaviour of the prey–predator model in the presence of non-linear predator harvesting. The model system shows complex dynamics with the change of different system parameters. We have established the positivity and boundedness of the solutions under a certain parametric condition with non-negative initial conditions. The existence and stability criterion of different equilibrium points are investigated in terms of system parameters. We have shown that the system undergoes through saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation under different parametric conditions. The structural changes or the system bifurcations predict the global dynamics of the model system. We have computed the first Lyapunov number to find the direction of the Hopf-bifurcating periodic solution of the system. Using numerical simulation we have studied here the effect of conversion efficiency, protection of the environment to the prey population and non-linear predator harvesting on the model dynamics. Finally the paper is ended with some conclusions. © 2021 Informa UK Limited, trading as Taylor & Francis Group.