A delay predator–prey model is formulated with continuous threshold prey harvesting and Holling response function of type III. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. The positive invariance of the non-negative orthant is proved and the uniform boundedness of the trajectories. Stability of equilibria is investigated and the existence of some local bifurcations is established: saddle-node bifurcation, Hopf bifurcation. We use optimal control theory to provide the correct approach to natural resource management. Results are also obtained for optimal harvesting. Numerical simulations are given to illustrate the results.