<p>This article investigates a diffusive predator–prey system with harvest delay and reproduction delay. A mathematical analysis of the existence of the positive equilibrium points is provided. In the absence of delay, explicit conditions for Turing instability are derived. It is observed that the diffusion coefficient plays a crucial role in the formation of Turing patterns. Furthermore, to study the influence of delays on the system, we follow the method of stability switching curve. The crossing sets are determined for each wave number, which helps deduce the various Hopf bifurcation curves. It is observed that as the delay parameters pass through these curves, the stability of the coexisting equilibrium changes accordingly. From the stability curves, we can infer that the harvest delay has no significant impact on the dynamics of the system when the reproduction delay is low. While for moderate values of reproduction delay, harvest delay can induce a stability switching phenomenon. Moreover, such switching behavior is also observed when the reproduction delay is varied. To further understand these dynamic changes, the properties of Hopf bifurcation are discussed using normal form theory. Numerical simulations are conducted to sustain the theoretical findings.</p>