<p>In this paper, we investigate the dynamics of a predator–prey model that incorporates time delay effects, including prey nonlinear refuge, linear prey harvesting, and predator nonlinear harvesting. First, we analyze the model’s well-posedness and demonstrate its biological feasibility. Subsequently, through both theoretical and numerical analysis, we reveal that the model can undergo various bifurcation phenomena, including a codimension-one transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation, as well as codimension-two bifurcations such as cusp bifurcation, generalized Hopf bifurcation, and Bogdanov–Takens bifurcation. For the delayed model, we derive the Hopf bifurcation conditions induced by gestation delay and, through numerical simulations, find that as the gestation delay increases, the system can undergo period-doubling bifurcation. Moreover, we observe that variations in the refuge coefficient and the nonlinear harvesting parameter of predators in the delayed system can induce more complicated oscillations than in the non-delayed system. Our results suggest that appropriate adjustments to the refuge coefficient and predator nonlinear harvesting rate can help maintain system stability.</p>