<p>Studying the dynamical behavior of the time-delayed predator–prey model helps to predict the pattern of population change, thus maintaining ecological balance and promoting harmonious coexistence between humans and nature. The predator–prey model of a highland pasture with two-time delays was investigated in this work. Firstly, the shape of the critical manifold of the model and the existence of the equilibrium point are analyzed when the time delay is equal to zero. Using the geometric singular perturbation theory, the dynamical behavior in the first quadrant of the model is explored based on the distribution of the equilibrium points, including singular Hopf bifurcations, relaxation oscillations, homoclinic orbits, heteroclinic orbits, and canard explosion phenomena. Subsequently, we examined the local stability around the positive equilibrium point when the time delay is not equal to zero. We also derived parametric requirements for the Hopf bifurcation. Besides, the direction and stability of the bifurcation are analyzed by employing center manifold and normal form theory. Simultaneously, we performed numerical simulations to confirm our conclusions in every case.</p>