Missiles pose a major threat to aircraft in modern air combat. Advances in technology make them increasingly difficult to detect until they are close to the target and highly resistant to jamming. The evasion maneuver is the last line of defense for an aircraft. However, conventional rule-based evasion strategies are limited by computational demands and aerodynamic constraints, and existing learning-based approaches remain unconvincing for manned aircraft against modern missiles. To enhance aircraft survivability, this study investigates missile evasion inspired by the pursuit-evasion game between a gazelle and a cheetah and proposes a multi-stage reinforcement learning-based evasion strategy. The strategy learns a large azimuth policy to turn to evade, a small azimuth policy to keep moving away, and a short distance policy to perform agile aggressive maneuvers to avoid. One of the three policies is activated at each stage based on distance and azimuth. To evaluate performance, a high-fidelity simulation environment modeling an F-16 aircraft and missile under various conditions is used to compare the proposed approach with baseline strategies. Experimental results show that the proposed method achieves superior performance, enabling the F-16 aircraft to successfully avoid missiles with a probability of 80.89 percent for velocities ranging from 800 m/s to 1400 m/s, maximum overloads from 40 g to 50 g, detection distances from 5000 m to 15000 m, and random azimuths. When the missile is detected beyond 8000 m, the success ratio increases to 85.06 percent.