The spring-loaded inverted pendulum(SLIP) has been widely studied in both animals and robots. Generally, the majority of the relevant theoretical studies deal with elastic leg, the linear leg length-force relationship of which is obviously conflict with the biological observations. A planar spring-mass model with a nonlinear spring leg is presented to explore the intrinsic mechanism of legged locomotion with elastic component. The leg model is formulated via decoupling the stiffness coefficient and exponent of the leg compression in order that the unified stiffness can be scaled as convex, concave as well as linear profile. The apex return map of the SLIP runner is established to investigate dynamical behavior of the fixed point. The basin of attraction and Floquet Multiplier are introduced to evaluate the self-stability and initial state sensitivity of the SLIP model with different stiffness profiles. The numerical results show that larger stiffness exponent can increase top speed of stable running and also can enlarge the size of attraction domain of the fixed point. In addition, the parameter variation is conducted to detect the effect of parameter dependency, and demonstrates that on the fixed energy level and stiffness profile, the faster running speed with larger convergence rate of the stable fixed point under small local perturbation can be achieved via decreasing the angle of attack and increasing the stiffness coefficient. The perturbation recovery test is implemented to judge the ability of the model resisting large external disturbance. The result shows that the convex stiffness performs best in enhancing the robustness of SLIP runner negotiating irregular terrain. This research sheds light on the running performance of the SLIP runner with nonlinear leg spring from a theoretical perspective, and also guides the design and control of the bio-inspired legged robot.