An erbium-doped all-fiber laser model based on dual-peak filter was designed, and the numerical simulation of the dynamic characteristics of asynchronous dual-wavelength pulse mode-locking was carried out. Using the same noise as the initial condition, and setting the saturation energy of the gain fiber to 15 pJ, 40 pJ and 55 pJ, respectively, the simulation results show that the noise finally evolves into single-wavelength pulse mode-locking, asynchronous dual-wavelength pulse mode-locking, and asynchronous dual-wavelength pulse mode-locking in the form of soliton molecules, in which the evolution process of asynchronous dual-wavelength pulses goes through three stages: noise pulse generation, multi-pulse mode-locking and gain competition, and stable asynchronous dual-wavelength pulse mode-locking. Besides, the saturation energy of the gain fiber directly determines the evolution direction of the pulse in the gain competition, and the pulse frequency shifts caused by cross-phase modulation during the pulse collision process result in the time domain pulse time jitter.