Like many nonlinear dynamical systems, thermoacoustic engines (TAEs) exhibit hysteresis in the amplitude of self-excited acoustic oscillations when the temperature gradient implemented across the porous material is first increased and then decreased. This research numerically studies the hysteresis features of a quarter-wavelength standing-wave TAE. Computational Fluid Dynamics (CFD) is employed to investigate the effects of the stack parameters on the hysteresis characteristics. It is found that the lower critical temperature, the upper critical temperature, the difference between upper and lower temperatures, and the maximum pressure amplitude in the hysteresis loop are all dependent on the gap between stack plates and the stack position. This study deepens the understanding of the hysteresis phenomena reported in previous studies, providing useful guidance for reducing the lowest temperature gradient for the initiation of acoustic oscillations in TAEs.