This Ph.D. thesis defense talk studies first passage times in the near-continuum limit of birth-death processes, with the Schlögl model as the main example. Starting from the deterministic description and the backward equation for the mean first passage time (MFPT), the talk applies a discrete exponential (large deviation, or rare event) ansatz. It then treats large deviations for Markov jump processes, including a rare event ansatz for the rescaled Master equation and Freidlin-Wentzell large deviation theory. The main messages are that the Fokker-Planck (diffusion) approximation may give the opposite answer to the relative stability question, that the MFPT exponent is not necessarily piecewise-constant over the range of state variables, and that the MFPT may not be dominated by the most likely path.