  We revisit gravitational particle production from the Stokes phenomenon
viewpoint, which helps us make a systematic way to understand asymptotic
behavior of mode functions in time-dependent background. One of our purposes of
this work is to make the method more practical for evaluation of
non-perturbative particle production rate. In particular, with several examples
of time-dependent backgrounds, we introduce some approximation methods that
make the analysis more practical. Specifically, we consider particle production
in simple expanding backgrounds, preheating after $R^2$ inflation, and a
transition model with smoothly changing mass. As we find several technical
issues in analyzing the Stokes phenomenon of each example, we discuss how to
simplify the problems while showing the accuracy of analytic estimation under
the approximations we make.
