  In this paper, we develop a method of analyzing long transient dynamics in a
class of predator-prey models with two species of predators competing
explicitly for their common prey, where the prey evolves on a faster timescale
than the predators. In a parameter regime near a {\em{singular zero-Hopf
bifurcation}} of the coexistence equilibrium state, we assume that the system
under study exhibits bistability between a periodic attractor that bifurcates
from the singular Hopf point and another attractor, which could be a periodic
attractor or a point attractor, such that the invariant manifolds of the
coexistence equilibrium point play central roles in organizing the dynamics. To
find whether a solution that starts in a vicinity of the coexistence
equilibrium approaches the periodic attractor or the other attractor, we reduce
the equations to a suitable normal form, and examine the basin boundary near
the singular Hopf point. A key component of our study includes an analysis of
the long transient dynamics, characterized by their rapid oscillations with a
slow variation in amplitude, by applying a moving average technique. We obtain
a set of necessary and sufficient conditions on the initial values of a
solution near the coexistence equilibrium to determine whether it lies in the
basin of attraction of the periodic attractor. As a result of our analysis, we
devise a method of identifying early warning signals, significantly in advance,
of a future crisis that could lead to extinction of one of the predators. The
analysis is applied to the predator-prey model considered in [\emph{Discrete
and Continuous Dynamical Systems - B} 2021, 26(10), pp. 5251-5279] and we find
that our theory is in good agreement with the numerical simulations carried out
for this model.
