  There is a growing awareness that catastrophic phenomena in biology and
medicine can be mathematically represented in terms of saddle-node
bifurcations. In particular, the term `tipping', or critical transition has in
recent years entered the discourse of the general public in relation to
ecology, medicine, and public health. The saddle-node bifurcation and its
associated theory of catastrophe as put forth by Thom and Zeeman has seen
applications in a wide range of fields including molecular biophysics,
mesoscopic physics, and climate science. In this paper, we investigate a simple
model of a non-autonomous system with a time-dependent parameter $p(\tau)$ and
its corresponding `dynamic' (time-dependent) saddle-node bifurcation by the
modern theory of non-autonomous dynamical systems. We show that the actual
point of no return for a system undergoing tipping can be significantly delayed
in comparison to the {\em breaking time} $\hat{\tau}$ at which the
corresponding autonomous system with a time-independent parameter $p_{a}=
p(\hat{\tau})$ undergoes a bifurcation. A dimensionless parameter
$\alpha=\lambda p_0^3V^{-2}$ is introduced, in which $\lambda$ is the curvature
of the autonomous saddle-node bifurcation according to parameter $p(\tau)$,
which has an initial value of $p_{0}$ and a constant rate of change $V$. We
find that the breaking time $\hat{\tau}$ is always less than the actual point
of no return $\tau^*$ after which the critical transition is irreversible;
specifically, the relation $\tau^*-\hat{\tau}\simeq 2.338(\lambda
V)^{-\frac{1}{3}}$ is analytically obtained. For a system with a small $\lambda
V$, there exists a significant window of opportunity $(\hat{\tau},\tau^*)$
during which rapid reversal of the environment can save the system from
catastrophe.
