  As a mathematical theory for the stochasstic, nonlinear dynamics of
individuals within a population, Delbr\"{u}ck-Gillespie process (DGP)
$n(t)\in\mathbb{Z}^N$, is a birth-death system with state-dependent rates which
contain the system size $V$ as a natural parameter. For large $V$, it is
intimately related to an autonomous, nonlinear ordinary differential equation
as well as a diffusion process. For nonlinear dynamical systems with multiple
attractors, the quasi-stationary and stationary behavior of such a birth-death
process can be underestood in terms of a separation of time scales by a
$T^*\sim e^{\alpha V}$ $(\alpha>0)$: a relatively fast, intra-basin diffusion
for $t\ll T^*$ and a much slower inter-basin Markov jump process for $t\gg
T^*$. In the present paper for one-dimensional systems, we study both
stationary behavior ($t=\infty$) in terms of invariant distribution
$p_n^{ss}(V)$, and finite time dynamics in terms of the mean first passsage
time (MFPT) $T_{n_1\rightarrow n_2}(V)$. We obtain an asymptotic expression of
MFPT in terms of the "stochastic potential" $\Phi(x,V)=-(1/V)\ln
p^{ss}_{xV}(V)$. We show in general no continuous diffusion process can provide
asymptotically accurate representations for both the MFPT and the $p_n^{ss}(V)$
for a DGP. When $n_1$ and $n_2$ belong to two different basins of attraction,
the MFPT yields the $T^*(V)$ in terms of $\Phi(x,V)\approx
\phi_0(x)+(1/V)\phi_1(x)$. For systems with a saddle-node bifurcation and
catastrophe, discontinuous "phase transition" emerges, which can be
characterized by $\Phi(x,V)$ in the limit of $V\rightarrow\infty$. In terms of
time scale separation, the relation between deterministic, local nonlinear
bifurcations and stochastic global phase transition is discussed. The
one-dimensional theory is a pedagogic first step toward a general theory of
DGP.
