  The present paper is devoted to the study of the long term dynamics of
diffusion processes modelling a single species that experiences both
demographic and environmental stochasticity. In our setting, the long term
dynamics of the diffusion process in the absence of demographic stochasticity
is determined by the sign of $\Lambda_0$, the external Lyapunov exponent, as
follows: $\Lambda_0<0$ implies (asymptotic) extinction and $\Lambda_0>0$
implies convergence to a unique positive stationary distribution $\mu_0$. If
the system is of size $\frac{1}{\epsilon^{2}}$ for small $\epsilon>0$ (the
intensity of demographic stochasticity), demographic effects will make the
extinction time finite almost surely. This suggests that to understand the
dynamics one should analyze the quasi-stationary distribution (QSD)
$\mu_\epsilon$ of the system. The existence and uniqueness of the QSD is
well-known under mild assumptions.
  We look at what happens when the population size is sent to infinity, i.e.,
when $\epsilon\to 0$. We show that the external Lyapunov exponent still plays a
key role: 1) If $\Lambda_0<0$, then $\mu_\epsilon\to \delta_0$, the mean
extinction time is of order $|\ln \epsilon|$ and the extinction rate associated
with the QSD $\mu_{\epsilon}$ has a lower bound of order
$\frac{1}{|\ln\epsilon|}$; 2) If $\Lambda_0>0$, then $\mu_\epsilon\to \mu_0$,
the mean extinction time is polynomial in $\frac{1}{\epsilon^{2}}$ and the
extinction rate is polynomial in $\epsilon^{2}$. Furthermore, when
$\Lambda_0>0$ we are able to show that the system exhibits multiscale dynamics:
at first the process quickly approaches the QSD $\mu_\epsilon$ and then, after
spending a polynomially long time there, it relaxes to the extinction state. We
give sharp asymptotics in $\epsilon$ for the time spent close to
$\mu_\epsilon$.
