  Extinction is the ultimate absorbing state of any stochastic birth-death
process, hence the time to extinction is an important characteristic of any
natural population. Here we consider logistic and logistic-like systems under
the combined effect of demographic and bounded environmental stochasticity.
Three phases are identified: an inactive phase where the mean time to
extinction $T$ increases logarithmically with the initial population size, an
active phase where $T$ grows exponentially with the carrying capacity $N$, and
temporal Griffiths phase, with power-law relationship between $T$ and $N$. The
system supports an exponential phase only when the noise is bounded, in which
case the continuum (diffusion) approximation breaks down within the Griffiths
phase. This breakdown is associated with a crossover between qualitatively
different survival statistics and decline modes. To study the power-law phase
we present a new WKB scheme which is applicable both in the diffusive and in
the non-diffusive regime.
