  Population dynamics reflects an underlying birth-death process, where the
rates associated with different events may depend on external environmental
conditions and on the population density. A whole family of simple and popular
deterministic models (like logistic growth) support a transcritical bifurcation
point between an extinction phase and an active phase. Here we provide a
comprehensive analysis of the phases of that system, taking into account both
the endogenous demographic noise (random birth and death events) and the effect
of environmental stochasticity that causes variations in birth and death rates.
Three phases are identified: in the inactive phase the mean time to extinction
$T$ is independent of the carrying capacity $N$, and scales logarithmically
with the initial population size. In the power-law phase $T \sim N^q$ and the
exponential phase $T \sim exp(\alpha N)$. All three phases and the transitions
between them are studied in detail. The breakdown of the continuum
approximation is identified inside the power-law phase, and the accompanied
changes in decline modes are analyzed. The applicability of the emerging
picture to the analysis of ecological timeseries and to the management of
conservation efforts is briefly discussed.
