  The ability of random environmental variation to stabilize competitor
coexistence was pointed out long ago and, in recent years, has received
considerable attention. Analyses have focused on variations in the
log-abundances of species, with mean logarithmic growth rates when rare,
$\mathbb{E}[r]$, used as metrics for persistence. However, invasion
probabilities and the times to extinction are not single-valued functions of
$\mathbb{E}[r]$ and, in some cases, decrease as $\mathbb{E}[r]$ increases.
Here, we present a synthesis of stochasticity-induced stabilization (SIS)
phenomena based on the ratio between the expected arithmetic growth $\mu$ and
its variance $g$. When the diffusion approximation holds, explicit formulas for
invasion probabilities and persistence times are single valued, monotonic
functions of $\mu/g$. The storage effect in the lottery model, together with
other well-known examples drawn from population genetics, microbiology and
ecology (including discrete and continuous dynamics, with overlapping and
non-overlapping generations), are placed together, reviewed, and explained
within this new, transparent theoretical framework. We also clarify the
relationships between life-history strategies and SIS, and study the dynamics
of extinction when SIS fails.
