  Temporal environmental noise (EN) is a prevalent natural phenomenon that
controls population and community dynamics, shaping the destiny of biological
species and genetic types. Conventional theoretical models often depict EN as a
Markovian process with an exponential distribution of correlation times,
resulting in two distinct qualitative dynamical categories: quenched
(pertaining to short demographic timescales) and annealed (pertaining to long
timescales). However, numerous empirical studies demonstrate a fat-tailed decay
of correlation times. Here, we study the consequences of power-law correlated
EN on the dynamics of isolated and competing populations. We reveal the
emergence of a novel intermediate phase that lies between the quenched and
annealed regimes. Within this phase, dynamics are primarily driven by rare, yet
not exceedingly rare, long periods of almost-steady environmental conditions.
For an isolated population, the time to extinction in this phase exhibits a
novel scaling with the abundance, and also a non-monotonic dependence on the
spectral exponent.
