Population dynamics in a time-varying environment with fat-tailed correlations

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 (long environmental timescales) and annealed (short environmental 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 analyze the intermediate region that lies between the quenched and annealed regimes and show that it can widen indefinitely. Within this region, dynamics is 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 region exhibits a power-law scaling with the logarithm of the abundance and also a nonmonotonic dependence on the spectral exponent.