Phase diagram for a logistic system under bounded stochasticity

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.