Phase Diagram for Logistic Systems 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 logisticlike 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 a temporal Griffiths phase, with a 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 nondiffusive regime.

Figure 1

A phase diagram for a logistic system under bounded stochasticity, presented in the $r_0-\sigma$ plane. In the inactive phase ($r_0<0$, red), the time to extinction scales like $\ln n$ where $N$ plays no role. In the active phase $r_0>\sigma$ (blue), the extinction time grows exponentially with $N$. Under pure demographic noise (along the $\sigma =0$ axis), the transition occurs at $r_0=0$. When $\sigma >0$, the logarithmic and the exponential phases are separated by a finite power-law region (temporal Griffiths phase, green). The dashed-dotted line indicates the failure of the continuum (diffusive) approximation and the crossover from soft to sharp decline.

Figure 2

In the temporal Griffiths phase $T\sim N^q$. The main figure shows $q$ vs $r_0/\sigma$ as obtained from the numerical solution of Eq. (8) (red open circles), in comparison with the asymptotic expressions for the diffusive regime [Eq. (11), purple line] and in the large-$r_0$ regime [Eq. (12), black line]. (Inset) Results for $T(N)$ as obtained from the numerical solution of the exact backward Kolmogorov equation for $r_0=0.003$ (blue circles), 0.025 (yellow), and 0.06 (green). By fitting these numerical results (full lines), one obtains the actual power $q$, and the outcomes are represented by blue $\mathsf{X}$s in the main figure (the $\mathsf{X}$s that correspond to the three specific cases depicted in the inset are marked by arrows). In general, the WKB predictions fit the numerical outcomes quite nicely, and the slight deviations in the low-$r_0$ region are due to the prefactors of the power law [in these cases, the numerical $T(N)$ graph fits perfectly the predictions of Eq. (4)]. All the results here were obtained for $\sigma =0.08,\tau =3/2,\nu =0.04$.

Figure 3

Typical trajectories (frequency vs time) for a system with $\tau =1$, $\sigma =0.11$, and $\nu =0.1$, where (a) $r_0=0.02$ and (b) $r_0=0.105$. The dashed line corresponds to $x^{*}$, the point where the mean (over environmental conditions) growth rate is zero. In (b) The population fluctuates most of its lifetime in a relatively narrow band around $x^{*}$, extinction happens due to the accumulation of rare sequences of bad years, and the decline time is logarithmic in $N$ (sharp decline). As $r_0$ becomes smaller (a), the fluctuations are comparable with $x^{*}$; hence, the decline time becomes a finite fraction of the lifetime (soft decline).