Extinction transitions in correlated external noise

We analyze the influence of long-range correlated (colored) external noise on extinction phase transitions in growth and spreading processes. Uncorrelated environmental noise (i.e., temporal disorder) was recently shown to give rise to an unusual infinite-noise critical point [Europhys. Lett. 112, 30002 (2015)]. It is characterized by enormous density fluctuations that increase without limit at criticality. As a result, a typical population decays much faster than the ensemble average, which is dominated by rare events. Using the logistic evolution equation as an example, we show here that positively correlated (red) environmental noise further enhances these effects. This means, the correlations accelerate the decay of a typical population but slow down the decay of the ensemble average. Moreover, the mean time to extinction of a population in the active, surviving phase grows slower than a power law with population size. To determine the complete critical behavior of the extinction transition, we establish a relation to fractional random walks, and we perform extensive Monte Carlo simulations.

Figure 1

Panels (a), (b), and (c) show examples of the fractional Gaussian noise $\mathrm{\Delta }{\mu }_n$ for the superdiffusive $\left(\gamma <1\right)$, uncorrelated $\left(\gamma =1\right)$, and subdiffusive $\left(\gamma >1\right)$ regimes, respectively. To emphasize the differences, we show $\text{sgn}\left(\mathrm{\Delta }{\mu }_n\right)$ in panels (d), (e), and (f). These plots show that positive correlations $\left(\gamma <1\right)$ make the noise less likely to alternate between positive and negative values, while negative correlations $\left(\gamma >1\right)$ cause the noise to alternate more frequently.

Figure 2

Density $\rho \left(t\right)$ for an individual noise configuration, plotted as $x=-ln\rho$ vs. time interval $n$. The initial condition is ${\rho }_0=0.5$; the parameters are $\mathrm{\Delta }t=2,{\lambda }_n=1$, and ${\mu }_n$ is a uncorrelated Gaussian random variable with average and variance both equal to 1.

Figure 3

Average density $\rho$ as a function of $t$ close to the critical point for $\gamma =1.5$. Below $\mu ={\mu }_c=128$ the population survives by reaching a stationary density, whereas it decays if $\mu >{\mu }_c$. Between survival and extinction there is a critical point at $\mu ={\mu }_c$. The solid black line shows a power-law fit whose exponent 0.75(1) agrees with the one predicted by Eq. (13). The data are averages over ${10}^7$ disorder configurations.

Figure 4

Probability distribution $P$ as function of $x=-ln\rho$ at criticality and at different times $t$, using $\sigma =1$ and $\mathrm{\Delta }t=2$. Panel (a) shows the singularity at $x=0$ for $\gamma =0.5$ (red noise or superdiffusive regime). In the double-logarithmic plot in panel (b), we can see that the singularity takes a power-law form. Panel (c) shows the uncorrelated case, $\gamma =1$, where $P$ follows the half Gaussian $2exp\left\{x^2/\left({\sigma }^2\mathrm{\Delta }{t}^{2}t\right)\right\}/\sqrt{2\pi \sigma \mathrm{\Delta }{t}^{2}t}$ (represented by the solid lines). Panel (d) displays this same data, but in a double-logarithmic scale. Panel (e) shows that the singularity is also present for $\gamma =1.5$ (subdiffusive regime or blue noise), however, $P$ gets depleted instead of diverging. Panel (f) displays the same data as in panel (e), however, in a double-logarithmic plot, again demonstrating the power-law behavior of the singularity. These simulations run up to $t=2^{16}\left(\approx 16\times {10}^3\right)$ and average over ${10}^7$ disorder realizations. Uncertainties are smaller than the symbol sizes.

Figure 5

(a) Typical density at criticality plotted as $\left[x\right]=-[ln\rho ]=-ln{\rho }_{\text{typ}}$ vs. time $t$ for several $\gamma$. (b) Average density $\left[\rho \right]$ vs. time $t$. In both cases the curves follow power laws for more than two decades in time for all values of $\gamma$ we have simulated. The inset, panel (c), compares $-ln{\rho }_{\text{typ}}$ and $-ln\left[\rho \right]$, for $\gamma =1.5$. Each curve averages over ${10}^6$ disorder configurations, making uncertainties about the size of the symbols.

Figure 6

Exponents $\delta$ and $\varphi$ extracted from the data in Fig. 5. The solid and dashed (green and red) lines are our theoretical predictions $\varphi =(2-\gamma )/2$ and $\delta =\gamma /2$. All exponents agree with the predictions within the uncertainty bars that are smaller than the symbol size. The small deviations in the exponent $\varphi$ for $\gamma \ge 1.3$ are also observed in fractional Brownian motion [25] and stem from corrections to scaling.

Figure 7

(a) Lifetime $\tau$ at the critical point as a function of the population size $N$ for several $\gamma$ using $\sigma =2$. The expected behavior Eq. (15) corresponds to a straight line. (b) Effect of the disorder strength $\sigma$ on the lifetime for $\gamma =1.5$. (c) Exponents were determined by fitting the lifetime to $\tau \sim {(lnN)}^{\eta }$. The uncertainties are smaller than the symbol size unless explicitly shown. The solid magenta line represents the prediction $\eta =2/(2-\gamma )$. Each curve is averaged over a number of disorder configurations that ranges from ${10}^4$ to ${10}^5$.

Figure 8

Off-critical simulations. Panel (a) displays $\left[\rho \right]$ as a function of time for $\gamma =1.5$ and several $\mu$. The stationary density ${\rho }_{\text{st}}$ is obtained fitting a constant to the long time behavior of $\left[\rho \right]$. Representing the critical curve as $A{t}^{-\delta }$, the dashed (green) and dash-dotted (red) lines show $1.9{\rho }_{\text{crit}}$ with one uncertainty in $\delta$. The crossover time $t_x$ (black circles) is evaluated as the average time between those curves and its uncertainty is the half width. In panels (b) and (c) we can see that ${\rho }_{\text{st}}$ and $t_x$ follow power laws in $|\mu -{\mu }_c|$ for several $\gamma$. In panel (b), uncertainties are much smaller than the symbol size. Uncertainties in panel (c) are of the order of the symbol size.

Figure 9

Critical exponents $\beta$ and ${\nu }_{\parallel }$ extracted from the data in Fig. 8. Simulations for $\gamma <1$ required times up to $2^{26}\approx 6.7\times {10}^7$ due the long crossover to the asymptotic behavior. All values agree with the theoretical predictions. The uncertainty in $\beta$ is of the order of ${10}^{-2}$.

Figure 10

Lifetime $\tau$ vs. population size $N$ plotted as $ln\tau$ vs. ${(lnN)}^{\gamma }$, such that straight lines indicates a behavior compatible with Eq. (23). For $\gamma =1.5$ [panel (a)], straight lines can be fitted over almost three decades in time and for $\mu$ within $2\sigma$ of ${\mu }_c\left(\sigma =1\right)$. However, for $\gamma =0.5$, we see a smaller range in time and $\mu$. The prefactor of ${[lnN]}^{\gamma }$ in Eq. (23) was also evaluated and plotted in Fig. (11).

Figure 11

Analysis of the prefactor $C$ of Eq. (23), $\tau \sim exp\left\{C{(lnN)}^{\gamma }\right\}$. Panels (a) and (b) shows $C$ extracted from the data in Fig. 10 vs. $|\mu -{\mu }_c|$. The prefactor $C$ is expected to behave as a power law $C=C^{\prime }{|\mu -{\mu }_c|}^{2-\gamma }$ [see Eq. (25)]. Panel (c) shows the prefactor exponent extracted from the data (blue circles) and our prediction $2-\gamma$ (solid magenta line). Uncertainties are of the order of the symbol size if not shown.

Figure 12

Scaling collapse of the lifetime in the temporal Griffiths phase. To achieve this collapse we employed very high (unphysical) disorder $\sigma =32$ and $\mathrm{\Delta }t={10}^{12}$. The correction to scaling ${\tau }_0\left(\mu \right)$ was obtained adjusting ${\tau }_0\left(\mu \right)=B\left(\mu \right)exp\left\{C{[lnN]}^{\gamma }\right\}$ in every curve. The grey background is a plot of $\tau =D_1{e}^{D_{2}z}$ with the appropriate parameters.