Early-warning signals of critical transition: Effect of extrinsic noise

Complex dynamical systems often have tipping points and exhibit catastrophic regime shift. Despite the notorious difficulty of predicting such transitions, accumulating studies have suggested the existence of generic early-warning signals (EWSs) preceding upcoming transitions. However, previous theories and models were based on the effect of the intrinsic noise (IN) when a system is approaching a critical point, and did not consider the pervasive environmental fluctuations or the extrinsic noise (EN). Here, we extend previous theory to investigate how the interplay of EN and IN affects EWSs. Stochastic simulations of model systems subject to both IN and EN have verified our theory and demonstrated that EN can dramatically alter and diminish the EWS. This effect is stronger with increasing amplitude and correlation time scale of the EN. In the presence of EN, the EWS can fail to predict or even give a false alarm of critical transitions.

Figure 1

Comparison of theory (lines) and simulation (symbols) for the insect outbreak model with $\mathrm{\Omega }=50,a=1/20,b=1.5$. (a,b) Total variance and $R\left(1\right)$ vs EN amplitude for white EN (blue) and adiabatic EN (orange). (c,d) Total variance and $R\left(1\right)$ vs EN correlation time scale with ${\sigma }_e=0.2,{\tau }_c=10$. The red solid lines show the theoretical results for white and adiabatic EN. Error bars are standard deviation of 50 simulations.

Figure 2

Comparison of theory (lines) and simulation (symbols) for system-size-dependent effect. Total variance (a) and $R\left(1\right)$ (b) vs system size for IN-only (blue) and with-EN (orange) cases in the insect outbreak model (with $a=1/20,b=1.5)$. Error bars are standard deviations of 50 simulations.

Figure 3

EWSs in a toggle switch model subject to various ENs: theoretical (solid lines) and simulation (symbols) results. Shown are the Fano factor of gene B, the Pearson correlation coefficient between A and B, and $R\left(1\right)$ of B, subject to different amplitudes of EN (a–c) and different correlation time scales of EN (d–f). At each $a_1$, numeric results are estimated from stationary long-time trajectories. Shadowed areas indicate the critical point. Parameters values are $n_1=n_2=2,a_2=10,a_0=0.05,\mathrm{\Omega }=20$. In (a–c), ${\tau }_c=100$, in (d–f), ${\sigma }_e=0.1$. Error bars are standard deviations of 50 simulations.

Figure 4

EWSs can give a false alarm in a simple negative feedback loop model with no critical transition. (a) Steady state expression level of gene B as a function of parameter $k_1$. (b–d) When subjected to EN, EWSs rise rapidly with the increase of control parameter $k_1$. The solid lines are theoretical results based on Eq. (6) and symbols with error bars are the mean of 50 stochastic simulations. Parameter values are $a_0=0.05,k_2=0.5,n=3,\mathrm{\Omega }=500$.

Figure 5

Comparison of EWSs derived from time series of the toggle switch model with and without EN. (a) Control parameter $a_1$ increases linearly with time from 2 to 18 in a time window of 1000. Trajectories of expression level of A and B with only IN are shown. Gray band marks the transition events. The smooth solid gray line through the time series is a Gaussian kernel smoothing function used to filter out longer-term trends. The arrow marks the width of the sliding window used to calculate EWSs. (b) Residue after subtracting the average of the Gaussian kernel. (c) Example of estimated Fano factor from (b) prior to the critical transition. Purple line: IN only. Blue line: with EN $\left({\sigma }_e=0.1,{\tau }_c=10\right)$. (d) Including EN dramatically diminishes two EWSs, but the Pearson correlation coefficient is relatively robust. Box plot of Kendall's rank correlation coefficient $\left({\rho }_{\mathrm{K}}\right)$ of 500 trials.