How Stochasticity Influences Leading Indicators of Critical Transitions

Abstract

Many complex systems exhibit critical transitions. Of considerable interest are bifurcations, small smooth changes in underlying drivers that produce abrupt shifts in system state. Before reaching the bifurcation point, the system gradually loses stability (‘critical slowing down’). Signals of critical slowing down may be detected through measurement of summary statistics, but how extrinsic and intrinsic noises influence statistical patterns prior to a transition is unclear. Here, we consider a range of stochastic models that exhibit transcritical, saddle-node and pitchfork bifurcations. Noise was assumed to be either intrinsic or extrinsic. We derived expressions for the stationary variance, autocorrelation and power spectrum for all cases. Trends in summary statistics signaling the approach of each bifurcation depend on the form of noise. For example, models with intrinsic stochasticity may predict an increase in or a decline in variance as the bifurcation parameter changes, whereas models with extrinsic noise applied additively predict an increase in variance. The ability to classify trends of summary statistics for a broad class of models enhances our understanding of how critical slowing down manifests in complex systems approaching a transition.

Appendix A: Variance Approaches Zero from the Right

Since v(r) is always positive, by definition $\lim \limits _{r\rightarrow 0^{+}}v(r) = 0$ if and only if for all $\varepsilon > 0$ there exists some $\delta >0$ so that $\frac{g(x^s,r)^2}{2 | f'(x^s, r)|} < \varepsilon$ for each $r\in (0,\delta )$. This can be restated as: $\lim \limits _{r\rightarrow 0^{+}}v(r) = 0$ if and only if for all $\varepsilon > 0$ there is $\delta > 0$ so that for each $r\in (0,\delta )$ we have

$$\begin{aligned} g(x^s,r)^2 < 2\varepsilon | f'(x^s, r)|. \end{aligned}$$

(19)

Consequently, if inequality (19) holds in some neighborhood of the bifurcation point for every $\varepsilon > 0$, then the variance will approach 0 as r approaches the bifurcation point. Since we already have (10), inequality (19) implies that $\lim \limits _{r\rightarrow 0^+}g(x^s,r)^2 = 0$ and that $g(x^s,r)^2$ approaches 0 more rapidly than $2|f'(x^s,r)|$. When (10) holds, both of these conditions on g are necessary conditions for $\lim \limits _{r\rightarrow 0^{+}}v(r) = 0$, but inequality (19) is both necessary and sufficient. Comparing Fig. 2 with Table 6, we see that for models with intrinsic noise, the O–U variance curve declines to zero faster than the resilience term, and in each of these cases $\lim \limits _{r\rightarrow 0^{+}}v(r) = 0$. The other case where $\lim \limits _{r\rightarrow 0^{+}}v(r) = 0$ is with mechanistic environmental noise for the transcritical bifurcation case, where the O–U variance term also approaches zero more rapidly than the resilience term.