Early warning signals: The charted and uncharted territories

The realization that complex systems such as ecological communities can collapse or shift regimes suddenly and without rapid external forcing poses a serious challenge to our understanding and management of the natural world. The potential to identify early warning signals that would allow researchers and managers to predict such events before they happen has therefore been an invaluable discovery that offers a way forward in spite of such seemingly unpredictable behavior. Research into early warning signals has demonstrated that it is possible to define and detect such early warning signals in advance of a transition in certain contexts. Here we describe the pattern emerging as research continues to explore just how far we can generalize these results. A core of examples emerges that shares three properties: the phenomenon of rapid regime shifts, a pattern of 'critical slowing down' that can be used to detect the approaching shift, and a mechanism of bifurcation driving the sudden change. As research has expanded beyond these core examples, it is becoming clear that not all systems that show regime shifts exhibit critical slowing down, or vice versa. Even when systems exhibit critical slowing down, statistical detection is a challenge. We review the literature that explores these edge cases and highlight the need for (a) new early warning behaviors that can be used in cases where rapid shifts do not exhibit critical slowing down, (b) the development of methods to identify which behavior might be an appropriate signal when encountering a novel system; bearing in mind that a positive indication for some systems is a negative indication in others, and (c) statistical methods that can distinguish between signatures of early warning behaviors and noise

Critical fluctuations of noisy period-doubling maps

We extend the theory of quasipotentials in dynamical systems by calculating, within a broad class of period-doubling maps, an exact potential for the critical fluctuations of pitchfork bifurcations in the weak noise limit. These far-from-equilibrium fluctuations are described by finite-size mean field theory, placing their static properties in the same universality class as the Ising model on a complete graph. We demonstrate that the effective system size of noisy period-doubling bifurcations exhibits universal scaling behavior along period-doubling routes to chaos.Comment: 11 pages, 5 figure

Limits to the detection of early warning signals of population collapse

Background/Question/Methods

The recog­ni­tion that ecosys­tems can undergo sud­den shifts to alter­nate, less desir­able sta­ble states has led to the desire to iden­tify early warn­ing signs of these impend­ing col­lapses. This search has been moti­vated by the math­e­mat­ics of bifur­ca­tions, in which sud­den shifts result not from direct per­tur­ba­tions to the state (i.e. the pop­u­la­tion abun­dance, through mech­a­nisms such as over-harvesting) but to a slowly chang­ing para­me­ter that impacts the sys­tem sta­bil­ity. While these col­lapses can­not be antic­i­pated by observ­ing only the mean dynam­ics (as described by a deter­min­is­tic model), signs of the impend­ing col­lapse are expressed in the ran­dom per­tur­ba­tions, or noise, inher­ent in real sys­tems. The math­e­mat­i­cal the­ory of early warn­ing signs exploits this fact by seek­ing to detect pat­terns such as “crit­i­cal slow­ing down” of these per­tur­ba­tions due to the grad­ual loss of sta­bil­ity which leads to a bifurcation.

While much atten­tion has been given to empha­siz­ing the exis­tence both of sud­den col­lapses and of signs of crit­i­cal slow­ing down, lit­tle atten­tion has been paid to its detec­tion. Faced with only finite data, any method risks both false alarms and failed detec­tion events. We believe that weigh­ing these risks must be the bur­den of man­age­ment pol­icy, while research must first pro­vide a reli­able way to quan­tify the rel­a­tive risks of each. We present a method which quan­ti­fies this risk and show how to decrease the uncer­tainty inher­ent in com­mon summary-statistic approaches through the use of a like­li­hood based mod­el­ing approach.

Results/Conclusions

We demon­strate that com­monly used cor­re­la­tion tests applied to sum­mary sta­tis­tics such as auto­cor­re­la­tion and vari­ance are both inap­pro­pri­ate and insuf­fi­cient tests of early warn­ing signals.

Our method esti­mates directly the para­me­ters of a gen­er­al­ized model of the bifur­ca­tion pos­tu­lated by early warn­ing sig­nals the­ory, with and with­out the pres­ence of a grad­ual change lead­ing towards col­lapse. Using Monte Carlo sim­u­la­tion we gen­er­ate the dis­tri­b­u­tion of warning-signal sta­tis­tics expected under each model. From this we can quan­tify the risk of false alarms and missed detec­tion. We then show how apply­ing this approach to the data directly rather than the sum­mary sta­tis­tic increases the power of detec­tion. We illus­trate the approach in both sim­u­lated and empir­i­cal data of sud­den eco­log­i­cal shifts

Spatial patterns of tree yield explained by endogenous forces through a correspondence between the Ising model and ecology.

Spatial patterning of periodic dynamics is a dramatic and ubiquitous ecological phenomenon arising in systems ranging from diseases to plants to mammals. The degree to which spatial correlations in cyclic dynamics are the result of endogenous factors related to local dynamics vs. exogenous forcing has been one of the central questions in ecology for nearly a century. With the goal of obtaining a robust explanation for correlations over space and time in dynamics that would apply to many systems, we base our analysis on the Ising model of statistical physics, which provides a fundamental mechanism of spatial patterning. We show, using 5 y of data on over 6,500 trees in a pistachio orchard, that annual nut production, in different years, exhibits both large-scale synchrony and self-similar, power-law decaying correlations consistent with the Ising model near criticality. Our approach demonstrates the possibility that short-range interactions can lead to long-range correlations over space and time of cyclic dynamics even in the presence of large environmental variability. We propose that root grafting could be the common mechanism leading to positive short-range interactions that explains the ubiquity of masting, correlated seed production over space through time, by trees