Limits to the detection of early warning signals of population collapse

Abstract

Abstract Background/Question/Methods The recog-ni-tion that ecosys-tems can undergo sud-den shifts to alter-nate, less desir-able sta-ble states has led to the desire to iden-tify early warn-ing signs of these impend-ing col-lapses. This search has been moti-vated by the math-e-mat-ics of bifur-ca-tions, in which sud-den shifts result not from direct per-tur-ba-tions to the state (i.e. the pop-u-la-tion abun-dance, through mech-a-nisms such as over-harvesting) but to a slowly chang-ing para-me-ter that impacts the sys-tem sta-bil-ity. While these col-lapses can-not be antic-i-pated by observ-ing only the mean dynam-ics (as described by a deter-min-is-tic model), signs of the impend-ing col-lapse are expressed in the ran-dom per-tur-ba-tions, or noise, inher-ent in real sys-tems. The math-e-mat-i-cal the-ory of early warn-ing signs exploits this fact by seek-ing to detect pat-terns such as “crit-i-cal slow-ing down” of these per-tur-ba-tions due to the grad-ual loss of sta-bil-ity which leads to a bifurcation.While much atten-tion has been given to empha-siz-ing the exis-tence both of sud-den col-lapses and of signs of crit-i-cal slow-ing down, lit-tle atten-tion has been paid to its detec-tion. Faced with only finite data, any method risks both false alarms and failed detec-tion events. We believe that weigh-ing these risks must be the bur-den of man-age-ment pol-icy, while research must first pro-vide a reli-able way to quan-tify the rel-a-tive risks of each. We present a method which quan-ti-fies this risk and show how to decrease the uncer-tainty inher-ent in com-mon summary-statistic approaches through the use of a like-li-hood based mod-el-ing approach. Results/Conclusions We demon-strate that com-monly used cor-re-la-tion tests applied to sum-mary sta-tis-tics such as auto-cor-re-la-tion and vari-ance are both inap-pro-pri-ate and insuf-fi-cient tests of early warn-ing signals.Our method esti-mates directly the para-me-ters of a gen-er-al-ized model of the bifur-ca-tion pos-tu-lated by early warn-ing sig-nals the-ory, with and with-out the pres-ence of a grad-ual change lead-ing towards col-lapse. Using Monte Carlo sim-u-la-tion we gen-er-ate the dis-tri-b-u-tion of warning-signal sta-tis-tics expected under each model. From this we can quan-tify the risk of false alarms and missed detec-tion. We then show how apply-ing this approach to the data directly rather than the sum-mary sta-tis-tic increases the power of detec-tion. We illus-trate the approach in both sim-u-lated and empir-i-cal data of sud-den eco-log-i-cal shifts.