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Figure S1. Thresholds and residuals. Regime assignment and residuals for each of the four individual models: phytoplankton (a), zooplankton (b), gelatinous plankton (c), and planktivorous fish (d). The first column (a–c) shows the temporal evolution of the threshold variables (phosphorus, fish, and fish, respectively) and their regime allocation: those points above the threshold (represented by the blue line, θ) are shown in red (high regime) while those below appear in black (low regime). The bivariate threshold of the fish model (d) is explained in Fig. 2 (see legend there for details). The following four columns show the inspection of residuals for the assumptions of independence (autocorrelation function), normality (Q-Q plot), and homoscedasticity (residuals vs. time and residuals vs. fitted values). Apart from visual inspection, Shapiro (normality) and Breusch-Pagan (homoscedasticity) tests were performed (values not shown). As commented earlier (see text), only jellyfish residuals violated the latter assumptions.

Figure S2. Predatory fish model. Plots showing the effect of zooplankton on predatory fish (a), observations vs. predictions (b), and residual statistics: autocorrelation (c) and normality (d).

Figure S3. Alternative gelatinous model partial effects and residuals. Plots showing the estimation of the threshold value (a), all the partial effects (b–e), regime assignment to threshold variable (f), residual statistics (g–h), and predictive performance (i) for the gelatinous plankton model when accounting for the two outliers detected in the residuals.

Figure S4. Observations (black) and out-of-sample predictions (purple) of the biomasses of phytoplankton (a), zooplankton (b), gelatinous plankton (c), and planktivorous fish (d).

Figure S5. Threshold estimation (a) and partial plots (b–d) for the phytoplankton cross validation models. Each plot consists of the overlaid results from the 36 subset models. The numbers in parentheses on the y-axis indicate the averaged estimated degrees of freedom. See also Fig. 2 legend for further details.

Figure S6. Threshold estimation (a) and partial plots (b–e) for the zooplankton cross validation models calculated as for PHY (see legend in Fig. S5).

Figure S7. Threshold estimation (a) and partial plots (b–e) for the jellyfish cross validation models calculated as for PHY (see legend in Fig. S5).

Figure S8. Threshold estimation (a) and partial plots (b–d) for the fish cross validation models calculated as for PHY (see legend in Fig. S5).

Figure S9. 1970s phase space plots for all the phosphorus scenarios. Illustration of the phase space trajectories for zooplankton and phytoplankton (a–b), and planktivorous fish and zooplankton (c–d) for all the phosphorus regimes during the 1970s major shift. The lower phosphorus regimes are shown to the left (scale of reds) and the higher to the right (greens). The legend shows the value of the slopes to the corresponding percentage of the mean increase or decrease.

Figure S10. 1970s phase space plots for all the phosphorus scenarios (fish observations). Same as Fig. S9 but using observations for fish biomass, not simulations.

Figure S11. 1970s phase space plots for all the fish scenarios. As Fig. S9 but for fish regimes, lower abundance to the left (reds) and higher to the right (greens).

Figure S12. Observations vs. simulations phase space plots. Phase space plots of consumer (driver) against resource (response) for the observations (left column, in black) and simulations (right column, in red) during the 90s regime shift. Standardized data from Fig. 4 are used. Numbers on the plots are years. Dashed lines are the slopes of the linear regression lines joining all the points.

Figure S13. 1990s phase space plots for all the phosphorus scenarios. Illustration of the phase space trajectories for gelatinous plankton and zooplankton (a–b), and zooplankton and phytoplankton (c–d) for all the phosphorus regimes during the 1990s second major shift. The lower phosphorus regimes are shown to the left (reds) and the higher to the right (greens). The legend shows the value of the slopes to the corresponding percentage of the mean increase or decrease.

Figure S14. 1990s phase space plots for all the fish scenarios. As Fig. S13 but for fish regimes, lower abundance to the left (reds) and higher to the right (greens).

Table S1. gCVs. Genuine Cross-validation scores (gCV) for the non-additive GAM model (Threshold GAM) and the fully additive equivalent for all the trophic levels.

Table S2. Predatory fish (PRE) models results. Intercept, estimated degrees of freedom (edf) and significance (P-value) of the covariate, and R-squared (R2) for the predatory fish model (see Fig. S2). It also includes a dummy variable accounting for the effect of an outlier (o1).

Table S3. GEL models results – accounting for outliers. Intercept, estimated degrees of freedom (edf) and significance (P-value) of the various effects, and R-squared (R2) for the gelatinous plankton model when including two dummy variables (o1 and o2) targeting the two outliers observed in the residuals (Fig. S1c). This model summary corresponds to Fig. S3.

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