Table 1.
Indicators considered in this paper.
Figure 1.
Simulation with the eutrophication model using noise added to recycling (eq. 10).
A. Time series of the state variable. Note that the sample includes data before and after the transition. B. Diffusion and S2 versus a. C. Diffusion and S2 versus time. D. Diffusion and S2 versus time for a short time interval near the transition.
Figure 2.
Simulation with the eutrophication model using additive noise (eq. 11).
A. time series of the state variable. Note that the sample includes data before and after the transition. B. Diffusion and S2 versus a. C. Diffusion and S2 versus time. D. Diffusion and S2 versus time for a short time interval near the transition.
Figure 3.
Simulation with the eutrophication model using noise added to recycling (eq. 10).
A. Time series of the state variable. Note that the sample includes data up to, but not after, the transition. B. Diffusion and S2 versus a. C. Diffusion and S2 versus time. D. Diffusion and S2 versus time for a short time interval near the transition.
Figure 4.
Monte Carlo simulations with the eutrophication model using noise added to recycling (eq. 10).
The samples ended at x = 2. A. Parameter standard deviations for drift, diffusion and S2 versus x. B. Drift ± standard deviation versus a. C. Diffusion ± standard deviation versus a. D. S2 ± standard deviation versus a.
Figure 5.
Time series plots with confidence bands (± standard deviation) for the eutrophication model, interpolated from the functions in Fig. 4.
A. Drift. B. Diffusion. C. Conditional variance S2.
Figure 6.
Simulation with the food web model (Supporting Information S2).
A. Time series of phytoplankton. Note that the sample includes data before and after the transition. B. Diffusion (times 1000) and S2 versus a. C. Diffusion (times 1000) and S2 versus time. D. Diffusion (times 1000) and S2 versus time for a short time interval near the transition.
Figure 7.
Simulation with the food web model (Supporting Information S2).
A. Time series of phytoplankton sampled up to x = 40, just before the transition. B. Diffusion (times 100) and S2 versus a. C. Diffusion (times 100) and S2 versus time. D. Diffusion (times 100) and S2 versus time for a short time interval near the transition.
Figure 8.
Results of multivariate nonparametric analysis of the same food web simulation depicted in Fig. 7.
For multivariate analysis, time series for planktivore, herbivore and phytoplankton were analyzed. Time series are presented for 100*diffusion (solid line) and conditional variance (dashed line). (A) Largest eigenvalue of the diffusion and conditional variance matrices. (B) Variance of planktivores from the diffusion and conditional variance matrices. (C) Variance of herbivores from the diffusion and conditional variance matrices. (D) Variance of phytoplankton from the diffusion and conditional variance matrices.
Figure 9.
Monte Carlo simulations with the Food Web model (Supporting Information S2) showing results for phytoplankton.
The samples included data for X ≤ 40, just below the transition. A. Parameter standard deviations for drift, diffusion and S2 versus a. B. Drift ± standard deviation versus a. C. Diffusion ± standard deviation versus a. D. S2 ± standard deviation versus a.
Figure 10.
Time series plots with confidence bands (± standard deviation) for the food web model, interpolated from the functions in Fig. 9.
A. Drift. B. Diffusion. C. Conditional variance S2.