Thresholds for Boreal Biome Transitions

Transcription

1 Supporting Online Material for: Thresholds for Boreal Biome Transitions Marten Scheffer 1, Marina Hirota 1, Milena Holmgren 2, Egbert H. Van Nes 1 and F. Stuart Chapin, III 3 1 Department of Aquatic Ecology and Water Quality Management, Wageningen University, P.O. Box 47, NL-6700 AA, Wageningen, The Netherlands 2 Resource Ecology Group, Wageningen University, P.O. Box 47, NL-6700 AA, Wageningen, The Netherlands 3 Institute of Arctic Biology, University of Alaska Fairbanks, Fairbanks, AK 99775, USA Contents 1. Effects of alternative stable states and their resilience on frequency distributions of states Multimodality of tree cover on separate continents Frequency distributions of tree cover on separate continents Permafrost associated to the different tree cover states Probability of forest as a function of temperature and precipitation Frequency distributions of temperature and precipitation Effect of rates of change and stochasticity on probability densities Computed stability landscapes, attractors and repellors References

2 1. Effects of alternative stable states and their resilience on frequency distributions of states If a system has only one stable state, stochastic perturbations and environmental variation will tend to cause indicators of the state to be unimodally distributed around that attractor. By contrast, if there are alternative attractors, and stochastic disturbances are strong and common enough, a collection of snapshots of the state will tend to be multimodal (Figure S1 B and C). In fact these frequency distributions reflect the approximate shapes of the basins of attraction around the alternative states. The most frequent states roughly indicate where the attractors are, while rarest intermediate states reflect the position of the repelling border between the basins of attraction. This information may therefore be used as an indication of the resilience of the alternative states. In the example (Figure S1) we illustrate the effect of a change in attractors through a simulation with a very slow change of a parameter (c) that brings the system from a situation with one stable state, through a phase with alternative stable states to a situation with another single stable state. Snapshots of many instances of such a system over a spatial gradient of a driving parameter is analogous (1), when it comes to interpreting the frequency distributions in terms of attractors and repellors. Prob. density Potential State of the system Control parameter (c) A Time State of the system B D State of the system Figure S1 Frequency distributions (B and C) of the state reveal the approximate positions of attractors and their basins of attraction (D and E) in simulations of a stochastically perturbed system with alternative attractors (A). The data in this example are generated with a model of overexploitation(2): 1 with different additive and multiplicative stochastic terms (3) (K=11). Over time, parameter c is linearly increasing causing the model to go from a situation with one stable state (around 10), through a phase with alternative stable states to a situation in which overexploited low biomass leads to a unique stable state (modified from: 4). C E 2

3 2. Multimodality of tree cover on separate continents Figure S2 Probability density functions for boreal Eurasia (a) and North America (b) constructed from a subsample (n = 10,000) of the arcsine transformed data (bars) described as the weighted sum (solid curve) of the three normal distributions (dashed). The overall distribution is best described by four modes for Eurasia and by three modes for North America (see Tables S1 and S2). Note that tree cover values are arcsine-squared-root transformed. Table S1. The BIC criterion of models with a different number of fitted density distributions. The model with the minimal BIC (in bold) has the optimum fit. clusters N. America Eurasia Total ) ) No convergence. 3

4 Table S2. Combinations of normal distributions fitted to the density distributions of arcsine transformed tree cover for North America (top table), Eurasia (middle table) and both continents combined (lower table). The optimum number of modes according to the Bayesian Information Criterion is four for Eurasia, three for North America and four for both continents combined (See Table S1). North America: k μ σ fraction Eurasia: k μ σ fraction All regions: k μ σ fraction

5 3. Frequency distributions of tree cover on separate continents Figure S3 Frequency distributions of tree cover in Eurasia (top 6 panels) and North America (lower 6 panels) for different ranges of mean July temperature averaged for the period Other is primarily shrub, herbaceous, and mosaic vegetation classes (see item 1 of Material and geoinformation processing section for the definition of the 4 classes, according to GLC2000 map). The distinct modes suggest four underlying stable states in Eurasia: forest, dense savanna-like woodland, sparse woodland, and treeless (tundra or steppe), separated by relatively rare and therefore apparently unstable states around 10, 30 and 60% tree cover. In North America there is no statistical evidence of the sparse woodland mode (Table S1). Note that tree cover values are arcsine-squared-root transformed. 5

6 4. Permafrost associated to the different tree cover states Figure S4 Frequency of permafrost classes in the pixels with alternative tree cover states. 6

7 5. Probability of forest as a function of temperature and precipitation ,..... Figure S5 - Logistic regression models of the probability that a 500x500m grid-cell has boreal forest (tree cover >= 60%) as a function of mean annual precipitation (P, panel a), mean July temperature (T, panel b) (both averaged for the period ), and the combination of the two (panel c). All terms in the logistic regression equations depicted in panels a, b and c are highly significant (p < 0.05). 7

8 6. Frequency distributions of temperature and precipitation Figure S6 Frequency distributions of mean annual precipitation (left) and mean July temperature averaged for the period in the grid-cells used for our analysis of tree cover. 8

9 7. Effect of rates of change and stochasticity on probability densities 10 A biomass grazing rate 3 B C D pdf potential biomass biomass biomass Figure S7 (adapted from (1)): The effect of the intrinsic rates of change and the level of stochastic perturbations on the probability distribution of states illustrated by means of a simple bi-stable model (2) of a grazed population (N) 1 where r is the growth rate, K the carrying capacity, c a maximum grazing rate, H the half-saturation of the Holling type II functional response, W is a normally distributed Wiener process, and the scaling factor is used to tune the slowness of the system. To obtain snapshots of this model in time we drew 1000 random initial conditions (between 0 and 10) and ran the model for 1100 steps. The first 100 steps were discarded, and after that each 100 steps one value was saved. We analysed these values using potential analysis (5, 6). The non-stochastic version of the model can have two alternative stable states over a range of conditions (A). The red dashed line indicates the used grazing rate. The probability density (pdf) and the estimated potentials (5) are calculated for (B) a slow system ( =0.03), (C) the default conditions ( = 1; c = 2.1; H = 1; K = 10; r = 1; = 0.05), and (D) is a highly stochastic system = 0.2. Both slowness of a system and stochasticity of the environment make the modes around the equilibria less pronounced, and this is reflected in the shapes of computed stability landscapes, computed from the generated data. 9

10 8. Computed stability landscapes, attractors and repellors Figure S8 Approximate position of attractors (solid blue curves) and repellors (dashed blue curves) represented in the main text (Fig. 2), interpreted from minima (solid dots) and maxima (open dots) in the computed stability landscapes (see section 3 on Computation of stability landscapes from the data) for the global data-set. Shading reflects the height of the stability landscape (lighter is higher). Figure S9 Approximate position of stable states and unstable repellors reflected by minima (solid dots) and maxima (open dots) in the computed stability landscapes (see Methods) for the separate continents. Shading reflects the height of the stability landscape (lighter is higher). 10

11 References 1. Van Nes EH, Holmgren M, Hirota M, & Scheffer M (2012) Response to comment on "Global resilience of tropical forest and Savanna to critical transitions". Science 336(6081). 2. May RM (1977) Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature 269(5628): Dakos V, et al. (2012) Early warnings of critical transitions: methods for time series. PLoS ONE 7(7):e Scheffer M, et al. (2012) Anticipating critical transitions. Science accepted. 5. Livina VN, Kwasniok F, & Lenton TM (2010) Potential analysis reveals changing number of climate states during the last 60 kyr. Climate of the Past 6(1): Hirota M, Holmgren M, Van Nes EH, & Scheffer M (2011) Global resilience of tropical forest and savanna to critical transitions. Science 334(6053):