Electronic Supporting Information for
Bistability, Spatial Interaction and the Distribution of Tropical
Forests and Savannas
Arie Staal1, Stefan C. Dekker2, Chi Xu3, and Egbert H. van Nes1
1
Aquatic Ecology and Water Quality Management Group, Department of Environmental Sciences,
Wageningen University, P.O. Box 47, 6700 AA Wageningen, The Netherlands
2
Department of Environmental Sciences, Copernicus Institute for Sustainable Development, Utrecht
University, P.O. Box 80115, 3508 TC Utrecht, The Netherlands
3
School of Life Sciences, Nanjing University, Nanjing 210093, China
E-mail addresses: [email protected], [email protected], [email protected], [email protected]
Appendices
S1: Effects of spatial interactions on local coexistence
S2: χ2 tests for probabilities of local coexistenceS3: Mapping inferred Maxwell points vs. local forestsavanna coexistence
S4: Potentials of tree cover
S5: Examples of local forest-savanna coexistence
S6: Google Earth files of results on 0.5° scale (online)
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Appendix S1: Effects of spatial interactions on local coexistence
Theoretical work (Van de Leemput and others 2015) has shown that if a system has alternative stable
states, spatial interactions destabilize local coexistence of the states. To illustrate this, we made a
simple model that describes alternative stable states in tree cover (adapted from Van Nes and others
2014). The model includes a tree cover-fire feedback and was parameterized such that it reasonably
fits the bimodal tree cover against mean annual precipitation (MAP) for South America and Africa.
Figure S1A shows the model equilibria against MAP, together with the stable and unstable equilibria
as inferred using potential analysis (Livina and others 2010; Hirota and others 2011) on 1‰ of the tree
cover data from South America and Africa. We combined both continents and excluded human-used
areas (see main text). The model equation for tree cover T (fraction) is:
𝑑𝑇
𝑑𝑡
=ℎ
𝑃
𝑃 +𝑃
𝑇
∗ 𝑟𝑚 ∗ 𝑇 ∗ (1 − 𝐾) − 𝑚𝑓 ∗ 𝑇 ∗
𝑝
ℎ𝑓
𝑝
ℎ𝑓 +𝑇 𝑝
where P is mean annual precipitation (mm); hP is the half saturation of the growth of T (250 mm y-1);
rm is the maximum growth rate of T (0.25 y-1); K is carrying capacity of T (1), mf is the maximum fireinduced mortality of T (0.18 y-1), hf is the half saturation for the fire-induced mortality of T (0.57); p is
an exponent in the function for fire-induced mortality of T (3.3). The model has bistability of forest (T
≈ 0.8) and savanna (0 < T < ~0.3) for P = 1040–3000 mm y-1. It has a Maxwell point at P = 1650.31
mm y-1. Note that this model can be useful for demonstrating general behavior of systems with
alternative stable states, but not for making quantitative predictions of, for example, the effect of
spatial interactions on hysteresis.
We first implemented the model for two interacting patches with some amount of tree cover. Spatial
interactions were modeled as the exchange of tree cover: each patch diffused a fraction d of its tree
cover to the neighboring patch each year. This diffusion had no effect on the bistability range: no
matter the value of d, initializing both patches for either forest or savanna anywhere in the bistability
range (P between 1040–3000 mm y-1) resulted in stable forest, respectively savanna. However, d did
affect at which P the two patches can exist in contrasting states in a stable way. The higher was d, the
smaller became the precipitation range at which forest and savanna could coexist (Figure S1B). This
continued until (from about d = 0.01 y-1) local coexistence was only possible at the Maxwell point. In
this case the system is said to follow the Maxwell convention (Gilmore 1979). (In this example, stable
local coexistence occurred at slightly lower P than the Maxwell point due to an artifact of using only
two patches.)
We implemented the model also for a larger grid of patches (401 × 50) and imposed a gradient in P
from 500–3500 mm y-1 (Figure S1C). We initialized the model at forest (T = 0.8), imposed high
2
diffusion so the Maxwell convention applies (d = 0.5) and ran a simulation of 2000 years. This
resulted in a traveling front of the forest-savanna boundary until a stable configuration was reached at
which savanna was present at the P values below the Maxwell point and at which forest was present at
the P values above the Maxwell point. The traveling front slowed down as the Maxwell point was
approached until the velocity became zero at the Maxwell point itself. This result is inevitable for any
initialization of the grid. All above calculations were done in GRIND for MATLAB.
Figure S1: The theoretical effect of spatial interactions on a bistable forest-savanna system. A) Potential
analysis indicates the stable (solid dots) and unstable points (open dots) of tree cover (South America and Africa
combined) for different levels of MAP. A simple model for tree cover with forest and savanna as alternative
stable states was fitted on these data. The model has fold bifurcation points at 1040 mm and 3000 mm. B) If the
model is implemented for two interacting patches with diffusion d (fraction of tree cover diffused to the other
patch in each year), the MAP range at which the patches can exist in contrasting states becomes narrower. C) If
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the model is implemented on a grid (401 × 50 cells; initial tree cover = 0.8) along a gradient of MAP, a stable
boundary between forest (blue) and savanna (red) settles at the Maxwell point if diffusion is sufficiently high
(here, d = 0.5; the model was ran for 2000 years).
As shown, a situation in which alternative stable states follow the Maxwell convention only occurs in
case there are strong spatial dynamics at play, which makes it an extreme scenario. At the other
extreme are systems with alternative stable states that are said to follow the delay convention. In such
cases spatial dynamics do not reduce hysteresis in the system at all, as with d = 0 in the above
demonstration. For forests and savannas, the distinction between the delay convention and the
Maxwell convention reflects the distinction between, on the one hand, these ecosystems having large
hysteresis, and, on the other hand, climate determinism. However, even if the Maxwell convention (or
climate determinism) applies, local conditions such as in soil differ. Consequently, the stability of
forests and savannas, including their tipping points and Maxwell points, would also differ. Such
spatial heterogeneity can be expected to increase the range of climatic conditions at which local
coexistence is found (Favier and others 2004; Van Nes and Scheffer 2005; Favier and others 2012).
Stochasticity in the ecological dynamics can also be expected to increase this range of conditions
(Villa Martín and others 2015). Furthermore, not all local coexistence is necessarily stable.
Disturbances to forests, for example, may temporarily cause local coexistence in tree cover. It should
be noted that such temporary local forest-savanna coexistence would be more probable at conditions
close to the Maxwell point than further from it, as in theory, the recovery to the most stable state slows
down when conditions approach the Maxwell point (Van de Leemput and others 2015).
We also used the model to demonstrate the combined effects of spatial interactions and spatial
heterogeneity in conditions on the occurrence of local forest-savanna coexistence for different values
of P (550 through 3550 mm y-1 in 100 mm y-1 increments). We implemented the model for a 32x32
grid of patches with and without spatial heterogeneity in those patches. Spatial heterogeneity was
included by randomly assigning a value for the growth rate rm from a normal distribution with mean
0.25 and standard deviation 0.03. We ran the model without spatial interactions (d = 0 y-1), with
moderate spatial interactions (d = 0.005 y-1) and with strong spatial interactions (d = 0.01 y-1). Initial
tree cover was assigned randomly between 0 and 1 for each patch. Here we assume that on a long term
random disturbances to the patches justify using random initial conditions. The grid had periodic
boundaries and the model was always run for 2000 years. After each run we determined whether there
was forest-savanna bimodality, indicating local coexistence, following the same method as for the
analyses on the MODIS tree cover data (see main text). We repeated this procedure 100 times and for
each parameterization we determined the fraction of runs after which forest and savanna coexisted.
4
Without spatial interactions between patches and spatial heterogeneity in growth rate the delay
convention applied, and the probability of forest-savanna coexistence was roughly equally distributed
throughout the bistable range (1050–2950 mm y-1) (Figure S2A). With spatial interactions added,
coexistence was only found around the Maxwell point (Figures S2B–C). In this case the Maxwell
convention applied. The pattern changed when spatial heterogeneity in growth rate was included. With
this spatial heterogeneity, the range of MAP at which forest and savanna coexist was consistently
wider than without spatial heterogeneity. Even at the strongest simulated spatial interactions there was
coexistence in a large part of the bistability range.
Figure S2: The modeled probability of local forest-savanna coexistence at different levels of spatial interaction
between patches and spatial heterogeneity in growth rates of tree cover. The same model was used as for Figure
S1. In the “no spatial interactions” runs (A and D), the diffusion term was set to 0. In the “moderate spatial
interactions” runs (B and E) the diffusion factor d was set to 0.005 y-1 and in the “strong spatial interactions”
runs (C and F) d was set to 0.01 y-1. To simulate spatial heterogeneity (D–F), the tree-cover growth rate rm in
each patch was sampled randomly from a normal distribution with mean 0.25 and standard deviation 0.03.
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Appendix S2: χ2 tests for probabilities of local coexistence
We determined whether the probability of local forest-savanna coexistence in the 0.5° cells varied
within the bistability ranges of MAP and MSI. With potential analyses on the tree cover data it was
inferred that the bistability ranges were 1080–2020 mm (MAP for South America; Figure 3), 1280–
2060 mm (MAP for Africa; Figure 3), 12–55% (MSI for South America; Figure 4) and 12–34% (MSI
for Africa; Figure 4). We binned the 0.5° cells within the bistability ranges (bin sizes 100 mm for
MAP and 3% for MSI) and performed χ2 tests for independence. Only MSI in Africa was not
significant on the 5% significance level (p = 0.213); other p values were 0.008 (for MAP in South
America), 0.047 (MAP in Africa) and 0.000 (MSI in South America). On these three bistability ranges
we performed post-hoc pairwise two-sided χ2 tests between the bin with highest probability of local
forest-savanna coexistence and the other bins. We considered the 95% confidence interval for the
MAP or MSI at which the probability of local coexistence peaks to be the range of bins that did not
differ significantly with the bin of the highest probability. We accounted for Type I errors by using
Keppel’s modification of the Bonferroni correction for α inflation:
𝛼𝑐𝑜𝑟𝑟 =
1 − (1 − 𝛼)𝑐
𝑐
where αcorr is the corrected significance level, α = 0.05 and c is the number of pairwise comparisons
(number of bins – 1). The p values of the pairwise χ2 tests are shown in Table S1.
Table S1: p values of the pairwise χ2 tests of bins of mean annual precipitation (MAP, in mm) within the
bistability ranges for South America and Africa, and those of Markham’s Seasonality Index (%) for South
America. Results tagged with letters indicate no significant difference with the reference bin, being the one with
the highest probability of local coexistence (where only a letter is given).
MAP (mm)
1100–1200
1200–1300
1300–1400
1400–1500
1500–1600
1600–1700
1700–1800
1800–1900
1900–2000
South America
0.001
0.013
0.110a
0.302a
a
0.081a
0.402a
0.889a
0.931a
Africa
N/A
N/A
0.001
0.040
0.224b
b
0.201b
0.313b
0.377b
MSI (%)
13–16
16–19
19–22
22–25
25–28
28–31
31–34
34–37
37–40
40–43
43–46
46–49
49–52
52–55
6
South America
0.001
0.014
0.001
0.258
0.000
0.000
0.000
0.005
0.000
0.022
0.020
0.315c
c
0.068
Appendix S3: Mapping inferred Maxwell points vs. local forest-savanna coexistence
Figure S3 shows how distant the cells at 0.5° resolution which displayed local forest-savanna
coexistence are from the estimated Maxwell points in MAP and MSI. It can be seen that the climatic
conditions at which local coexistence was found have a large range. Notably, in South America,
bimodal cells with a MAP around 7000 mm are observed (Figure S3A). Local forest-savanna
coexistence will not be stable at such extreme conditions, given that potential analysis indicated that
forest and savanna are only bistable in South America between 1000–2000 mm. It should, however, be
noted that although MAP and MSI are independent variables, there is an interaction effect between
them on the occurrence of bimodal cells. In Africa, for example, bimodal cells at low MSI (<30%) are
clustered around 1600–1700 mm MAP, whereas at high MSI (>30%) the MAP of bimodal cells ranges
roughly between 500–2500 mm (Figure S3B). Besides their interaction effect, the correlations
between the Maxwell points of MAP and MSI vary geographically. This further complicates the
disentanglement of the effects that annual precipitation and its seasonality have on the stability and
distribution of tropical forest and savanna as alternative stable states.
Figure S3: The cells (0.5°) with local forest-savanna coexistence and the estimated Maxwell points of MAP and
MSI for South America and Africa. A and B: the MAP and MSI of local forest-savanna coexistence (black dots)
with lines indicating the estimated Maxwell points for MAP (red) and MSI (blue). C: maps with the locations
where local forest-savanna coexistence was detected (dark) with isolines for the Maxwell points of MAP (red)
and MSI (blue).
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Appendix S4: Potentials of tree cover
Figure S4: Relative potentials of tree cover for different values of mean annual precipitation, Markham’s
Seasonality Index and the number of dry and wet years. Red indicates low potential (high stability) and yellow
indicates high potential (low stability).
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Appendix S5: Examples of local forest-savanna coexistence
A
B
C
D
B
B
Figure S5: Different causes of ‘local forest-savanna coexistence’ on a 0.5° scale. A) A fishbone
pattern of low tree cover within forest in the State of Rondônia in Brazil indicates deforestation as
cause of the detection of local-scale bimodality in tree cover; B) Forest-savanna boundary at the
Sipaliwini savanna at the Suriname-Brazil border. The round shapes of the savanna boundary indicate
the influence of fire on the boundary location; C) Gallery forests within savanna in Cameroon
indicate that fine-scale landscape relief enhances forest-savanna coexistence; D) Savanna patches in
the Congo rainforest near its southern edge in the Democratic Republic of the Congo.
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Appendix S6: Google Earth files of results on 0.5° scale (online)
There are four accompanying .kmz files, which can be viewed in Google Earth. “Google Earth South
America coexistence.kmz” displays the locations of the 0.5° cells in South America that were found to
have local coexistence of forest and savanna. “Google Earth South America all categories.kmz”
displays for South America the results of Figure 5. Likewise two files for Africa have been added.
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