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The authors have declared that no competing interests exist.

Recent studies have interpreted patterns of remotely sensed tree cover as evidence that forest with intermediate tree cover might be unstable in the tropics, as it will tip into either a closed forest or a more open savanna state. Here we show that across all continents the frequency of wildfires rises sharply as tree cover falls below ~40%. Using a simple empirical model, we hypothesize that the steepness of this pattern causes intermediate tree cover (30‒60%) to be unstable for a broad range of assumptions on tree growth and fire-driven mortality. We show that across all continents, observed frequency distributions of tropical tree cover are consistent with this hypothesis. We argue that percolation of fire through an open landscape may explain the remarkably universal rise of fire frequency around a critical tree cover, but we show that simple percolation models cannot predict the actual threshold quantitatively. The fire-driven instability of intermediate states implies that tree cover will not change smoothly with climate or other stressors and shifts between closed forest and a state of low tree cover will likely tend to be relatively sharp and difficult to reverse.

The emerging idea that tropical forest and savanna may be alternative stable states over a range of climatic conditions [

Here we use remotely sensed data on fire frequencies at 500 m resolution, tropical tree cover and climatic variables to develop a simple model that we use to evaluate whether the fire feedback hypothesis is consistent with observed patterns of tree cover and fire, and present simulations that provide a mechanistic explanation of those patterns.

Mean annual precipitation (MAP) and tree cover explain much of the variation in fire frequency (

A. All tropics, B. South America, C. Africa and D. Australia and Asia.

A. The dots represent the average probability that a grid cell (500x500 m) catches fire per year as a function of tree cover (for each 1% bin). For parameters of the fitted line, see ^{-1}) (blue line) is equal to the ^{-1}) due to fire (red line) the system is in equilibrium. The equilibrium at intermediate tree cover is unstable. The probability density in the lower panel is produced by exposing the model to a stochastic environment. The parameters of the model _{fire} = 0.0004, and applying additive, normally distributed noise with a standard deviation of 0.003 using the Fokker-Planck equation ([

Below each figure the corresponding frequency of each tree cover class of 1% is given. A. All tropics, B. South America, C. Africa and D. Australia and Asia. Fitted line with the best AIC:

The universality of the sharp drop in fire frequency above a critical tree cover is consistent with the idea that percolation might play a role in determining the impact of fire on landscapes [

Fire in grass spreads within connected grass areas; trees do not burn. The areas indicate the ranges between the 5^{th} and 95^{th} percentiles of the average probability of fire calculated in 100 independent runs. Yellow area: randomly dispersed overlapping circles of grass in a continuum of trees; blue area: randomly dispersed overlapping circles of trees in a continuum of grass.

To address the question under which conditions a drop in fire probability above a critical tree cover could cause intermediate tree cover to be unstable, resulting in alternative stable states of low and high cover, we need to consider the role of fire in the overall dynamic equilibrium of tree cover. Various modeling approaches have been developed for this, ranging from simple [

Obviously, fire-induced tree mortality is highly stochastic and depends on a range of factors. For instance, most tree species in the savanna biome are typically less tall [

The intersection points of the growth and the loss curves represent equilibria where growth balances average loss. It can be seen that the intersection point around the threshold where loss due to fire drops, is an unstable equilibrium, as any perturbation from this specific tree cover will result in either increased tree cover towards the closed forest state, or decrease towards a very low tree cover. The existence of such an unstable equilibrium is explained by a positive feedback causing self-propagating change away from the unstable point [

The universality of the sharp change in tropical fire frequency around ~40% tree cover that we find is striking. Also striking is the observation that across the global tropics intermediate tree cover is systematically rare [

Our simulations of the expected effects of percolation on fire frequencies illustrate the fascinating possibility that the steepness of the drop in fire frequency around a certain tree cover results from a generic fire percolation phenomenon. However, our analysis also shows that the actual tree cover at which such a percolation would happen cannot be predicted from the kind of models discussed in the literature, as the outcome depends strongly on the choice of simplifying assumptions. Nonetheless, our results do confirm that the universality of the patterns of fire and tree cover we find across the tropics are consistent with percolation as an explanation, provided that conditions such as geometry of tree distributions and their capacity to act as firebreaks are roughly universal too.

Clearly, even if the characteristics affecting percolation would be more or less invariable, there will be other relevant aspects that vary between regions. For instance, fire probabilities differ markedly between continents. The causes of those differences are still poorly understood, but may include a range of factors related to both ecology [

Our inferred critical cover for tropical forest should not be confused with another possible critical cover resulting from large-scale forest-rainfall feedbacks. Forests can enhance regional rainfall, implying that a certain level of forest loss could change regional climate to a point where it becomes unfavorable for forests themselves [

We used the standard MODIS burned-area product MCD45 Collection 5 [

We tested for climatic, topographic and anthropogenic effects on the probability of fire using twelve relevant variables. Specifically, mean annual precipitation (MAP), precipitation of the wettest quarter (PWQ) and precipitation of the driest quarter (PDQ) at 1 km resolution, which were downloaded from the WorldClim website [^{2} where different livestock species are converted to a mean standard weight of 250 kg per individual. All spatial data were resampled to a consistent resolution of 0.1×0.1°, after which we took a sample of 1% of the data points (n = 2737).

The net change in tree cover (

As growth function we use the generalized logistic growth function of Richards (growth rate ^{-1}), in which the shape of the density dependence can be adjusted by adding one extra parameter, the power

The loss due to fire is proportional to the probability of fire (_{fire} _{fire}). The power γ is by default set to 1 but can be used to evaluate sensitivity to the model definition.

Alternatively, we assume that the relative loss of tree cover when catching fire is proportional to the tree cover:

The annual probability of catching fire as a function of the tree cover (_{fire}

Asymmetric optimum function ‘Double Hill function” (powers _{3} and _{5}, half-saturation _{2} and _{4}):

Sigmoidal Hill function (power _{3} and half-saturation _{2}):

Mirrored Hill function (power _{3} and half-saturation _{2}):

Standard logistic regression (parameters _{1} and _{2}):

Logistic regression with optimum (parameters _{1,} _{2} and _{3}):

These functions are not mechanistic, but are simply meant for obtaining a good fit. The parameters _{1-5} determine the shape of the functions and are fitted using the procedure described above. We selected the most parsimonious model using the Akaike Information Criterion (AIC) assuming a binomial distribution for the fire frequency (

Imagine savanna to be a very large lattice of grass. At random, a site of the lattice can be occupied by trees with a probability

In this approach, circles (or other shapes) are randomly distributed in a continuum of another state. We considered two possibilities: circular trees being randomly dispersed on a continuous space of grass, or circular grass patches being randomly dispersed on a continuum of trees. For computational convenience, we approximated continuum percolation by drawing overlapping circles with a radius of 20 units at random positions on a fine lattice of 1000x1000 units. We continued drawing these overlapping circles until we reached a certain tree cover. We repeated these simulations considering the continuum to be trees.

In all models, we calculated the average probability that any patch burns if a randomly chosen grass patch ignites. First, we determined the sizes of all clusters of connected grass patches _{i}. The probability that a randomly ignited cell belongs to cluster _{g} grass cells that belong to that cluster (= _{i}_{g}_{i}_{av}) equals:

This figure is not intended to be a predictive model, but we try to explain the differences in fire frequency between these continents.

We did not perform multiple regression because of covariations among variables. All variables are divided in 100 bins (except SPID and SPIW, which are discontinuous). The area of the circles indicates the frequency of observation within each bin (see legend). A. Altitude (m), B. Tree cover (%), C. Mean Annual Precipitation (MAP) (mm yr^{-1}), D. Precipitation of Wettest Quarter (PWQ) (mm yr^{-1}), E. Precipitation of Driest Quarter (PDQ) (mm yr^{-1}), F. Coefficient of variation of annual precipitation (mm yr^{-1}), G. Markham’s seasonality index (MSI) (-), H. Percentage of severely wet years (SPIW) (%), I. Percentage of severely dry years (SPID) (%), J. Livestock density in number of livestock units (km^{-2}), K. Human rural population density per grid cell ^{10}log (x+1) (-), L. Human population density per grid cell ^{10}log (x+1) (-).

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The grayed areas approximate the range of logistic growth functions where alternative stable states are possible. a: MAP<500 mm yr^{-1}; b: MAP between 500 and 1000 mm yr^{-1}; c: MAP between 1000 and 1500 mm y^{-1}, the maximum probability of fire here is 0.27 yr^{-1}; d: MAP between 1500 and 2000 mm y^{-1;} e: MAP between 2000 and 2500 mm yr^{-1}; f: MAP > 2500 mm yr^{-1}.

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The frequency distribution shows how often the precipitation class occurs. The lines are fitted logistic curves with optimum (for parameter values see

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The drop due to the percolation point is dependent on the assumptions about the connectivity between the cells. Yellow: “square grid”: fire can spread in 4 directions in the lattice, cyan: “hexagonal grid” fire can spead in 6 directions in the lattice; blue “8-neighbors” like the quare grid but fire can also spead in 4 diagonal directions.

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These intersection are either stable (solid circle) or unstable (open circle) equilibria. At the y-axis there is an additional unstable trivial equilibrium (open circle). In all these cases, the model has alternative stable states (see also ^{β}) (^{γ} (_{B}^{α} (α = 0 (blue line), 0.5 (green line), 1 (red line)) for two levels of growth rate (r = 0.0001 (purple line), 0.0002 (cyan line)). For other parameters see

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Only the six best models based on AIC are shown. ^{-1}); tree cover = tree cover (%); PWQ = Precipitation of Wettest Quarter (mm yr^{-1}); PDQ = Precipitation of Driest Quarter (mm yr^{-1}), std. precip. = standard deviation of annual precipitation (mm yr^{-1}), cv precip. = coefficient of variation of annual precipitation (mm yr^{-1}), MSI = Markham’s seasonality index (-), TLU = Livestock density in number of livestock units (km^{-2}). _{1}‒p_{5} differ for each of these formulas, and are simply meant to describe empirical patterns.

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The range of the steepest drop is defined as the area where the fire frequency is between 25% and 75% of the maximum.

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The data analyzed in this paper were downloaded from the following publicly available websites.

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This work was carried out under the programme of the Netherlands Earth System Science Centre (NESSC) and received funding from the European Union’s Horizon 2020 research and innovation Programme under the Marie Sklodowska-Curie grant agreement No 643073 (ITN CRITICS). A.S. was supported by SENSE Research School. S.H. acknowledges support by the EU FP7 projects BACCHUS (grant agreement no. 603445) and LUC4C (grant ag. No. 603542). The data reported in this paper were extracted from the publicly available sites of MODIS, WorldClim, CRU, HYDE and Gridded livestock of the world (see