Are strong fire–vegetation feedbacks needed to explain the spatial distribution of tropical tree cover?

Introduction

Fire constrains the current areal extent of tropical forest (Bond et al., 2005; Sankaran et al., 2005), partly by killing trees (e.g. Pinard et al., 1999). Tropical forest trees suffer fire-induced mortality, largely through heating of the vascular cambium (Michaletz & Johnson, 2007) killing above-ground biomass, which may or may not lead to whole-plant mortality (Hoffmann et al., 2012). The future extent of the forest may then be affected by change in fire activity, from two contrasting anthropogenic influences: global climate change and regional land-use change (Coe et al., 2013).

Some clues on future change are available from the current spatial pattern of fire-driven tropical tree mortality (in particular, the decrease in fire from mesic savanna through to tropical forest interiors). Fire behaviour is affected through a series of ‘switches’, depending on fuel connectivity, biomass and moisture (Bradstock, 2010). The current pattern arises partly through two main controls on fire (Archibald et al., 2009): climate and tree cover (other factors such as distance to roads, topography, grazing and the fire tolerance of particular species also play important roles). Wind can also affect the rate of spread (and hence tree mortality), although slow-moving tropical forest fires have high tree mortality rates due to the long heating time involved (Cochrane, 2003).

Climate affects the biomass, type, spatial connectivity and moisture content of grass or leaf-litter fuels (Cochrane, 2003; Krawchuk & Moritz, 2011; Granzow-de la Cerda et al., 2012; Pausas & Ribeiro, 2013; Bowman et al., 2014). In the dry tropics (regions of low vegetation productivity), lack of rainfall limits the biomass or connectivity of grass fuel. In mesic savanna and forest (regions of high productivity), however, the spread of fire is restricted by the moisture content of the fuel (through a limited fuel-drying season). Peak fire, then, tends to occur in regions with intermediate levels of vegetation productivity (van der Werf et al., 2008; Krawchuk & Moritz, 2011; Prentice et al., 2011): the ‘intermediate fire–productivity’ hypothesis (Pausas & Ribeiro, 2013; Bowman et al., 2014). This represents a nonlinear, ‘humped’ large-scale relationship between fire and productivity (see e.g. Fig. 1). This is also seen in the response of fire to climate variability (van der Werf et al., 2008). In dry regions, more rain (especially during the growing season) tends to enhance fire (e.g. in arid central Australia; Murphy et al., 2013). In wet regions, more rain (especially a shorter drying season) reduces fire (e.g. in the Amazon; Brando et al., 2014). At intermediate locations, neither biomass nor fuel moisture is limiting, so the fire–rainfall correlation is weaker.

Tree cover itself also limits the biomass of grass fuels and how dry they are. A closed canopy can suppress light-sensitive C4 grass fuels and maintain higher humidity (Pueyo et al., 2010; Staver et al., 2011a). Both of these effects suppress fire and hence permit greater vegetation growth (positive feedbacks). Many studies have established compelling evidence for important feedback of tree cover on fire activity local to the forest boundary (Cochrane, 2003; Ray et al., 2005; Dantas et al., 2013; Silverio et al., 2013). A sufficiently strong feedback of tree cover on fire can permit high or intermediate tree cover to be alternative stable states for the same climate conditions (Staver et al., 2011a; Moncrieff et al., 2014); mosaic patterns of tree cover are strongly suggestive of this possibility, at least over smaller spatial scales (Bond & Parr, 2010). Thus, the ‘humped’ relationship between fire and productivity (Fig. 1) could partly be controlled by feedback of tree cover on fire.

Separating the influences of climate and tree cover on fire is not straightforward. First, their spatial patterns are partly degenerate: along a rainfall gradient from mesic savanna to forest, increasing rainfall leads to both increased tree cover and increased fuel moisture, both of which suppress fire. For example, Archibald et al. (2009) showed a relatively abrupt decrease in area burned above 40% tree cover in South Africa, apparently suggestive of strong feedback of tree cover on fire. However, their analysis of the main controls over the study region predominantly highlighted environmental factors, such as climate (length of the dry season and annual rainfall) and grazing, rather than tree cover. Uncertainty about how to quantify environmental controls on fire and tree cover is another problem. Neither tree productivity nor the different forms of mortality (both drought and fire-related) are simple functions of rainfall, and many environmental factors contribute (Lehmann et al., 2014). Hence, only approximate estimates are possible (e.g. Murphy & Bowman, 2012). Finally, global observations of fire-induced vegetation mortality are not available. Fire counts or area burned tend to be inferred from observations, but the tree mortality per area burned is not constant (Hoffmann et al., 2012).

It has recently been suggested (Hirota et al., 2011; Staver et al., 2011b; Moncrieff et al., 2014) that feedback of tree cover on fire is strong enough to permit forest and savanna to be alternative stable states across large areas of the tropics. This been inferred over Africa using both observations (Hirota et al., 2011; Staver et al., 2011b) and a model (Moncrieff et al., 2014), albeit for quite different parts of the continent. One key piece of observational evidence here is that the statistical distribution of tree cover in the tropics is trimodal (with peaks at low, intermediate and high tree cover). It has been suggested (Hirota et al., 2011) that this is strong evidence for alternative stable states – an artificial effect of satellite data processing is another possibility (Hanan et al., 2014), although it has been suggested that this may not be a major issue (Staver & Hansen, 2015). A second piece of observed evidence is that both intermediate and high (or intermediate and low) tree cover is found for similar levels of precipitation (Hirota et al., 2011). If the chosen precipitation indices were sufficiently accurate measures of the climate control on tree growth, this would represent evidence for alternative stable states (for given climate conditions).

A key question is whether tree-cover feedback is strong enough to dominate fire behaviour over large spatial scales. For sufficiently high or low rainfall, forest cover is expected to be determined by climate. In the vicinity of the forest boundary, tree-cover feedback can dominate, allowing forest and savanna to be alternative stable states. The spatial scale over which alternative stable states are possible is uncertain, but depends partly on the strength of the feedback of tree cover on fire.

The present study revisits the evidence that feedback of tree cover on fire is strong enough to permit forest and savanna to be alternative stable states across large areas of the tropics, in a similar spirit to the work of Hanan et al. (2014). We ask: are these results only consistent with strong large-scale tree-cover feedback? Could they also be consistent with strong climate control? This is done via a simple model of vegetation competition. We first discuss the basic vegetation competition equation used. We then examine the role of two nonlinear effects that occur even without feedback of tree cover on fire: in the equilibrium solution of the logistic vegetation equation and in the climate-driven intermediate fire–productivity hypothesis. Then, the role of uncertainty in quantifying environmental controls on tree cover is discussed.

Methods

Our study analyses the commonly used logistic equation, which corresponds to the Lotka–Volterra equations reduced to a single dominant fire-sensitive tree species (Mallet, 2012). This section describes how the equilibrium behaviour of this equation may be visualized on plots of mortality versus productivity (or spreading rate) and the two nonlinearities in this equilibrium.

The logistic vegetation equation

The logistic equation for growth, mortality and self-competition of a single dominant tree species is given by

(1)

where T is tree cover and dT/dt its rate of change with time. The first term on the right-hand side represents spreading productivity (βT) limited by self-competition (at large tree cover, trees of the same type compete with each other for resources). This is balanced by mortality (the second term). The mortality coefficient, m, represents the rate of tree loss from mortality, and will depend on fire (as well as other factors such as drought and windthrow). Both m and β are functions of climate (the dependence of m on tree cover, T, is excluded in our study).

Equation 1 may be rearranged simply to give its stable equilibrium solution for β ≥ m (this restriction is necessary to prevent negative T):

(2)

(the other solution, T = 0, is unstable for this case). That is, equilibrium tree fraction is determined by the ratio of mortality to productivity.

If β < m, the equilibrium solution is T = 0, which is stable for this case. Consequently, the stable equilibrium solution (see Fig. 2) of the logistic equation (equation 1) is a nonlinear function of β in two ways. First, there is a jump in the slope at β = m where T goes from the T = 0 regime to the positive T regime. Second, as β grows further, the slope gradually decreases because the self-competition term in equation 1 introduces a T² dependence.

Alternatively, equation 2 may be re-arranged to give

(3)

This means that if we plot the mortality coefficient m against the productivity coefficient β (both coefficients will in general vary spatially) then isopleths of constant T_eq are straight lines passing through the origin (see the straight black lines in Fig. 3a). The isopleth T_eq = 0 follows the line m = β.

Optimization procedure: the relationship between mortality and productivity coefficients that gives a trimodal distribution

This section details the optimization procedure outlined in the Results section. The aim of this optimization is to determine a large-scale relationship between m, the mortality coefficient, and β, the productivity coefficient (expressing m as a function of β). This procedure determines the best relationship (yielding a trimodal distribution of the vegetation cover) via a series of iterations. The details are included for completeness. This method contains some subjective choices, but is sufficient for our aim: to show that feedback of tree cover on fire is not the only explanation for the trimodal distribution of tree cover.

For each iteration, a ‘large-scale’ relationship between mortality and productivity coefficients was chosen randomly as follows. Eight values of β were chosen, equally spaced over the range from minimum to maximum β (0 to 1). For each of these eight β values, a value of m was chosen randomly, in the interval from 0 to 0.5 (up to half the maximum value of β). Linear interpolation between the above eight values of β is used to obtain the ‘large-scale’ value of m at any intermediate value of β. Therefore, for each tropical location, a ‘large-scale’ value of m is obtained, based on the specific value of β for that location.

Having obtained a ‘large-scale’ value of m for a given location in the tropics, some ‘small-scale’ random noise is added. This was drawn from a normal distribution, with standard deviation equal to 15% of the ‘large-scale’ value of m. This was chosen to give a plausible level of noise in Fig. 3(c).

Given these values of productivity and mortality coefficients, the equilibrium tree fraction is calculated for each grid location. The resulting statistical distribution of tree fraction is approximately scored according to whether or not it is trimodal. This score was given by a measure of the ‘peak'-to-‘trough’ ratio

(4)

where f_s is the mean frequency of points in a ‘savanna’ interval (tree fractions 0.05 to 0.3) and f_f is the mean frequency of points in a ‘forest’ interval (tree fractions 0.7 to 0.9). f_{dip_lo} is the mean frequency in the interval between ‘treeless’ and ‘savanna’ (tree fractions between 0.025 and 0.05) and f_{dip_hi} is the mean frequency in the interval between ‘savanna’ and ‘forest’ (tree fractions between 0.3 and 0.7). Distributions where the difference between f_s and f_f was larger than their mean were discarded (score set to zero); as were distributions where the difference between f_{dip_lo} and f_{dip_hi} was larger than their mean.

These steps are repeated multiple times, with the top-scoring model chosen.

Results

Uncertainty in the environmental drivers of tree fraction

Here we discuss the observation that both intermediate and high (or intermediate and low) tree cover may be found at different locations in the tropics with similar measures of rainfall (Hirota et al., 2011; Staver et al., 2011b). We emphasize the point (also made qualitatively by Murphy & Bowman, 2012; van Nes et al., 2014) that we do not have a precise measure of environmental control on tree cover. Hence, even if tree cover was completely determined by environmental conditions (including rainfall) we would still find sets of locations with differing tree cover but (for example) the same mean annual precipitation.

We illustrate this point in Fig. 4, first using results from HadGEM2-ES (Fig. 4a). Our analysis follows that of Hirota et al. (2011) (their Fig. 1b), but for a highly idealized case: if tree cover is perfectly determined by the NPP of broadleaf trees as calculated by HadGEM2-ES (i.e. with no feedback of tree cover on fire). To do this, each grid-cell in HadGEM2-ES is assigned a label (either ‘forest’, ‘savanna’ or ‘treeless’), depending only on the NPP of broadleaf trees for that location. Locations with NPP greater than the 79th percentile of tropical NPP were labelled ‘forest’. Hence, 21% of the tropics was labelled ‘forest’. This percentile was estimated from the proportion in the ‘forest’ mode in Fig. 1(b) of Hirota et al. (2011). Similarly, locations with NPP less than the 21st percentile were labelled ‘treeless’ and the rest ‘savanna’.

As in the observed result (Hirota et al., 2011), Fig. 4(a) shows locations with the same mean annual precipitation but different ‘tree cover’ labels. In our idealized analysis this happens purely because mean annual precipitation is not a perfect predictor of broadleaf NPP (due to spatial variations in seasonal precipitation cycles, temperature, solar irradiance and soil type). The overlap between ‘savanna’ and ‘forest’ and ‘treeless’ and ‘savanna’ curves is significant, albeit less extensive than in the observations of Hirota et al. (2011). Weaker overlap in Fig. 4(a) is expected because the idealized case in Fig. 4 neglects key uncertainties.

One issue neglected in Fig. 4(a) is scatter in the relationship between tree mortality (including both fire and drought mortality) and mean annual precipitation. We therefore repeated this analysis, using exactly the same method (and the same rainfall data as in Fig. 4a) but with tree fraction calculated by the simple model (as in Fig. 3c) used in place of NPP from HadGEM2. This introduces additional scatter (through scatter in the relationship between tree mortality and precipitation; see Fig. 3c). Consequently, the overlap between the ‘forest’, ‘savanna’ and ‘treeless’ curves is increased (see Fig. 4b), and is similar to the observations of Hirota et al. (2011). Other factors could increase this overlap further, such as drought mortality, variable disturbances, soil quality and depth, different tree species and observational error.

Our results do not rule out feedback of tree cover on fire as driving the results of Hirota et al. (2011); they simply present a quantitatively plausible alternative. More detailed measures of climate, including rainfall seasonality, have been included in other studies (Staver et al., 2011b; Murphy & Bowman, 2012). However, no matter how sophisticated the measure of the environmental control on tree cover is, some error will remain. It is in principle not possible to rule out the hypothesis that this error is the reason for finding different levels of tree cover at the same measure of ‘climate’ (Murphy & Bowman, 2012, discuss this ‘chicken and egg’ problem).

Discussion

This study shows that some observations previously presented as evidence for large-scale alternative stable states (Hirota et al., 2011; Staver et al., 2011b) may in fact arise without strong feedback of tree cover on fire. They may instead be caused by nonlinearities in the equilibrium solution for spread of vegetation and in the relationship between tree mortality and productivity (if this relationship is driven by rainfall gradients), along with uncertainties in the environmental controls on tree cover. These nonlinearities arise from fundamental processes, although they are challenging to quantify. While not ruling out tree-cover feedbacks as driving the aforementioned observations, we show that the evidence for them doing so is not as strong as previously thought (in a similar spirit to the study of Hanan et al., 2014). Plenty of evidence exists for feedbacks of tree cover on fire being important, supporting alternative stable states in the vicinity of forest boundaries. However, it is harder to determine the relative roles of climate and influences of tree cover on fire-induced mortality over larger spatial scales.

It is important to understand and quantify these different influences more accurately, to both project and monitor the effects of anthropogenic influence on tropical tree cover. For example, it is unclear to what extent the nonlinear relationship in Fig. 1 is driven by climate (rainfall) gradients as opposed to vegetation feedbacks. It has been suggested that a way forward is to tune DGVMs to observed data (Murphy & Bowman, 2012). Any approach must be based on understanding which aspects discriminate between the influences of climate and tree cover (and other uncertainties). We showed that some aspects of nonlinear vegetation competition and varying mortality may be understood via plots of mortality against productivity (e.g. Fig. 3a,c).

The most obvious discriminant between the influences of climate and tree cover is via internal variability (e.g. van der Werf et al., 2008), as variability in tree cover (particularly re-growth) has a long time-scale. It would also be valuable to apply the analysis of Archibald et al. (2009) including that behind their Fig. 3(b), to other regions, along with further analysis of climate drivers. While observations of productivity are improving, another key gap is global monitoring of tree mortality rates: in particular its interannual variability, including the relationship with both productivity and fire variability. This is important, since the extent of tropical forests is thought to be limited by fire (Bond et al., 2005). The response to extreme dry years would give a clearer idea of forest sustainability under potential future drought. Satellite retrievals are possible (Flores et al., 2014), but represent a large data-processing challenge. It would also seem valuable to monitor the correlation between ‘fire danger potential’ and fire explored by van der Werf et al. (2008): this is currently low in the moist forest interior but could alter with future climate change. Improved monitoring of forest biomass (e.g. the Biomass satellite) will also provide a valuable tool for monitoring forest sustainability.

Potential effects of competition from grasses

The single-species logistic equation used above (equation 1) neglects any effect of competition from grasses on trees (it is unclear if this is important). It is shown below that if grasses are sufficiently vigorous to (mostly) fill the space not occupied by trees, then the distribution of tree fraction may still be modelled by a single-species logistic equation – leaving our conclusions unaltered.

The effect of competition from grasses on trees may be introduced by modifying equation 1 as follows:

(5)

where G is the area fraction occupied by grass and c_g is a competition coefficient. For vigorous grass growth, we can put G = (1 – T), giving

(6)

which may be rearranged to give

(7)

This is identical to equation 1 except that the productivity is multiplied by the constant (1 – c_g). Such a scaling of productivity would not affect the arguments presented below.

Topkill versus whole-plant mortality

Here we show that our conclusions regarding the trimodal distribution of tree fraction should not be altered if resprouting following fire is included. The simple model in the Methods does not allow for the possibility of resprouting following topkill. Here we explore this by introducing a second variable: the area fraction of topkilled trees that eventually resprout, T_t. This gives the following two equations:

(8)

(9)

In addition to the processes in equation 1, these equations represent competition from topkilled trees on healthy trees (the c_tT_t term), the rate of topkill that doesn't lead to whole-plant mortality (m_tT) and the resprouting rate of topkilled trees (rT_t). c_t, m_t and r are coefficients. Here, T is the area fraction of healthy trees and m_wT is the total rate of whole-plant mortality (including fire and other forms).

At equilibrium (setting the rates of change of T and T_t to zero), we can relate T to T_t:

(10)

Substituting this into the equilibrium form of equation 8 gives

(11)

Rearranging this gives

(12)

We now separate whole-plant mortality into fire and other effects. We also assume that the topkill rate is roughly proportional to the whole-plant fire mortality rate:

(13)

We also assume that the resprouting rate is roughly proportional to the productivity coefficient

(14)

Substituting these into equation 12 gives

(15)

which may be rearranged to give:

(16)

That is, if m_f were plotted against β, the contours of constant T would again appear as straight lines. That is, to achieve constant T fire-related mortality (m_f) must increase linearly with productivity (β). This means that our qualitative conclusions about the trimodal distribution of tree fraction would still apply.