Fire, resprouting and variability: a recipe for grass–tree coexistence in savanna

Conceptual definition

The model rephrases the grass–tree coexistence question as: why do grasses not eliminate trees, and why do trees not thicken up to form forests that would exclude grasses? We propose that the storage effect promotes the persistence of trees at low densities through variations in seedling establishment and adult recruitment against a background of low adult mortality. We believe that the storage effect operates in savanna because (i) seedling establishment rates depend on rainfall, which is highly variable in savanna; (ii) grass fires, which vary considerably in intensity in savanna, can prevent tree recruitment; and (iii) savanna trees are long-lived. It follows that understanding grass–tree coexistence requires an understanding of grass fire behaviour, fire-induced tree damage, tree recruitment and seedling establishment (Fig. 1). In this paper we use the term establishment to refer to the seed to seedling transition, and the term recruitment to refer to the seedling to adult transition. The model is based largely on data and assumptions from southern African studies of savannas that burn relatively frequently, but similar savannas that also burn relatively frequently are characteristic of large areas of Africa, South America, Asia and Australia. We do not consider savannas heavily impacted by herbivores, although we believe that the demographic problems of trees escaping from fire resemble the problems of trees escaping from browsing (Pellew 1983; Dublin et al. 1990; Prins & van der Jeugd 1993).

Grass fires occur in savanna ecosystems because grass production in the wet season is followed by an extended dry season leading to a continuous cover of fuel, and there is a ready source of ignitions (lightning and human). We hypothesize that savannas exist under conditions where fires are intense enough to limit the recruitment rates of trees, but not so intense to prevent recruitment (as in grasslands) or so mild to not limit recruitment (as in surface fires in forests). Variation in fire intensity can be attributed to variations in grass standing crop, grass moisture content (which varies with species and season), air temperature, humidity and wind speed (Trollope 1982; Cheney et al. 1993; Cheney & Sullivan 1997; Trollope 1998). Spatial variation in fire intensity may therefore be due to patchy grass production (Chidumayo 1997), patchy herbivory (Coughenour 1991), the effects of tree neighbourhoods on grass production (Mordelet & Menaut 1995) and grass moisture contents (Vetaas 1992; Webber 1997). The temporal variation in fire intensity may be due to both interannual variation of rainfall and variation in the timing of ignition events and hence fuel conditions (Trollope 1982; Cheney & Sullivan 1997). In our model (Fig. 1) rainfall is the primary determinant of grass production. Local site variables, notably soil characteristics and nutrient availability, obviously also influence grass production (Scholes & Walker 1993), as does grass species composition (Trollope et al. 1989), but such subtlety is not our concern. Grass standing crop increases during the wet growing season and decreases as the dry season progresses, due to herbivory and decomposition; the moisture content also decreases as the dry season progresses. The realized fire intensity is therefore dependent on the grass standing crop and grass moisture content on the day of the fire, the species of grass (as different species have different moisture contents), as well as the temperature, humidity and wind speed on the day of the fire (Fig. 1). It follows that the intensity of a fire regime is strongly dependent on the seasonal distribution of ignition events.

Understanding the variation in tree recruitment needs not only an understanding of variation in fire intensity but also an understanding of the life history of savanna trees. Savanna trees only recruit into the adult population once they escape the zone of influence of grass fires. The ability of stems that are killed in a fire to resprout is a key life-history trait that promotes the persistence of trees in savanna (Walter 1971; Bond & van Wilgen 1996; Gignoux et al. 1997; Trollope 1998). Tree seedlings may persist as suppressed juveniles (called ‘gullivers’ by Bond & van Wilgen 1996) for many years because such stems continue to resprout repeatedly after being burnt back by fires. We model the frequency of escape of gullivers from the flame zone into the adult population by simulating how fire intensity and tree size influence the likelihood of stem mortality (Fig. 1). Taller, thicker stems and stems with thicker bark have a higher chance of surviving a fire of a given intensity (Wright et al. 1976; Moreno & Oechel 1993; Gignoux et al. 1997; Trollope 1998; Williams et al. 1999). Hence the frequency of gulliver escape depends strongly on stem growth rates and the frequency and intensity of fire (Trollope 1984).

Gulliver banks are maintained by both resprouting and seedling establishment. Little is known about the regeneration niches of savanna tree species. It is generally believed that the seedlings of many savanna species are shade intolerant (Smith & Shackleton 1988), and high grass biomass can suppress tree seedlings (Brown & Booysen 1967; Walker et al. 1981; Knoop & Walker 1985; Harrington 1991). Other evidence suggests that establishment is facilitated by the presence of grasses (Brown & Archer 1989; Holmgren et al. 1997; Davis et al. 1998), and that some savanna species are shade tolerant (Smith & Walker 1983; O’Connor 1995; Hoffmann 1996). What is clear is that most savanna germinants cannot tolerate droughts during the wet growing season (du Toit 1965; Medina & Silva 1990; Harrington 1991; Hodgkinson 1991; O’Connor 1995; Hoffmann 1996). It seems, therefore, that the likelihood of wet season droughts will strongly influence seedling establishment patterns and hence grass–tree coexistence (Fig. 1).

Operational definition

We developed an individual-based, spatially explicit simulation model of grass and tree dynamics because this class of model allows flexible simulation of a wide range of ecological processes. In particular the individual-based approach allows us to keep track of the size and fate of individual tree stems as influenced by their neighbourhoods. Because of the large differences in the size of grasses and trees, we do not model individual grass tufts but model grass patches. Because we are interested in capturing the heterogeneity introduced by tree neighbourhoods on grass patches, we chose a spatial grain of 1 m² and assumed that only one stem can occupy each 1-m² site. The area the model simulates can be varied, but for this study we use a 1-ha area (100 ×  100 cells). We chose to use an annual time step because we postulate that interannual variation in rainfall is the key source of variation, and because most of the data we had access to was annual data. However, as discussed above, fuel properties vary considerably within a year; we deal with this problem by allowing the day of fire ignition to be a random variable of defined moments. Below we discuss the assumptions and the functions used to simulate the ecological processes described above; we then describe how we combine these functions to define a dynamic model.

Rainfall

Mean annual rainfall is variable in savanna systems; this variability can be divided into two components. The first component is stochastic variation; the second component is long-term periodicity. We use a sine wave function, which captures both these components of variability, to generate rainfall:

(eqn 1)

Here R is the annual rainfall (mm); x is a normally distributed random number defined by the mean (R_x; mm) and standard deviation (R_sd; mm) of annual rainfall; s is the effect of the long-term periodicity of rainfall (mm); l is the frequency of periodicity; and y is the simulation year.

Grass production

Most authors use linear regression to describe the relationship between rainfall and grass production, as this produces the best fit to the data from savanna regions (O’Connor 1985; Scholes & Walker 1993). Grass production can be written as:

where G_p is the predicted above-ground production (kg ha^− 1), R is the annual rainfall (mm), and g_g is the growth coefficient. Using data from southern Africa we estimated G_p=  3.369 ×  R (P <  0.0001, d.f. =  71; Fig. 2). Grass production can be negatively (Grunow et al. 1980; O’Connor 1985; Mordelet & Menaut 1995) or positively influenced by tree neighbourhoods (Belsky et al. 1989; Weltzin & Coughenour 1990). Grass production beneath tree canopies can be boosted by almost 300% or suppressed by over 50% (Mordelet & Menaut 1995). To account for the effect of stem neighbourhoods on grass production we can write:

Here c is a coefficient that describes the effect of the stem neighbourhood on grass production at site (i,j) and C_(i,j) is an array that describes whether the site (i,j) is in a stem neighbourhood (C_(i,j)=  1) or not (C_(i,j)=  0); stem neighbourhoods are defined below (equation 12). The grass standing crop at the end of the growing season (G_(i,j)) is therefore:

where G_{y – 1(i,j)} is the amount of grass material that is carried over from the previous year (cf. equation 6). The levels of herbivory, the rate of grass decomposition and whether a fire has occurred will determine how much grass is carried over from one year to the next.

Herbivory and decomposition

While herbivores are a ubiquitous feature of savanna ecosystems and influence savanna dynamics in many ways (Cumming 1982; Pellew 1983; Dublin et al. 1990; Prins & van der Jeugd 1993; Scholes & Walker 1993), we are primarily interested in herbivores’ ability to manipulate fuel loads; for this reason we do not consider browsing. Following Danckwerts (1982) we assume that herbivores can reduce the grass standing crop as a linear function of time since production:

where G_f(i,j) is the grass standing crop on the day of ignition, t_i is the ignition day (days since the start of the dry season), a is the grazing rate (kg ha^− 1 day^− 1), and G_(i,j) is the grass standing crop at the end of the growing season. Note that this function implies that growth and consumption are treated as discrete events in the model, and ignores spatial and temporal heterogeneity in grazing. To estimate the amount of grass carried over from one year to the next (G_{y – 1(i,j)}), we could write:

Here u is the decomposition rate (kg ha^− 1 day^− 1); t_d is the length of the dry season in days; b is the completeness of the burn; and B_(i,j) is an array that describes whether site (i,j) is burnt or not.

Grass moisture content

The moisture content of the fuel influences fire intensity. Grass growing in a tree’s neighbourhood may be moister and retain moisture levels for longer into the dry season (Weltzin & Coughenour 1990; Vetaas 1992; Webber 1997). If we assume that the moisture content of grass decays exponentially into the dry season (Cheney & Sullivan 1997), we could write the moisture of grass on the day of ignition as:

Here M_f(i,j) is the moisture content (%) at location (i,j) t_i days after the start of the dry season; M_o and M_c are the moisture contents of grass (outside and inside the tree neighbourhoods) at the end of the growing season; and d_oand d_c are the drying rates for outside and inside the tree neighbourhood. The array C_(i,j) records if a cell is inside a tree neighbourhood (see equation 12 for the definition of tree neighbourhood).

Fire intensity

We use empirically derived relationships to predict fire intensity, as these statistical models provide, given the information available, a better prediction of fire intensity than physical fire models (Trollope 1998). The statistical model was developed using 200 monitored fires in South African savannas (P <  0.01, d.f. =  196, R²=  0.60; Trollope 1998). The model was tested against independent fire behaviour data and accounted for 56% of the variation in fire intensity (Trollope 1998). The multiple regression model is:

(eqn 8)

Here Q_(i,j) is the fire intensity (kJ s^− 1 m^− 1) at site (i,j); G_f(i,j) is grass standing crop (kg ha^− 1) at site (i,j); M_f(i,j) is fuel moisture (%) at site (i,j); H is the relative humidity (%); and W is wind speed (m s^− 1). The empirical model therefore proposes that the realized fire intensity is dependent on the grass standing crop, the grass moisture content, the relative humidity, and the wind speed on the day of the fire. For the simulations we assume that humidity and wind speed are normally distributed random numbers defined by the site’s mean and standard deviation of humidity (H_x,H_sd;H≥ 0) and wind speed (W_x,W_sd;W≥  0).

Stem mortality

The probability of stem mortality (or ‘topkill’) in a fire is a function of stem height and fire intensity (Trollope 1984). We used data on the survival rates of 7400 stems of 76 species in 40 fires of known intensities (W.S.W. Trollope, A.L.F. Potgieter and N. Zambatis, unpublished data) to estimate a logistic regression model of the probability of stem mortality (P <  0.01, R²=  0.48, d.f. =  7397):

(eqn 9)

Here p_t is the probability of stem mortality; h is stem height (m); and Q is the fire intensity (kJ s^− 1 m^− 1; equation 8). Interestingly Williams et al. (1999), using data from a single intense fire in a tropical savanna in northern Australia, found that p_t was a quadratic function of tree size, with larger and small trees suffering highest p_t. The susceptibility of large trees to topkill was attributed to termite damage to large trees (Williams et al. 1999).

Stem resprouting

Stems that have been topkilled usually resprout from rootstocks. Savanna species have very high probabilities of resprouting (Lacey et al. 1982; Trollope 1982, 1984; Boo et al. 1997; Gignoux et al. 1997). Resprouting ability is generally thought to increase with stem size (Wright et al. 1976; Moreno & Oechel 1993) but, in some tree species, decreases again in the larger size classes (Trollope 1974; Hodgkinson 1998; K. Maze and W.J. Bond, unpublished data). The effect of stem size on the probability of resprouting (p_r) can be written as:

where p_max is the maximum probability of resprouting, h_0.5 is the stem height (h) at which there is a 50% chance of resprouting, and v_r is a constant that describes how rapidly the probability of resprouting changes with stem height. Recent germinants do not have the root reserves to resprout (Moreno & Oechel 1993); we therefore assume that stems less than the resprouting height (h_r) cannot resprout. The model only allows one stem per site; it therefore does not consider the resprouting of multiple stems. Because the probability of fire survival is influence by tree height and not stem number, we do not consider this an important limitation.

Tree mortality

Recent carbon dating evidence suggests that savanna trees can be more than 1000 years old (CSIR, personal communication). Rates of mortality due to stress are therefore expected to be low. Andersen et al. (1998) reported annual mortality rates of 0.01 for Australian savannas; Trapnell (1959) reported mortality rates of 0.04 in Zimbabwean miombo woodlands; and Shackleton (1997) reported mortality rates of 0.05 in South African savannas. In the model trees of maximum height (h_max) face a p_m chance of mortality each year.

Stem growth rates

The stem mortality and stem resprouting functions all use stem height information. Very little data exist on height or diameter growth of savanna stems. We know that stems initially grow rapidly in height but subsequently growth slows (K. Maze and W.J. Bond, unpublished data); using this information we can describe stem growth using a difference equation:

(eqn 11)

where g_s is the growth rate of stems (cm year^− 1), h_max is the maximum stem height (m), and h_y−1 is the stem height in the previous year. Stem growth rates and maximum tree heights in savannas are known to be influenced by moisture and nutrient availability (Shackleton 1997), although we do not explicitly consider these effects here.

Stem neighbourhoods

A stem’s neighbourhood has two components. First, a stem’s canopy creates a moister and shadier environment; secondly, the laterally spreading roots influence the soil moisture and soil nutrient status. The root- and canopy-defined neighbourhoods do not always overlap (Vetaas 1992), but we assume, nevertheless, that the diameter of the neighbourhood (n; m) increases as a linear function of tree size:

where g_n is a growth coefficient, h is stem height (m), and k is a constant. We do not have data on below-ground neighbourhoods and we therefore use data on the relationships between stem height and canopy diameter (W.S.W. Trollope and A.L.F. Potgieter, unpublished data) to estimate k and g_n.

Seed production

Following Ribbens et al. (1994), we define seed production as a function of tree size, such that:

Here F is tree fecundity (seeds year^− 1) and f is the number of seeds dispersed by a tree of a reference height (h_std); stems smaller than the height of reproductive maturity (h_f) do not produce seeds. Data on the seed production of savanna trees (Tybirk et al. 1993) is used to estimate h_f and f.

Seed dispersal

Many savanna tree species are dispersed passively and by animals (bird, ungulates, rodents, termites and ants; Brown & Archer 1989; Tybirk et al. 1993; Miller 1994). Previous demographic models of savanna (Menaut et al. 1990; Hochberg et al. 1994; Jeltsch et al. 1996, 1998) have emphasized the importance of tree clumps and hence the role of local vs. long-distance dispersal. The importance of local vs. long-distance dispersal motivated us to model dispersal as a stratified process that explicitly considers both local and long-distance dispersal (Higgins & Richardson 1999). We can use a mixture of two exponential distributions to describe a probability density function of dispersal distances d(x):

Here p_l is the proportion of seeds that are dispersed short distances (described by the parameter β_l) and (1−p_l) is the proportion of seeds that are dispersed longer distances (described by the parameter β_f).

Seed bank decay

Seed banks of savanna tree species are not very long-lived. Those that do not suffer predation by insects and rodents either decay rapidly, germinate or lose viability (Tybirk et al. 1993). We summarize all these processes by assuming that a constant proportion of seeds decay each year (s_d). We could not find any published estimates of s_d, although anecdotal evidence suggests that the decay rate is relatively high (Skoglund 1992; Tybirk et al. 1993; Miller 1994; Chidumayo & Frost 1996).

Seedling establishment

Little is known about the regeneration niches of savanna tree species, although we know that many seedlings are shade-intolerant and that high grass biomass can suppress their recruitment, while seedlings of other species are shade-tolerant and little influenced by grass biomass. What is clear is that one of the major factors limiting establishment is the availability of moisture: droughts during the wet season of more than 30 days can lead to seedling mortality (Medina & Silva 1990; Chidumayo & Frost 1996; Hoffman 1996). If we assume that the number of rainfall events during the wet season is positively correlated with annual rainfall, then the probability of wet season drought should decrease with annual rainfall. We can express the probability of wet season drought (p_d) as:

(eqn 15)

Here R_0.5 is the annual rainfall at which there is a 0.5 chance of a wet season drought; and v_d is a constant that describes the rate at which the probability of wet season drought changes with rainfall (R). If there is no wet season drought then the probability of establishment (p_e) is:

Here we account for the observations that some species require light for establishment (c_t=  1), while others are shade tolerant (c_t=  0). If the light conditions are suitable (equation 16.2) then the probability of establishment is a function of the grass standing crop. In equation 16.2G_0.5 is the grass standing crop at which the probability of establishment (p_e) is 0.5, G_(i,j) is the grass standing crop and v_e is the rate at which the probability of establishment changes with grass standing crop.

Implementation

Each year the model sequentially simulates the following ecological processes: rainfall, tree growth, seed dispersal, grass production, potential fire intensity and fire spread, stem mortality due to fires, resprouting, adult tree mortality, seedling establishment, and seed bank decay. Rainfall is generated using equation 1. Tree heights are incremented each year (equation 11) and tree neighbourhoods are subsequently calculated from the tree height (equation 12). The number of seeds produced by each stem is calculated (equation 13) and these are available for dispersal. Each seed is dispersed individually and the distance each seed moves is a random number defined by the mixture distribution (equation 14) . This distance and a randomly selected direction are used to calculate the location of each seed. Grass production is calculated as a function of the rainfall and neighbourhood state (equation 3). Estimating potential fire intensity (equation 8) requires the estimation of grass standing crop, grass moisture content, relative humidity and wind speed on the day of the fire. We assume that ignition can occur on any day during the dry season. The relative humidity and wind speed on the day of the fire are generated by assuming that daily humidity and wind speed are normally distributed random numbers. The grass standing crop and grass moisture content on the day of the fire can be estimated by using equations 5 and 7 and by assuming that ignition occurs t_i days into the dry season. In the current version of the model one fire ignition occurs per year. We assume that t_i is a normally distributed random number with a mean (I_x) and standard deviation (I_sd) characteristic of the temporal distribution of ignition events. This method of generating ignition events emphasizes the temporal distribution of ignition events, i.e. there is an emphasis on fire intensity rather than fire frequency. The fire spread algorithm is analogous to that proposed by Turner & Romme (1994). We assume that fires can spread if a threshold fire intensity is exceeded; this threshold has been estimated as 150 kJ s^− 1 m^− 1 in savanna systems (van Wilgen & Scholes 1997). The fire spread algorithm allows a fire to spread to neighbouring cells if a neighbouring cell’s potential fire intensity exceeds the threshold. The fire spread algorithm is not influenced by wind or topography. The way we model fire ensures that fuel properties rather than ignition frequency determines the modelled fire frequency. The average fire intensity in the nine 1-m² cells in a tree’s neighbourhood and the tree’s height are used to estimate the probability of stem mortality (equation 9). The likelihood of a dead stem resprouting is estimated as a function of stem height (equation 10). Adult trees face a p_m probability of death each year. Seeds can only germinate and establish if there is no wet season drought (equation 15). If there is no drought then a seed can establish, provided the grass biomass is low enough, light conditions are suitable (equation 16) and the site is unoccupied by a tree stem. The seed bank is decayed by a constant proportion (s_d) each year.

The nature of grass–tree coexistence

For this section we initialized the model with the best parameter estimates available. Sources of these, largely southern African, parameter estimates are discussed under operational definition, and the parameter values used are listed in Table 1. We then varied the base parameterization (Table 1) to simulate four sites representing a rainfall gradient from arid to mesic savanna (Table 2). In southern Africa this gradient is associated with a change from palatable to relatively unpalatable grass; and we simulate this by decreasing the grazing rate with increasing rainfall. We also assume that (i) stem growth rates will increase with rainfall and (ii) that the periodicity and stochasticity in rainfall (equation 1) changes from arid to mesic savanna. The fact that the coefficient of variation of rainfall tends to decrease with increasing rainfall is simulated by changing the stochasticity and strength of periodicity of rainfall. The relative contribution of the periodicity vs. stochasticity in rainfall to the coefficient of variation of rainfall is varied to simulate situations where rainfall cycles are not present. While we do not claim that these parameterizations are full representations of the differences between arid and mesic savannas, they do illustrate the different kinds of dynamics the model can produce.

Running the model with the parameter settings listed in Table 2 generates coexistence between grasses and trees at all four sites (Fig. 3), in that trees persisted but did not reach 100% cover. The tree dynamics at all sites were characterized by long periods of slow decline in adult stems punctuated by occasional recruitment events. The frequency of recruitment events and the ratio of gulliver (non-reproductive) to adult stems and the stem densities vary across the rainfall gradient; these differences are best explained by examining the mean and variance in rates of establishment, recruitment and mortality (Fig. 4). First, low mortality rates at all sites explains the slow rate of decline of adult numbers in the absence of recruitment. The low gulliver relative to adult stem numbers at the arid site is due to low seedling establishment rates (due to the high frequency of wet season droughts, cf. equation 15) and the fact that when establishment does occur at the arid site it often leads to tree recruitment (due to the low frequency of high-intensity fires caused by low fuel loads). At the semi-arid and semi-mesic sites establishment rates are relatively high (due to less frequent drought and low grass standing crop) but recruitment into the adult stage is lower and more variable (due to relatively intense fires). The combination of high establishment rates and low recruitment explains the accumulation of large numbers of gullivers at both these sites. The high variance in gulliver stem numbers at the semi-arid site is due to relatively high establishment rates, coupled with a slow growth rate that prevents many of the smaller gulliver stems from resprouting after topkill. At the semi-mesic site gulliver stem numbers accumulate because they are large enough to resprout but too small to recruit frequently. At the mesic site establishment rates are lower due to the negative effect of high grass standing crop (produced by the higher rainfall) on seedling establishment, but the higher growth rates of stems means that recruitment rates are maintained. The lower coefficient of variation of rainfall at the mesic site accounts for the lower variance in establishment and recruitment rates and hence the more constant adult population size.

While the patterns in establishment, recruitment and mortality are instructive, the challenge is to understand in more general terms the factors responsible for generating coexistence. It is established in the coexistence literature that varying environments are themselves not recipes for coexistence: some interaction between environmental variability and species behaviour is needed for variability to promote coexistence (Turelli & Gillespie 1980; Chesson & Warner 1981; Chesson & Huntly 1989). It follows that understanding coexistence in heterogeneous systems requires understanding the interaction between environmental variability and life history. One such interaction occurs when adult survival is high and recruitment rates are variable; and it is this combination of factors that constitutes the storage effect (Warner & Chesson 1985). The low adult mortality and variable seedling establishment and recruitment rates produced by our model (Fig. 4) are consistent with the storage mechanism of coexistence (Warner & Chesson 1985). Moreover, partitioning out the contribution of the storage effect to the growth rate of the population (following Warner & Chesson 1985) shows that the population growth rate for trees was negative or zero when the storage effect was excluded ( − 0.0019, − 0.0019, − 0.0020, 8.0E − 05; for the arid to mesic sites, respectively, cf. Table 2) and positive with the storage effect included (0.0024, 0.0050, 0.0071, 0.0055), suggesting that the storage effect is essential for the persistence of trees in the model system. The reasons for the relatively high and constant adult survival rates are clear: savanna trees are long lived and have a low likelihood of suffering fire-induced stem mortality (equation 9). The relatively high variance in tree seedling establishment and recruitment rates can be related, respectively, to the variations in rainfall and fire intensity. In more arid systems variation enters at the seedling establishment and recruitment phase, whereas in more mesic systems variation enters primarily at the tree recruitment phase (Fig. 4). In agreement with this result, high variance in recruitment rates has been reported in Australian savannas (Harrington 1991). Harrington (1991) attributed the high variance in recruitment rates to the rarity of synchronization between adequate moisture conditions for seedling establishment and fires of intensities low enough to allow recruitment. The effect of the rarity of such synchronization is most easily detected in the model runs from the arid site (Fig. 3).

The sensitivity of grass–tree coexistence to environmental change

Exploring the sensitivity of the coexistence state to a range of potential influencing variables can help us understand the factors that influence grass–tree coexistence as well as its susceptibility to environmental change. The model we have constructed, however, contains 48 parameters [40 are listed in Table 1; the fire intensity and the stem mortality models (equations 8 and 9) contain an additional eight parameters]: consequently an exhaustive sensitivity analysis is not feasible here. We do, however, explore the sensitivity by varying key parameters from the base parameter estimates as defined in Table 1. For the sensitivity simulation runs the model was run for 2000 simulation years, a single run was used for each parameter setting, and the mean and standard deviation of tree density in the last 500 simulation years is reported.

We first vary the mean annual rainfall from 200 to 2000 mm (Fig. 5a; constant humidity); this shows that the trees can coexist with grasses between 500 and 1600 mm mean annual rainfall. At low rainfall the model predicts that trees are limited by moisture conditions for establishment, whereas at high rainfall trees are limited by fire intensity (in agreement with Trollope 1980). Trollope (1974, 1980) has shown that, in arid savanna, fire cannot control tree densities but can keep trees in the browse zone. Our model predicted highest tree densities at lower rainfall, suggesting that an additional factor such as browsing (which is excluded here) is needed to control tree densities in more arid areas. Browsing was also regarded as important in regulating tree densities in east African savannas (Pellew 1983; Prins & van der Jeugd 1993). However, because other parameters (e.g. tree growth rates, humidity and wet season drought) co-vary with rainfall, the rainfall range that allowed coexistence here can only be taken as a rough guide to the rainfall limits of savanna. For instance, if we increase humidity as we increase rainfall to simulate less flammable fuel conditions (Fig. 5a; increasing humidity) the model does not predict tree exclusion at higher rainfall. None the less, savanna ecosystems occur over a similar range of mean annual precipitation to the range predicted here (c. 300–1800 mm; Scholes & Walker 1993; O’Connor & Bredenkamp 1997; Scholes 1997). Hence both our model and empirical data suggest that rainfall is a key determinant of grass–tree ratios; although other factors must also influence this ratio.

We have established that variability in recruitment rates and low adult mortality rates (Fig. 4) allow trees to coexist with grasses, but the key to understanding the coexistence mechanism is determining what generates this variability. The model does not appear to be sensitive to the variability (Fig. 5b; the coefficient of variation in rainfall is increased by increasing s and R_sd) or the stochasticity (Fig. 5c; the contribution of R_sd is increased and the contribution of s to the coefficient of variation in rainfall is decreased) in rainfall. However, removing variance in fire intensity (by changing variation in rainfall, relative humidity, wind speed and fire ignition day) can lead to the exclusion of trees, whereas increasing this variance favours trees (Fig. 5d). Hence variable fire intensities provide opportunities for tree stems to escape the flame zone, where they are most susceptible to fire, and recruit into the more fire-resistant size classes. In other words, variance in fire intensity produces the variance in recruitment rates that is necessary for the storage effect to operate. Our model therefore suggests that a fire-mediated recruitment bottleneck (Walter 1971; Trollope 1974; Bond & van Wilgen 1996; Gignoux et al. 1997; Andersen et al. 1998) is central to understanding how fire mediates coexistence of grasses and trees. Other factors do, however, influence the regeneration niche and hence the tightness of the recruitment bottleneck. Very low seed production can suppress tree numbers, but the effect of seed production on tree density rapidly asymptotes (f;Fig. 5e). The effect of grass competition on seedling establishment (G_0.5; Fig. 5f) does not appear to influence tree density. The likelihood of wet season drought (R_0.5; Fig. 5g) does influence tree density; this parameter is likely to be more important in arid systems, where variation in recruitment is controlled by variance in establishment conditions rather than variance in recruitment conditions (3, 4). Interestingly, although the presence of moister subcanopy grass is likely to facilitate the coexistence of grasses and trees by buffering tree stems against fire intensity, increasing the moisture content of the subcanopy grasses does not lead to increased tree numbers (the subconopy moisture content is increased and the subcanopy drying rate is decreased relative to the between canopy values; Fig. 5h). This suggests that the spatial component of heterogeneity in fire intensity is not as important as the temporal one (Fig. 5d). However, the effects of tree canopies on grass production could introduce additional spatial heterogeneity; such effects were not considered here (c was set to 1 for these simulation runs).

The rate of stem growth (Fig. 5i) and the maximum likelihood of a damaged stem resprouting (Fig. 5j) strongly influenced tree dominance. Stem growth rates of more than 50 cm year^− 1 are needed for trees to persist, while stem growth rates greater than 70 cm year^− 1 lead to tree dominance. The paucity of existing data on stem growth rates and the sensitivity of the model to this parameter provides motivation for collecting stem growth rate data. Similarly, a probability of resprouting in excess of 0.6 is needed for tree persistence, while a probability of resprouting of 0.99 leads to tree dominance. Data suggest that resprouting probabilities in savanna are typically greater than 0.8 in savanna (Trollope 1974, 1998; Boo et al. 1997; Gignoux et al. 1997). The rate of adult mortality due to factors other than fire is another factor that strongly influences tree persistence; the model suggested that low adult mortality rates ( <  0.05) are necessary for tree persistence (Fig. 5k). In apparent contradiction to this model prediction, annual mortality rates, which may include the effects of fire, of c. 0.04–0.05 have been reported in southern African savannas (Trapnell 1959; Shackleton 1997). However, Andersen et al. (1998) reported mortality rates of 0.01 for Australian savannas, and Dublin et al. (1990) used 0.01 in a model based on field data from East Africa. Interestingly, elephants have been responsible for tree morality rates of 0.18 in Zimbabwean savannas (Thomson 1975); suggesting that their role as ecosystem modifiers should not be disregarded.

Increasing grass production negatively influences tree density (Fig. 5l) by making it more difficult for seedlings to establish, and by effectively increasing the fire intensity and therefore reducing escape opportunities into fire-resistant size classes. The grazing rate has the opposite effect (Fig. 5m), in general agreement with observations that high grazing rates promote bush encroachment (Archer et al. 1988; Skarpe 1991; Archer 1995). Our model therefore suggests that bush encroachment occurs due to increased tree recruitment caused by reductions in standing crop and hence fire intensity. This contradicts the competitive release mechanism of bush encroachment (Walker & Noy-Meir 1982; Stuart-Hill & Tainton 1989; Jeltsch et al. 1997), whereby the decreased grass standing crop as a result of grazing reduces competition between grasses and trees and thus increases opportunities for tree recruitment. Recent empirical studies also challenge the competitive release mechanism by showing that, in resource-limited systems, establishment and recruitment are limited more by resource availability than competition (Davis et al. 1998). In our model, grass standing crop had only a weak direct effect on tree recruitment (Fig. 5f), i.e. the effect of grazing on trees is manifested through the effect of grazing on grass standing crop and hence fire intensities. Our model’s mechanism is consistent with the hypothesis that bush encroachment is constrained by soil moisture availability and fire intensity rather than grass competition for soil moisture (du Toit 1967; Harrington 1991). In addition, heavy grazing can favour less flammable and less productive grasses, further decreasing fire intensities (Trollope 1998).

Received 11 March 1999revision accepted 9 September 1999