Ecology, 91(6), 2010, pp. 1682–1692
Ó 2010 by the Ecological Society of America
The stability of African savannas: insights from the indirect
estimation of the parameters of a dynamic model
STEVEN I. HIGGINS,1,5 SIMON SCHEITER,2
1
AND
MAHESH SANKARAN3,4
Institut für Physische Geographie, Goethe Universität Frankfurt am Main, Altenhoeferallee 1, Frankfurt am Main 60438 Germany
2
Biodiversity and Climate Research Centre (LOEWE BiK-F), Senckenberganlage 25, Frankfurt am Main D-60325 Germany
3
Faculty of Biological Sciences, 9.18 LC Miall Building, University of Leeds LS2 9JT United Kingdom
4
National Centre for Biological Sciences, Tata Institute of Fundamental Research, Bellary Road, Bangalore 560065 India
Abstract. Savannas are characterized by a competitive tension between grasses and trees,
and theoretical models illustrate how this competitive tension is influenced by resource
availability, competition for these resources, and disturbances. How this universe of
theoretical possibilities translates into the real world is, however, poorly understood. In this
paper we indirectly parameterize a theoretical model of savanna dynamics with the aim of
gaining insights as to how the grass–tree balance changes across a broad biogeographical
gradient. We use data on the abundance of trees in African savannas and Markov chain
Monte Carlo methods to estimate the model parameters. The analysis shows that grasses and
trees can coexist over a broad range of rainfall regimes. Further, our results indicate that
savannas may be regulated by either asymptotically stable dynamics (in the absence of fire) or
by stable limit cycles (in the presence of fire). Rainfall does not influence which of these two
classes of dynamics occurs. We conclude that, even though fire might not be necessary for
grass–tree coexistence, it nonetheless is an important modifier of grass : tree ratios.
Key words: African savanna; biogeographical patterns; disturbance gradient; grass and tree
coexistence; indirect parameter estimation; Markov chain Monte Carlo method; resource gradient; tropical
savanna.
INTRODUCTION
The relative abundance of grasses and trees strongly
defines the structure and function of tropical savannas.
Although the factors that influence this balance are wellknown, consensus on the details of how these factors
shape savannas remains elusive (House et al. 2003,
Sankaran et al. 2004). A recent contribution to this
debate was provided by Sankaran et al. (2005) who
investigated the factors that determine tree cover
observed at 854 savanna sites in Africa. Sankaran and
colleagues statistically examined these data and showed
(1) that tree cover is not simply related to resource
availability, and (2) that at sites that receive ,650 mm
mean annual precipitation (MAP), tree cover is constrained linearly with moisture availability, while at sites
that receive .650 mm MAP canopy closure is possible,
but disturbances such as fire can prevent canopy closure
and are therefore necessary to prevent the exclusion of
grasses. Sankaran et al. (2005) interpreted these data to
mean that arid (,650 mm MAP) savannas are stable in
the sense that fire is not necessary for grass–tree
coexistence and that mesic savannas (.650 mm MAP)
are unstable in that fire is needed for grass–tree
coexistence.
Manuscript received 22 July 2008; revised 10 June 2009;
accepted 17 June 2009; final version received 29 September
2009. Corresponding Editor (ad hoc): B. T. Bestelmeyer.
5 E-mail: [email protected]
This interpretation of the empirical data was influenced by the state of existing theoretical models. Two
kinds of theoretical models, resource-based and disturbance-based models, have been influential in shaping
thinking about savannas. Resource-based models show
that when the effects of grass–tree competition are
strong (when the product of the per capita effect of
competition and competitor density is high) competitive
exclusion is possible (Walker and Noy-Meier 1982, van
Langevelde et al. 2003). Disturbance-based models show
that the disturbance regimes common to savanna
regions can prevent competitive exclusion even when
grass–tree competition is strong (Higgins et al. 2000,
Gardner 2006). Sankaran et al. (2005) argued that the
data on tree cover in Africa could be explained by
assuming that when resource levels are low, grasses and
trees can coexist because both are below critical
densities. However, when resource levels are high, tree
leaf area index can be high enough to exert a highly
asymmetric light competitive effect on grasses. The
consequence of this is that grasses may be excluded at
resource-rich sites unless fire can prevent trees from
reaching the critical densities beyond which competitive
exclusion is inevitable. While this argument appears
plausible, it is something that existing theoretical models
of savannas (e.g., Higgins et al. 2000, van Langevelde et
al. 2003) do not predict.
The data analysis and discussion from Sankaran et al.
(2005) calls for a formal integration of resource- and
1682
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COMPETITION AND DISTURBANCE IN SAVANNAS
1683
FIG. 1. Conceptual illustration of the model’s structure emphasizing the separation of above- and belowground biomass of
grasses and trees.
disturbance-based theories of savannas. A recently
developed model of savanna dynamics integrates resource- and disturbance-based theories (Higgins et al.
2007b, Scheiter and Higgins 2007), and thereby shows
that coexistence between grasses and trees is possible
under a broad range of conditions. Specifically, coexistence in the model does not rely on a rooting niche
separation mechanism as assumed by, for example,
Walker and Noy-Meier (1982), Anderies et al. (2002),
and van Langevelde et al. (2003), nor does it rely on the
demographic bottlenecks invoked by Higgins et al.
(2000) and Gardner (2006). The main difference between
this model and previous models is that it simulates the
different roles that roots and shoots play. That is, the
model separates root competition from shoot competition and acknowledges that disturbances such as fire and
herbivory can remove shoot biomass, but not root
biomass. The resulting model predicts that grasses and
trees can coexist under a surprisingly broad range of
assumptions regarding growth, competition, and disturbance rates; something that competing models fail to
achieve (Scheiter and Higgins 2007).
One way of evaluating this model would be to
examine whether it can describe empirical patterns of
savanna plant abundance. In this paper we examine
whether the model can describe the data of Sankaran et
al. (2005) on tree abundance for the African continent.
We emphasize that this analysis merely serves (1) to
illustrate whether the model is flexible enough to
describe observed patterns and (2) to gain insights into
how competition for belowground and aboveground
resources interacts with fire and herbivory to define
regional patterns of tree abundance. In this paper, we
use an indirect method to estimate the model’s
parameters. In principle, this method allows the
estimation of unobserved parameters (growth rates,
decomposition rates, and competition coefficients) by
finding the combination of model parameters that
reduce the deviations between observed parameters
(measured tree abundances) and projected parameters
(modeled tree abundance). We use this indirectly
parameterized model to explore the role of fire in
regulating tree–grass coexistence and the stability of
savannas.
MODEL DESCRIPTION
We use a simplified version of the model proposed by
Higgins et al. (2007b) and analyzed by Scheiter and
Higgins (2007). The model is highly aggregated; that is,
spatial heterogeneity (Jeltsch et al. 1996), individual tree
size (Higgins et al. 2000, Gardner 2006), and the
stochastic nature of fire and tree recruitment (Higgins
et al. 2000, Gardner 2006) are not explicitly simulated,
while fire return interval and rainfall are implemented as
deterministic and externally forced variables.
The model simulates two hypothetical species, a grass
species and a tree species. That is, the model does not
distinguish between shrubs and trees. For each species
two biomass compartments, roots and shoots, are
simulated. This allows us to simulate the fact that fire
cannot consume roots, and allows a separation of
belowground and aboveground competition (Fig. 1).
The model consists of four linked equations, one for
each biomass compartment:
GStþ1 ¼ GSt þ gG GRt ð1 GSt xWG WSt Þ cGt GSt
dg GSt zGSt
ð1Þ
GRtþ1 ¼ GRt þ gG GSt ð1 GRt aWG WRt Þ dG GRt ð2Þ
WStþ1 ¼ WSt þ gW WRt ð1 WSt Þ cWt WSt dW WSt
ð3Þ
WRtþ1 ¼ WRt þ gW WSt ð1 WRt aGW GRt Þ dW WRt :
ð4Þ
Here, GSt, GRt, WSt, and WRt represent the grass
shoot biomass, grass root biomass, woody shoot
biomass, and the woody root biomass at time t; gG,
gW and dG, dW are the growth and decomposition rates
of grasses and trees; xWG is a coefficient that describes
the competitive effects of tree shoot biomass on grass
shoot growth; aWG and aGW describe the competitive
effects of tree root biomass on grass root growth and
vice versa; cGt and cWt describe the proportion of grass
and tree shoot biomass removed by a fire (defined in
Eqs. 6 and 7); z describes the proportion of grass shoot
biomass removed by grazing.
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STEVEN I. HIGGINS ET AL.
The equations simulate how the size of each biomass
compartment in time t þ 1 is influenced by the size of the
compartment in time t, growth, competition, and by
losses due to decomposition, fire, and/or herbivory. For
this paper we consider a time step to be two months.
This achieves a balance between computational efficiency and an adequate representation of the system’s
dynamics. That is, time steps of two months allow the
model to simulate a year using only six time steps, yet
allow several time steps between fire events for resource
competition between grasses and trees. The paragraphs
that follow describe the processes represented in Eqs.
1–4.
The maximum amount that a biomass compartment
can grow in a time step is determined by the product of
the growth rate (g) and the size of the complementary
biomass compartment. For example, in Eq. 1 growth of
the grass shoot compartment, GS, is influenced by the
size of the grass root compartment, GR. That is, the
model assumes a tight interdependence of shoot growth
on belowground biomass uptake and conversely of root
growth on aboveground biomass. This assumption is
intuitively attractive because of the different functions
that belowground and aboveground biomass compartments fulfill. Belowground biomass is responsible for
water and nutrient acquisition, whereas shoot biomass
harvests light energy, which is used to produce carbon
by photosynthesis. Furthermore, there is empirical
evidence for high transfer rates between shoots and
roots; thus, over 60% of the carbon fixed by photosynthesis can be allocated to roots (Law et al. 1999), while
75% of the nitrogen acquired by roots can be allocated
to shoots (Poorter et al. 1990). Although other ways of
linking above- and belowground biomass compartments
are conceivable, Scheiter and Higgins (2007) show that
how these compartments are linked does not change the
qualitative nature of this model’s dynamics.
To examine how patterns of plant abundance change
over a rainfall gradient, we define the growth rate
parameters (gG, gW ) as being functions of rainfall such
that
1
:
ð5Þ
g¼
1 þ ð~g=RÞ
Here g̃ is a coefficient that describes the sigmoidal
relationship between rainfall and growth. This functional relationship is an adequate description of the
relationship between growth and rainfall and is consistent with empirical data (e.g., Whittaker 1975). The
model further assumes that growth rates differ for trees
and grasses (gW, gG and their associated parameters g̃W,
g̃G differ).
In Eqs. 1–4 the terms in parentheses simulate how the
growth of each compartment is influenced by different
sources of intra- and interspecific competition. For
intraspecific competition we simulate that the growth of
each compartment is negatively influenced by its size.
For example, in Eq. 1, the growth of the grass shoot
Ecology, Vol. 91, No. 6
compartment is negatively influenced by GSt. Belowground interspecific competition in savannas is assumed
to be for soil moisture (Walter 1971, Walker and NoyMeier 1982) and/or for soil nitrogen (Mordelet et al.
1997). The simplest way to represent belowground
interspecific competition in the model is to assume that
growth of the grass root compartment is negatively
influenced by the size of the tree root compartment
(WRt in Eq. 2) and growth of the tree root compartment
is negatively influenced by the size of the grass root
compartment (GRt in Eq. 4). The intensity of these
interspecific competitive effects is described by the
coefficients aWG and aGW that are analogous to the
competition coefficients in the Lotka-Volterra equations
(a . 0 implies competition; a . 1 implies that
interspecific competition is stronger than intraspecific
competition). Trees have been reported to facilitate
grasses in accessing soil moisture in savannas, although
it remains unclear whether this facilitatory process
dominates or is overwhelmed by competitive processes
(Ludwig et al. 2003). An aWG , 0 would imply that trees
facilitate grasses in the acquisition of soil resources.
Hence, although facilitation might be unlikely, it is a
process that the model can simulate.
Aboveground interspecific competition is assumed to
be asymmetrical and in favor of trees. Specifically, we
assume that trees, because of their greater potential
height, shade grasses, but that the inverse does not
occur. The intensity of tree on grass light competition is
described by the parameter x WG (Eq. 1). This assumption ignores the fact that when trees are in the seedling
stage they are subject to shading effects induced by
grasses (Higgins et al. 2000). Considering this grass on
tree shading effect would require a substantially more
complex model.
The death of living tissue (that we, in the tradition of
systems models, call decomposition) reduces the size of
each biomass compartment by a constant rate; these
decomposition rates differ for trees and grasses (parameters dW and dG in Eqs. 1–4). Additional losses can be
caused by herbivory. In this paper we consider only
grazing losses (see Scheiter and Higgins [2007] for the
consideration of grazing and browsing losses). Grazing
is modeled in the same way as decomposition; that is, a
constant proportion (z) of grass shoot biomass (GS) is
removed each time step (Eq. 1).
Fire also causes losses in tree and grass shoot biomass.
Two fire models are considered, both assume that fires
are periodic. The return interval of fires is defined by the
integer parameter F. The simpler model assumes that
when a fire occurs all shoot biomass of both trees and
grasses are consumed (ct ¼ cWt ¼ cGt). The proportion of
shoot biomass consumed by fire (ct) is thus
1 for t mod F ¼ 0
ct ¼
ð6Þ
0 for t mod F 6¼ 0:
The more complex model assumes that grass shoot
biomass is the fuel for fires (i.e., it is assumed that
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COMPETITION AND DISTURBANCE IN SAVANNAS
woody shoot biomass does not make a significant
contribution to the fuel load in savannas; Liedloff and
Cook 2007, Higgins et al. 2008). An asymptotic
relationship between fuel load (grass shoot biomass,
GSt) and the grass and tree shoot biomass consumed by
fire as reported in Higgins et al. (2000) is assumed. We
use the function
8
b
>
< GSt
for t mod F ¼ 0
ct ¼ ab þ GSbt
ð7Þ
>
:
0
for t mod F 6¼ 0
to describe this relationship. The parameters a and b in
Eq. 7 describe the shape of the asymptotic function and
we assume that these two parameters differ for grasses
and trees (the parameters aG and bG and aW and bW,
respectively, define cGt and cWt). Fig. 6C illustrates ct as
a function of grass fuel load (GSt) for grasses and trees.
Analyses of the model show that stable coexistence,
unstable coexistence, or competitive exclusion of either
grasses or trees are possible outcomes of this model
system (Scheiter and Higgins 2007; also see Appendix).
Which outcome is realized is determined by the
competition and fire parameters. Scheiter and Higgins
(2007) show that the model is not reliant on a rooting
niche mechanism for grass–tree coexistence and that the
inclusion of fire allows coexistence even when both
belowground (a WG and aGW ) and aboveground (x)
competition are intense.
PARAMETER ESTIMATION
Approach
Each model parameter summarizes the outcome of
several aggregated ecological processes. The parameters
describing such aggregated processes are difficult or even
impossible to directly estimate from field measurements.
This is a general problem, encountered by many types of
ecological models. One solution to this problem is
indirect parameter estimation (Freckleton and Watkinson 2000, Wiegand et al. 2003, Nelson et al. 2004). Here
a process-based model is used to find the combination of
model parameters that best match some or several
observed patterns from the real system. A lack of fit
between model and data would suggest that the system is
not well-described by the model. A good fit between the
model and data suggests that the model can describe the
system. However, a good fit is only useful if the solution
is unique; many indirect estimation procedures are
plagued by the problem that more than one parameter
combination might fit the data equally well.
This problem of ‘‘non-uniqueness’’ in the parameter
estimates can be addressed by using ecological understanding to constrain the values that the parameters can
assume (Nelson et al. 2004). Constraining parameter
values might be based on general biological principles
(e.g., since a mortality rate must be a number in the
interval 0–1, we could constrain the mortality rates to be
in this interval) or based on data from previous studies
1685
(e.g., a meta-analysis might suggest that mortality rates
for savanna grasses are typically between 0.02 and 0.10;
we could use this information to constrain this
parameter to this interval). Bayesian statistics provide
a formal and established way to combine such prior or
expert knowledge of where the parameters are thought
to lie (the prior densities, pr(/)) and the information
contained in the data (the likelihood pr(data j /)) to
estimate the parameters (the posterior densities,
pr(/ j data)). The Bayesian approach additionally provides a robust method for estimating uncertainty (in
Bayesian terminology the credible intervals) in the
parameter estimates.
Markov chain Monte Carlo (MCMC) methods
provide a general method for estimating the posterior
densities of parameters. The MCMC is a means to
generate samples from the posterior distribution. Each
step in MCMC algorithm proposes a set of parameters
to explain the data; parameter combinations that are
relatively likely are added to the sample and combinations that are less likely are rejected from the sample.
The resulting sample numerically describes the multivariate posterior distribution of the parameters. Albert
(2007) and McCarthy (2007) provide detailed and
accessible introductions to Bayesian and MCMC
methods.
DATA
AND
IMPLEMENTATION
We use data from 197 sites scattered across Africa.
This data set is a subset of the data set used by Sankaran
et al. (2005) where aboveground tree biomass estimates
can, on the basis of basal area measurements, be made.
We restrict ourselves to these data because they provide
a better description of abundance than does the canopycover data analyzed by Sankaran et al. (2005). The data
cover a rainfall gradient from 200 to 1200 mm mean
annual precipitation (MAP). We assume that the
observed tree biomass at each site is in equilibrium (a
fixed point or a stable limit cycle) with the site’s MAP
and fire return interval.
We used these data to estimate, using a MCMC, the
posterior distributions of the growth (g̃G, g̃W ), decomposition (dG, dW ), and competition (x WG, aWG, aGW )
parameters. The model is forced with the rainfall (Ri )
and with the fire return interval (Fi ) of each site i. The
site rainfall (Ri ) is known, but the fire return interval
(Fi ) is optimized for each site. We ignore grazing
(parameter z in Eq. 1 is set to zero) and we use the
simpler fire model (Eq. 6) to facilitate the fitting process.
The consequences of these two simplifications are
considered in Results and discussion: The nature of
grass–tree coexistence.
We estimate the likelihood of the data by assuming
that the errors in tree abundance are normally distributed and use informed Gaussian priors for all parameters (the priors are shown in Fig. 3 and in Table 1). The
priors for the competition parameters (x WG, aWG, aGW )
describe our expectation that the competition parame-
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STEVEN I. HIGGINS ET AL.
Ecology, Vol. 91, No. 6
TABLE 1. Mean and standard deviation of the Gaussian priors used to estimate the model parameters, as well as mean and 95%
credible intervals of the estimates of the posterior distributions of these parameters.
Prior
Posterior estimates
Parameter
Symbol
Mean
SD
Mean
Lower BCI§
Upper BCI§
Growth coefficient, grass (5)
Growth coefficient, tree (5)
Decomposition rate, grass (1, 2)
Decomposition rate, tree (3, 4)
Intensity of competition for light (1)
Tree on grass root competition (1) Grass on tree root competition (3)à
g̃G
g̃W
dG
dW
xWG
aWG
aGW
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.25
0.10
0.10
0.10
0.25
0.25
0.25
0.87
1.23
0.077
0.081
1.07
0.74
1.16
0.49
0.74
0.055
0.055
0.92
0.48
1.02
1.29
1.80
0.10
0.11
1.19
1.00
1.29
Note: The number following each parameter refers to a numbered equation in the Model description section.
The competitive effect of tree root biomass on grass root growth.
à The competitive effect of grass root biomass on tree root growth.
§ Bayesian credible intervals.
ters are approximately 1.0. The priors for the growth
and decomposition parameters were selected to ensure
that the system’s dynamics are fast enough that a stable
state is achieved within 2000 time steps. Transient
dynamics were further avoided by randomly selecting
the initial abundances of GS, GR, WS, and WR for each
simulation. As stated in the Model description section,
we have assumed that each time step represents two
months. The length of the time steps is, for this study, of
minor importance because we compare the model’s
asymptotic state to the empirical abundance data. That
is, because we compare the model’s asymptotic state to
data, we are able to provide only relative and not
absolute estimates of the growth and decomposition
rates. The parameter estimates might therefore provide
inaccurate representations of the system’s transient
dynamics.
The model predicts relative abundance in the interval
0–1. For comparison to data and to facilitate the
interpretation of model output we transform these
values using linear scaling factors (1 unit of tree
abundance ¼ 120 Mg/ha [1 Mg/ha ¼ 1000 kg/ha]; 1 unit
of grass abundance ¼ 12 Mg/ha). These scaling factors
are consistent with aboveground biomass data presented
in O’Connor (1985), Higgins et al. (1999, 2000, 2007a),
and Grace et al. (2006).
The model was implemented in C and compiled as an
R package (R Development Core Team 2007), which
allows one to execute the model from within R. We use
our own R implementation of the Delayed Rejection
Adaptive Metropolis algorithm (DRAM; Haario et al.
2006) for running the MCMCs. We express all
parameters on a log scale and we define g̃W ¼ b1g̃G
and estimate b1 and g̃G to improve convergence of the
MCMC. Similarly we define dW ¼ b2dG and estimate b2
and dG. There is no generally accepted diagnostic that
can prove that a chain has converged to the posterior
distribution (Geyer 1992). We tested convergence by
confirming that replicate chains initiated with different
starting values all converged to the same region in
parameter space and by confirming that the correlation
structure in the estimated parameters was weak. For the
FIG. 2. (A) Modeled vs. observed aboveground tree
biomass for 197 sites scattered across Africa. (B) Empirical
data plotted against rainfall. The broken line is the model’s
predicted tree biomass in the absence of fire across the rainfall
gradient. Open circles in both panels indicate data points where
the model parameterization, in the absence of fire, underpredicts the empirically estimated tree biomass.
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COMPETITION AND DISTURBANCE IN SAVANNAS
1687
FIG. 3. Prior and posterior probability density functions of the parameter estimates. Only one prior distribution is shown in
each of the three panels. This is because grasses and trees are assumed to have the same prior distributions, and all competition
parameters are assumed to have the same prior distributions. The parameters are defined in Eqs. 1–5. The posterior densities are
displayed using a kernel smoother. Table 1 lists the parameters describing the prior and posterior distributions: x represents
competition among shoots for light, and a represents competition among roots.
estimation of the posterior distributions of the parameters we use a single long (1.5 3 105 ) chain.
RESULTS
AND
DISCUSSION
Parameter estimates
The model is flexible enough to replicate the
empirically estimated tree biomass at the majority of
the sampled sites (solid points in Fig. 2). Several sites
(open symbols in Fig. 2) have higher estimated biomass
than the model parameterization is capable of predicting. When fire is excluded, the model predicts a rainfalldefined upper limit of tree biomass (dotted line in Fig.
2B).
The posterior distributions of the parameter estimates
show that all parameters are well-defined (Fig. 3).
Particularly well-defined are the light competition
parameter (x WG) and the grass on tree root competition
parameter (aGW ), whereas the tree on grass root
competition parameter (a WG) has a relatively wide
credible interval.
The parameter estimates suggest that grass growth
rates (Fig. 4A) are higher than tree rates. The growth
rate parameters (g̃G and g̃W, Eq. 5) for grasses and trees
were, respectively, 0.87 (credible interval ¼ 0.49, 1.29)
and 1.23 (credible interval ¼ 0.74, 1.80; Fig. 3). These
parameter estimates suggest that grass growth is more
responsive than tree growth to rainfall. This is qualitatively consistent with empirical knowledge from savannas (Scholes and Walker 1993) and other ecosystems
(Shipley 2006) and can be understood as being a
consequence of architectural differences between trees
and grasses. The decomposition rates of grasses and
trees were similar (dG ¼ 0.077, credible interval ¼ 0.055,
0.10; dW ¼ 0.081, credible interval ¼ 0.055, 0.011; Fig. 3).
Root competition was found to be asymmetrical, with
grass root biomass having a stronger effect on tree root
biomass growth (aGW ¼ 1.07, credible interval ¼ 0.92,
1.19; Fig. 3) than tree root biomass has on grass root
biomass growth (a WG ¼ 0.74, credible interval ¼ 0.48,
1.00; Fig. 3). This is consistent with the empirical studies
that have shown that grasses are more effective
competitors for soil water than trees, although other
empirical studies suggest that aWG ’ aGW ’ 1 (see
Scholes and Archer [1997] for a review). Light competition was predicted to be intense (x WG ¼ 1.16, credible
interval ¼ 1.02, 1.29; Fig. 3), which is consistent with the
architectural differences between trees and grasses
(Sankaran et al. 2005, Scheiter and Higgins 2009).
The maximum level of competition that a competitor
can tolerate varies across the rainfall gradient (Fig. 4B;
for the definition of the maximum competition levels see
Appendix). Trees cannot persist in the presence of
grasses at rainfall levels much below 200 mm mean
annual precipitation (MAP); however, above this
rainfall threshold the level of competition trees can
tolerate is not influenced by rainfall (Fig. 4B). Grasses,
by contrast, can tolerate high levels of tree competition
at rainfall levels below 200 mm MAP, but the level of
competition they can tolerate decreases gradually as
rainfall increases. Above a threshold of 1200 mm MAP,
the parameter estimates suggest that grasses can, in the
absence of fire, no longer persist (Fig. 4C). Our
confidence in the position of this threshold is, however,
low. We assess the confidence by using the posterior
densities of the parameter estimates to propagate the
error in the maximum rainfall level for which aWG ,
amax
WG . The credible intervals of this rainfall threshold are
821 and 3142 mm (Fig. 4D). Hence we are uncertain
where this threshold level lies, but we are certain that the
threshold lies above 821 mm MAP.
The nature of grass–tree interactions
Scheiter and Higgins (2007) define the zero-growth
isoclines for the model for the case without fire (the
equations defining these isoclines are described in the
Appendix). Plotting these isoclines using the parameter
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STEVEN I. HIGGINS ET AL.
Ecology, Vol. 91, No. 6
FIG. 4. (A) The growth rate function (Eq. 5) selected by the MCMC (Markov chain Monte Carlo) procedure for grasses and
trees, in relation to annual rainfall. The growth rates describe the proportional change in biomass in a time step. (B) The maximum
root competition coefficient that grasses and trees can tolerate as a function of rainfall. Calculation of the maximum root
competition levels is described in the Appendix. (C) The predicted tree and grass biomasses as a function of rainfall in the absence
of fire. Black lines depict the predicted biomass in the presence of competitors; gray lines depict the predicted biomass in the
absence of competitors. (D) The uncertainty in the estimate of the maximum rainfall level at which grass–tree coexistence is possible
in the absence of fire is estimated by using the posterior densities of the parameter estimates to propagate error in the estimate of the
maximum rainfall for which aWG , amax
WG . The vertical dotted lines indicate the credible interval in this rainfall threshold.
estimates for three different points along a rainfall
gradient (arid ¼ 350 mm MAP, semiarid ¼ 700 mm
MAP, and mesic 1400 mm MAP) illustrates how the
model’s behavior changes as a function of rainfall. The
mesic site is defined to receive 1400 mm MAP (1200
MAP is the highest rainfall site in our data set); this
allows us to examine how the model behaves beyond the
domain of the data. These analyses show, in the absence
of fire, that the dynamics are characterized by a stable
equilibrium point at the arid and semiarid sites (Fig. 5).
At sites above a precipitation threshold the model
predicts that, in the absence of fire, trees exclude grasses
(mesic site, Fig. 5).
When fire is introduced, the solution is, in all cases, a
stable oscillator (Fig. 5). Fire prevents the tree
abundance from reaching its rainfall-defined equilibrium
point; instead the system trajectory oscillates around a
stable point of lower tree abundance and higher grass
abundance. In the case of the mesic site in Fig. 5, fire
allows grasses to persist in the system by reducing trees
to abundances below which they can exert a critical
competitive effect on grasses. Hence, in mesic systems,
fire does not, as is the case in arid and semiarid systems,
merely modify the position of the equilibrium point, but
it qualitatively changes the outcome from competitive
exclusion to coexistence and this coexistence is characterized by a stable-limit cycle.
This result shows that at more mesic sites the role of
fire shifts from being a modifier of the structure of
savannas to being a factor that allows the savanna state
to exist. Sankaran et al. (2005) interpret tree cover data
from Africa to suggest that for sites receiving more than
650 mm MAP, fire is needed to maintain the ‘‘savanna’’
state (i.e., discontinuous woody strata with a contiguous
grass understory) by preventing canopy closure. Our
model and parameter estimates suggest that grasses may
nevertheless persist in the understory, albeit at lower
biomass levels, up to 1200 mm MAP even in the absence
of fires. Above 1200 mm MAP, fires are needed to
prevent grasses from being competitively excluded from
the system. Although, as discussed in the previous
section, our confidence in the position of this rainfall
threshold is low; the credible intervals suggest that it lies
above 821 mm MAP.
The model parameterization predicts that fire plays a
similar role in African savannas receiving between 200
and 1200 mm MAP. This is well-illustrated by plotting
the predicted tree abundance as a function of fire
June 2010
COMPETITION AND DISTURBANCE IN SAVANNAS
FIG. 5. Phase planes of the model’s dynamics in arid (350
mm mean annual precipitation [MAP]), semiarid (700 mm
MAP), and mesic (1400 mm MAP) sites in the presence and
absence of fire. The isoclines are for the case without fire, and
the trajectory (gray line) is for the case with fire. The open
symbol is the stable point for the system without fire, and the
solid symbol is a point within the stable limit cycle that occurs
in the presence of fire. Note that at the mesic site the isoclines
do not intersect, and the stable point (in the absence of fire) is at
zero grass biomass.
1689
FIG. 6. (A, B) The effect of different fire return intervals
(parameter F in Eqs. 6 and 7) on tree abundance (measured as
biomass) along a rainfall gradient using (A) the simple fire
model (Eq. 6) and (B) the more realistic (Eq. 7) fire model. (C)
The assumed response between fuel load and aboveground
biomass consumption by fire for grasses and trees used in the
more realistic fire model. The parameters a and b in Eq. 7 are
0.1 and 3.0 for grasses and 0.1 and 1.0 for trees.
frequency (Fig. 6A). This plot suggests that high fire
frequencies are capable of drawing tree biomass close to
zero at any point on this rainfall gradient. The caveats
are that as rainfall increases more frequent fires are
required to draw tree abundance away from the rainfall-
1690
STEVEN I. HIGGINS ET AL.
FIG. 7. The effect of different levels of grazing (parameter
z, Eq. 1) on tree abundance (measured as biomass) in the
absence of fire.
defined tree abundance (Fig. 6A) and that fire is
necessary for grass persistence above some threshold in
precipitation (Fig. 4C, D). Our analysis is therefore
consistent with the view that fire is not necessary for
grass–tree coexistence over a broad range of conditions
even though it has a strong influence on tree abundance
(Fig. 6A).
We used the more realistic fire model (described by
Eq. 7, and parameterized as illustrated in Fig. 6C) to
explore whether the conclusions drawn in the previous
paragraph are an artifact of the assumptions of the
Ecology, Vol. 91, No. 6
simple fire model (Eq. 6). Using the more realistic fire
sub-model, it is apparent that although fire can be used
to prevent trees from excluding grasses at high rainfall
sites, this often requires the use of more frequent fires.
Furthermore, at high rainfall sites high fire frequencies
cannot draw tree biomass down to zero (Fig. 6B). This
analysis also reveals that there are thresholds beyond
which a given fire frequency ceases to be effective in
modifying the tree biomass. For example, above 1200
mm MAP, fires that occur triennially or less frequently
have no effect on tree biomass (Fig. 6B).
To explore the role of herbivory we ran simulations
that include grazing (z . 0; Eq. 2). These simulations
show that grazing favors trees and generates a ‘‘bushencroached’’ savanna (Fig. 7). This observation is
consistent with functional definitions of bush encroachment that define systems as being bush encroached if the
observed tree biomass is higher than the tree biomass
that is defined by resource levels and competition with
grasses. Using this definition, we can describe the
potential for grazing-induced bush encroachment as
the deviation between the tree biomass that is defined by
rainfall levels and grass competitors and that defined by
high grazing levels (Fig. 7). Under exceptionally high
grazing levels, grasses are lost from the system and the
tree biomass approaches the rainfall-defined, grasscompetitor-free levels that are illustrated in Fig. 4C.
This analysis suggests that the potential for grazinginduced bush encroachment is highest at intermediate
rainfall sites (Fig. 7).
As a final analysis we examine the extent to which the
observed biomass deviates from the potential biomass
defined by rainfall (Fig. 8A). Although these deviations
become increasingly negative with increasing rainfall,
FIG. 8. The deviation between observed tree biomass and rainfall-defined potential biomass as a function of (A) rainfall and (B)
percentage sand. The solid line in panel (A) is the maximum deviation the model is capable of predicting as a function of rainfall.
The solid lines in panel (B) are the 0.9 and 0.1 quantiles of the deviations as a function of percentage sand. Note that data on soil
properties were not available for all sites; i.e., the number of points in panel (B) is a subset of the points plotted in panel (A).
June 2010
COMPETITION AND DISTURBANCE IN SAVANNAS
the maximum potential deviation predicted by the model
(solid line in Fig. 8A) is observed in the data for sites
spanning the entire rainfall gradient. This suggests that
the relative effect of fire is equally important at arid and
mesic sites and implies that fire can explain almost all
variance in deviations from the climate potential. The
exceptions are the points above the broken line in Fig.
8A (these points correspond to the open circles in Fig.
2). We tested (using the soil texture and soil nitrogen,
phosphorous, and carbon contents provided by Sankaran et al. [2005]) whether soil properties can explain these
exceptions. A quantile regression (Koenker 2008)
reveals that although the lower quantile of the deviations is not influenced by soil texture, the upper quantile
of the deviations is defined by soil texture (Fig. 8B).
That is, the coarser the soil texture, the smaller the
deviation between potential tree biomass and observed
tree biomass. This suggests that trees growing in clay
soils have difficulty attaining the rainfall-defined potential biomass, whereas trees growing in sandy soils attain
biomasses approaching and even exceeding the biomass
that the model, using rainfall alone, predicts (points
above the broken line in Fig. 8B). This finding is
consistent with Walker and Noy-Meier (1982) who used
a rooting niche model to predict that tree biomass
should increase with increasing soil sand content.
CONCLUSIONS
This study used a heuristic model of savanna
dynamics to explain the continental scale patterns of
tree abundance observed in African savannas. The fitted
model shows that grass–tree coexistence is possible
across a broad rainfall gradient (200–1200 mm MAP).
This is true even in the absence of fire. At sites receiving
higher levels of rainfall, fire becomes necessary for
grasses to persist in the system. The model additionally
shows that fire can reduce tree biomass across the entire
gradient and while the absolute effect of fire is greater at
mesic sites, the relative effect of fire does not change
across the gradient. Finally, the solutions generated by
the model are, irrespective of rainfall, stable points for
solutions without fire and stable oscillators for solutions
with fire. That is, whether the solution is a stable
oscillator or stable point is influenced solely by the
presence or absence of fire.
Several data points in the empirical data had higher
tree abundance than the model could predict. We can
explain these points by showing that at more sandy sites
tree abundances can exceed the tree abundances that the
model, using rainfall alone, can predict. This finding is
consistent with the view that trees are more effective
competitors for soil water at sandy sites (Walker and
Noy-Meier 1982). The model further shows that the
effects of grazing on tree abundance, and therefore the
potential for bush encroachment, is highest at sites
receiving intermediate levels of rainfall. Taken together,
our findings support the view promoted by Frost et al.
(1986) that rainfall and soil properties are the primary
1691
drivers of savannas, but that fire and herbivory are
secondary drivers.
Our analysis is based on indirect parameter estimation. This means that our interpretation is influenced by
a combination of the information in the empirical data,
the information contained in the model structure, and
our prior estimates of parameters. Additional data
interpreted with this model might yield different
insights. We expect that these additional analyses will
contradict details of the quantitative predictions made
by this analysis. The real test of the utility of this
analysis is whether the following predictions are true: (1)
fire is not needed for grass–tree coexistence at sites
receiving between 200 and 1200 mm MAP (the
precipitation range examined here), even though fire
can substantially modify the relative abundance of
grasses and trees; (2) the relative importance of fire in
savannas is not influenced by rainfall; (3) fire determines
whether the asymptotic dynamics of savannas are
described by a fixed point or by a stable oscillator; and
(4) the asymmetry of competition between grasses and
trees for soil moisture is contingent on soil texture.
ACKNOWLEDGMENTS
We acknowledge the financial support of the Robert Bosch
Foundation, the National Science Foundation (Biocomplexity
in the Environment Grant EAR 0120630 to Niall Hanan), as
well as the Biodiversity and Climate Research Centre (which is
supported by the ‘‘LOEWE—Landes-Offensive zur Entwicklung Wissenschaftlich-ökonomischer Exzellenz’’ program
of Hesse’s Ministry of Higher Education, Research and the
Arts).
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APPENDIX
A description of the zero growth isoclines for the model system (Fig. 5) as well as the maximum levels of root competition grasses
and trees can tolerate (Fig. 4) (Ecological Archives E091-115-A1).