OIKOS 85: 451-466. Copenhagen 1999
Detecting process from snapshot pattern: lessons from tree spacing
in the southern Kalahari
Florian Jeltsch, Kirk Moloney and Suzanne J. Milton
Jeltsch, F., Moloney, K. and Milton, S. J. 1999. Detecting process from snapshot
pattern: lessons from tree spacing in the southern Kalahari. - Oikos 85: 451-466.
The spatial distribution of plants is often thought to be an indicator of underlying
biotic and abiotic processes. However, there are relatively few examples of spatial
patterns being analysed to detect an underlying ecological process. Using the spacing
of savanna trees in the southern Kalahari as an example, we applied methods of
computer simulation modelling and point pattern analysis in an evaluation of their
potential for identifying relevant pattern generating processes from snapshot pattern.
We compared real tree patterns from the southern Kalahari. derived from aerial
photographs, with patterns produced from computer simulation experiments in an
investigation of the following questions: does the present pattern of tree distributions
allow us to characterize ( I ) the relative importance of the major driving forces (e.g.,
competition for moisture, grass fire. herbivory). (2) the spatial dimensions and
structures of the underlying processes, and (3) the actual dynamic status of the
ecological system (a phase of decline, increase or constancy with respect to tree
abundance)?
The simulation experiments are based on a well established. spatially explicit,
grid-based model that simulates the vegetation dynamics of the major life forms
under a realistic rainfall scenario of the southern Kalahari and under the impact of
grass fires, herbivory and the formation of localized clumps with increased tree seed
availability. For a realistic range of parameters the simulation model produces
long-term coexistence of trees and grasses with tree densities that correspond with
densities observed in the field. Both real tree distributions derived from aerial
photographs and tree pattern produced by the model are characterized by a tendency
towards even spacing at small scales, clumping at intermediate scales and randomness
or clumping at large scales. However, increasing the spatio-temporal correlation in
the formation of seed patches in the model caused an increase in the tendency
towards clumping in the tree distribution whereas an increase in seed patch numbers
led to a decrease in clumping. Within single simulation runs the tree pattern could
change in response to the variable rainfall sequences and the corresponding differences in grass fire frequency: periods of slightly increasing tree numbers caused by
higher precipitation were characterized by an increase in tree clumping whereas
periods of slightly decreasing tree numbers showed a tendency towards random or
even tree spacing. Simulating the transition of an open savanna to a savanna
woodland showed that the tree pattern in the transitional phase can be diagnostic of
the underlying process: If the transition was caused by improved moisture conditions
the transitional phase was characterized by increased clumping in the tree pattern. In
contrast, a transition caused by an increase in the number of localized tree seed
patches led to a characteristic even spacing of trees.
Even though the simulated savanna clearly showed non-equilibrium dynamics, simulation results indicate that the tree population in the simulated area of the southern
Kalahari is in a state of long-term tree-grass coexistence with the persisting structure
of an open savanna system.
F. Jeltsch, UFZ-Centre f o r Environmental Research Leipzig-Hulle, Dept of'Ecologica1
Modelling. P.O. Box 2. 0-04301 Leipzig, Germany (pori@oesa.ufz,&). - K. Moloney,
Iowa Stares &it.., Dept of Botany, Ames, IA 5001 1-10220, USA. S. J. Milton, Univ.
of Cape To~vn.FitzPtrtrick Inst.. Rondehosch 7700, South Africa.
-
Accepted 23 September 1998
Copyright O OIKOS 1999
ISSN 0030-1299
Printed in Ireland - all r~ghtsreserved
For a long time (e.g. Gleason 1920, Greig-Smith and
Chadwick 1965), spatial patterns in the distribution of
individual plants have attracted much interest as a
presumed. but rarely proven, indicator of underlying
biotic and abiotic ecological processes, including processes such as regulatory mechanisms within the plant
community (e.g. Kenkel 1988, 1993, Hughes 1988,
Skarpe 1991), self-organization triggered by recurring
environmental stresses (e.g. Paine and Levin 1981,
Mueller-Dombois 1987, Sato and Iwasa 1993, Jeltsch
and Wissel 1994, Ratz 1995). or disturbances caused by
biotic agents (e.g. Klaas 1996. Moloney and Levin
1996. Lobo et al. in press). Although Watt (1947)
initiated an intensive discussion in the literature about
'pattern and process' in plant communities (e.g. Pickett
and White 1985, Glenn-Lewin and van der Maarel
1992. Levin 1992, van der Maarel 1996) there are
relatively few examples of spatial patterns being
analysed to detect an underlying ecological process (e.g.
Skarpe 1991. Jeltsch and Wissel 1994. Haase 1995,
Peterson and Squiers 1995, Ratz 1995, Thiery et al.
1995, Jeltsch et al. 1997a, b, Lobo et al. in press). More
commonly, studies examining the relationship between
pattern and process have focused on determining the
role that existing patterns play in modifying ecological
dynamics (Silvertown 1992, Glenn and Collins 1993,
Reichmann 1993, Cain et al. 1995. Kareiva and Wennergren 1995).
Detecting underlying processes from a pattern requires. in principle, three steps: (1) characterization of
the spatial pattern, (2) development of hypotheses
about the underlying processes generating the observed
pattern. and (3) evaluation of the hypothesis. Even
though these steps are easy to identify, each may be
difficult to accomplish.
The characterization of pattern in plant distributions
(step 1) is closely linked to spatial pattern analysis
techniques developed in the last four decades (Whittaker 1951. 1975, Clark and Evans 1954, Ripley 1976,
1981, Ludwig and Goodall 1978, Greig-Smith 1979.
1983, Ver Hoef et al. 1993, Haase 1995). Although
these methods have evolved continuously, problems in
their implementation remain (e.g. Ludwig and Goodall
1978, Haase 1995) and an ever-increasing variety of
methods, which give different results when analysing
the same patterns. are now in use (Haase 1995).
The primary question to be answered by the statistical analysis techniques used in characterizing the distribution of individual plants is whether the pattern is
random, clumped (aggregated) or regular (evenspaced). The underlying process for regular spacing of
even-sized individuals of the same species has often
been hypothesized (step 2) to be the result of densitydependent mortality caused by intraspecific competition
for an evenly spaced resource (Philips and MacMahon
1981, Skarpe 1991). Aggregated patterns have been
explained in terms of regeneration ecology, e.g. short-
range seed dispersal, vegetative reproduction, or the
occurrence of safe sites (Harper 1977, Augspurger 1984,
Beatty 1984), or in terms of the patchy distribution of
resources. The hypothesis for random distribution is
either that there are no significant spatial interactions,
or that the pattern represents a transitional state in a
population shifting from an aggregated distributional
pattern to a regular pattern (Prentice and Werger 1985,
Skarpe 1991). Although these hypotheses of pattern
producing processes are widely accepted (Skarpe 1991).
they are mainly oriented towards a static view of the
plant community (Haase 1995). The interpretation of
the distribution pattern of plants gets more complicated
if dynamic changes in time and space are considered
(Haase 1995). However, long-term data of dynamic
change in the spatial patterns of plant communities are
seldom available and are often difficult to collect. Thus
the question arises: what can be learned about processes occurring in plant communities on the basis of
static snapshots of spatial patterns obtained at only one
time?
Step (3). the critical evaluation of the hypotheses
generated in step (2) for the pattern-generating processes. is often neglected. One reason is the difficulty of
conducting replicated experiments on the relevant spatial and temporal scales. Spatially explicit computer
simulation models can be a helpful tool in this regard.
The consequences of the hypothetically relevant processes can be tested systematically in simulation experiments and the resulting spatial patterns can be
compared to plant distributions found in the field
(Jeltsch and Wissel 1994, Jeltsch et al. 1997a).
In this paper, we will examine tree spacing in semiarid savannas as an appealing example for applying
pattern analysis techniques in an evaluation of their
potential for identifying relevant pattern-generating
processes, such as seed dispersal, competition for moisture, grass fires and herbivory. Changes in the spatial
patterning of tree distributions in savannas are relatively slow and the patterns themselves can easily be
detected from aerial photographs. In addition. the
abundance and spatial distribution of savanna trees is
of significant ecological importance (Pianka and Huey
1971, Belsky et al. 1989, 1993, Maclean 1993, Belsky
1994. Milton and Dean 1995, Jeltsch et al. 1996). In the
following we will compare real tree patterns from the
southern Kalahari, derived from aerial photographs,
with patterns produced from computer simulation experiments in an investigation of the following question:
what can be learned about the underlying ecological
processes by analysing the spatial distributions of trees
in the southern Kalahari? More specifically: does the
present pattern of tree distributions allow us to characterize (a) the actual dynamical status of the ecological
system (a phase of decline, increase or constancy with
respect to tree abundance), (b) the relative importance
of the major driving forces, and (c) the spatial dimensions and structures of the underlying processes?
Site description
The investigations deal with the Kalahari Gemsbok
National Park (KGNP) in the South African part of
the southern Kalahari bordering on Botswana (situated
between 24"15' S and 26"30f S, and 20'00' E and 20°45'
E). The vegetation of the KGNP, being largely representative of the southern Kalahari, is protected from
the over-utilization common in adjacent farming areas
and is well preserved (Leistner and Werger 1973, van
Rooyen et al. 1990). The average annual rainfall ranges
from 209 mm in the south of the park to 220 mm in the
north and falls mainly during thunderstorms in mid- to
late summer (January to April). Monthly means of
daily minimum and maximum air temperatures at Twee
Rivieren in the southern part of the KGNP range
between 19.5"C and 37.4"C in January and between
1.2"C and 22.2"C in July (van Rooyen 1984). The
vegetation is described as the western form of Kalahari
thornveld (Acocks 1953, Leistner 1967) - an open
savanna with trees (Acacia erioloha, A . huernatoxylon,
A . mrllifera, Bosciu albitrunca) scattered within a matrix of low shrubs (Rhigozum trichotornum, Monechrna
spp.) and grasses (Schn~idtiaknlihuriensis, Stipagrostis
spp.). Geophytes and ephemerals are typical elements
of the vegetation cover in wet years.
Methods
Six maps of the spatial distribution of trees, each
covering an area of 50 ha. were digitized from aerial
photographs of the KGNP (South African Surveys and
Land Information Bureau, unpubl. data). The mapped
sites were chosen because they represented relatively
homogenous areas of the savanna. which allowed for a
comparison with tree distributions produced by our
computer simulation model. Tree distribution patterns
derived from aerial photographs and from simulation
experiments were analysed and compared using statistical point pattern analysis (see below).
Point pattern analysis
We used Ripley's L-function analysis (Ripley 1976,
1981, Cressie 1991, Bailey and Gatrell 1995) to characterize the spatial patterns of savanna trees across a
range of scales h in the aerial photographs of the
KGNP and in our savanna model output. The scale h
corresponds to a circular area of radius h around
individual points of the pattern being analysed. All of
the analyses were based on the general function L(h).
which is used to characterize the degree of clustering or
hyper-dispersion of a set of points relative to a randomly distributed set of the same number of points. We
use the standard formulas given in Bailey and Gatrell
(1995): 120-121) for the calculation of L(h).
OIKOS 85.3 (1999)
Values of L(h) are interpreted as follows: if a pattern
is spatially random at scale 11 then the expected value of
L(h) is 0, otherwise L(h) < 0 for an evenly distributed
(hyper-dispersed) pattern and L(h) > 0 for a clustered
pattern. In order to interpret the significance of L(h)
across a range of scales, it is necessary to test for
significant departures from the null hypothesis of a
random pattern. This is generally done through the
estimation of confidence intervals using Monte Carlo
simulations (Ripley 1981, Cressie 1991, Bailey and
Gatrell 1995). Here, we estimated 95'% confidence intervals around the expected value of L(h) and used these
to test for significant departures from a random pattern. The confidence intervals were constructed by randomizing the positions of the points in the pattern
being analysed 19 times (Haase 1995). We then determined values of L(h) for the randomized patterns.
Significant clustering in a pattern was indicated for
values of L(h) that were greater than the maximum
L(h) obtained through the randomization procedure
and significant repulsion was indicated by L(k) values
that were less than the smallest value obtained through
randomization (cf. Haase 1995). L(h) patterns falling
between the confidence intervals were considered to be
spatially random.
The model
The spatially explicit simulation model is based on the
ecological dynamics occurring in the South African
part of the southern Kalahari represented by the Kalahari Gemsbok National Park (KGNP). Results from
earlier versions of the model have been presented elsewhere (Jeltsch et al. 1996, 1998). The model has since
been modified to allow for a systematic exploration of
the tree distribution pattern developing under a realistic
rainfall scenario of the modelled environment.
Earlier versions of the model focussed on the general
question of long-term tree-grass coexistence in semiarid savannas. In the current version of the model we
use a realistic rainfall scenario based on 32 yr of rainfall
data from the KGNP to systematically explore the
process of pattern formation under the impact of various pattern-generating processes. such as grass fires,
patchy seed dispersal by mammals, and herbivory.
A description of the current structure of the model
will be presented in the following with further details in
Jeltsch et al. (1996, 1998).
General model structure
The savanna model presented here is a spatially explicit, grid-based simulation model representing an area
of 50 ha, subdivided into a grid of 20000 5 m x 5 m
cells. The modelled grid cell is large enough to accom453
modate a mature tree canopy and is designed to simulate, in annual timesteps, those processes that are crucial to the spatial vegetation dynamics of the savanna
system, i.e. (1) rainfall and moisture availability, (2)
vegetation dynamics of the relevant life forms, (3) grass
fires, (4) grazing, and (5) patchy tree seed dispersal by
mammals. On the level of individual grid cells, the
model determines whether trees, shrubs, perennial
grasses and herbs, or annuals, or whether a mixture of
some of these life forms is present. In addition, the
model keeps track of the ages of individual trees and
distinguishes between three levels of potential productivity for perennial plant species other than trees. Productivity levels are classified as either high. moderate or
low.
and subsoil moisture class M,) for each of the two soil
layers, namely class 3 ( > 150 mm for top- and subsoil
layer), 2 (51-150 mm for top- and subsoil layer), 1
(21-50 mm for topsoil layer, < 50 mm for subsoil
layer), and 0 ( < 21 mm for topsoil layer, 0 mm for
subsoil layer). The 32 yr of rainfall data were analysed
separately and the resulting 32 combinations of topand subsoil moisture classes (MI, M,) occurred in the
simulation runs in a random order. They initiated the
moisture availability in each cell at the beginning of
each simulated year of a simulation run. Within the
year the plant-available moisture in each cell was further modified by other processes, e.g. water reduction
by the vegetation (see below).
Rainfall and moisture availability
The dynamics of soil moisture availability, which
strongly influence vegetation dynamics, are produced in
the savanna model by a submodel based on 32 yr of
daily rainfall data from Twee Rivieren, which is located
in the south of the KGNP (South African weather
bureau, unpubl. data). The soil moisture submodel
aggregates daily rainfall data into longer rain events
(E,). Single rain events are separated by at least three
days without rainfall and are only considered if they
consist of more than 10 mm precipitation. Isolated rain
events of less than 10 mm are ecologically ineffective in
the Kalahari (Leistner 1967, Skarpe 1990). Total soil
moisture available to plants during each year of a
model run is calculated to be 50% of the total rainfall
arriving in rain events during that year, with the remaining 50% lost due to processes such as evaporation
from free water on the soil surface after rain showers,
evaporation from the soil, runoff, deep percolation, and
vegetation interception (Whitmore 1971, Bate et al.
1982, Skarpe 1990, Jeltsch et al. 1996, 1997b). Available
moisture is distributed into a topsoil ( < 4 0 cm; topsoil
moisture m,) and a subsoil layer ( > 4 0 cm; subsoil
moisture m,) following eqs.1 and 2.
m, =
1 0.5 x Minimum (E,, 40 mm)
E.
(2)
Eqs. 1 and 2 are based on the empirical finding that
on sandy Kalahari soil rainfall infiltration is about 10
mm deep per 1 mm rainfall (Leistner 1967, Jeltsch et al.
1996). Thus, only rain events with more than 40 mm
precipitation contribute to the moisture in the subsoil
layer, which is below 40 cm depth. In a next step the
resulting amounts of top- and subsoil moisture are
classified into four classes (topsoil moisture class MI,
454
Vegetation dynamics
Perennial grasses and herbs, shrubs and annuals
We distinguish between three different levels of potential productivity to characterize the condition of a
vegetation patch dominated by shrubs or perennial
grasses and herbs. The transitions between these levels
depend on the plant-available moisture in the grid cell.
If the moisture combination is insufficient, the potential
productivity deteriorates, whereas it is preserved or
improves if sufficient moisture is available (Jeltsch et al.
1996, 1997b). Annuals are either present or absent in a
grid cell. Colonization of empty space (i.e. grid cells)
and extinction (i.e. grid cells becoming empty of a
certain life form) depend on the plant available moisture. The establishment probability P,(M,, M,) has its
maximum value P,(,,,,,under optimal moisture conditions (M, = 3; M, = 3) and decreases exponentially with
deteriorating moisture conditions (eq. 3). I, and I, give
the relative influence of topsoil moisture or subsoil
moisture, respectively, on the described process (Table
1). The extinction probability is modelled in an
analogous way with its highest value P,,,,,,
at worst
moisture conditions (MI = 0; M , = 0) and decreasing
values with improving moisture availability (eq. 4).
Table 1. Standard set of variables for life forms. P,(,,,, = the
establishment probability under optimal moisture conditions;
P,,,,,,
= the highest extinction probability under worst moisture conditions; I, and I, give the relative influence of topsoil
or subsoil moisture, respectively.
Life form
Pecmax)
Perennial grass
Shrub
Annual
Tree < 2 yr
Tree 2-3 yr
Tree 4-5 yr
Tree 6-7 yr
Tree 8-9 yr
0.5
0.6
1 .O
0.6
Pdcmaxl
It
1%
OlKOS 85.3 (1999)
The establishment probability is spatially homogenous for the herbaceous vegetation (homogenous seed
dispersal), whereas shrubs only colonize empty space
in the immediate neighbourhood of established plants
(short-ranged seed dispersal, vegetative reproduction).
Parameter estimates are based on information from
the southern Kalahari (for details see Jeltsch et al.
1996). The herbaceous vegetation is more dependent
on topsoil moisture than on subsoil moisture whereas
shrubs show the opposite dependency, reflecting differences in rooting depth (see Table 1).
Competition for space occurs via establishment in
empty grid cells. If an empty grid cell is successfully
colonized by more than one life form. or if
tree seedlings establish in an occupied grid cell (see
below) competition for moisture occurs. The moistlire
available to one life form or to the tree seedlings
is reduced by the water uptake of the other vegetation which is present in the grid cell (Jeltsch et
al. 1996). Lateral root extension of woody life forms
also reduces soil moisture in neighbouring grid
cells (cf. Scholes and Walker 1993, Belsky 1994).
The extinction probability of the life forms increases as a result of these reductions in soil moisture
(eq. 4).
Trees
Trees (as well as tree seeds and seedlings) are modelled in an individual-based approach (DeAngelis and
Gross 1992. Uchmanski and Grimm 1996) because
this life form is of focal interest and individuals are
easy to identify. The establishment of tree seedlings in
a grid cell depends on the availability of tree seeds
and moisture. The moisture dependency of establishment is described by eq. 3 and Table 1. The general
pattern of seed availability is produced initially by an
exponential decrease of tree seed numbers with increasing distance d from mature savanna trees:
Seeds per cell = S e
d.
(5)
S is the number of germinable seeds beneath the
crown of the mature tree (S=50 for A. erioloba in
the KGNP. Jeltsch et al. 1996). In addition to the
general pattern of seed dispersal, there is an additional pattern of patchy seed dispersal produced by
mammals as described below. We did not include a
seed bank in the model since few African savanna
trees produce significant seed banks (Tybirk et al.
1994) mainly due to parasitism.
Tree seedling mortality follows eq. 4 with a linear
decrease of maximal mortality (PC,,,,,) with age
(Table 1). If at least one tree seedling or sapling survives the first 10 yr, one sub-mature tree survives to
dominate the grid cell (implicit self-thinning). After a
5-yr period of maturation the tree starts to produce
seeds until death. Extremely dry conditions
((M,,M,) = (0.0)) lead to drought stress which causes
a reduction in tree life span (Jeltsch et al. 1996).
This life span is modelled to be a maximum of 250
yr, with a linearly increasing mortality starting at
an age of 120 yr. The tree parameters are based on
life-history attributes of A. eriolobu (Jeltsch et al.
1996).
Grass fire
The probability of the occurrence of a grass fire increases with available grass fuel (Frost and Robertson
1987) and is thus modelled to increase proportionately to the number of grass-dominated grid cells in a
given year. Tree- and shrub-dominated grid cells
are only ignited and killed with a small probability
if they are adjacent to grass-dominated cells, whereas
all grass cells are modelled to burn in case of fire.
Burning of grass cells causes a transition to a poor
condition without any extinction. Only 5% of the
shrub-dominated cells in the neighbourhood of
grass cells are killed, due to the fact that most
shrubs resprout after fire (Trollope 1982, Frost and
Robertson 1987). Tree mortality by fire is fixed at
lo%, which is in the range of empirically observed
values for A. rrioloba in the K G N P (Jeltsch et al.
1996).
Grazing
Grazing is defined here as the consumption of herbaceous biomass combined with trampling. Only detrimental effects of grazing are considered here. Grazing
is modelled for individual cells. and is assumed to
decrease the level of potential productivity of the
herbaceous vegetation within the current timestep. In
addition, grazing and trampling reduce establishment
probabilities. Grazing intensity is described by the
parameter C which describes the probability for a
transition to the next lower level of potential productivity in a grazed grid cell. Establishment probabilities
(given by eq. 3) are reduced by the factor ( I - C ) in
grazed grid cells. Because of a low grazing pressure in
the K G N P except for at watering points (van Rooyen
et al. 1990) we choose G = 0.1, which represents a
lightly grazed scenario (compare Jeltsch et al. 1996).
Spatial heterogeneities in grazing are simulated by
calculating the transition for each cell separately.
In each timestep seed patches of a size of 5 m x 5 m
are distributed in the modelled landscape. The number
of patches was held fixed during individual simulation
runs but were varied between different simulations. The
number of tree seeds (S,,,,,) in seed patches was increased as a function of the actual number of seed
producing trees (N,) in the modelled area of 50 ha:
wash
term
adult
dung
none
Fig. 1. Frequency of tree seedling establishment in combination with smallscale heterogeneities in the K G N P (Jeltsch et al.
1998). 28 transects of 20 m width and 50 m length were
subdivided into 5 m x 5 m patches. Tree seedlings as well as
adult trees ('adult') and several smallscale heterogeneities were
recorded: Water channels caused by topographic features
('wash'), termite heaps ('term'), and antelope dung accumulations ('dung'). 'None' represents transect positions without
heterogeneities. Other types of heterogeneities were investigated as well (i.e. rodent burrows, diggings, depressions made
by antelopes that roll on their backs to clean their hair with
sand) but did not contain tree seedlings. Results of this study
show that patchy distribution of tree seeds in herbivore dung
is the most important factor for the development of a clumped
distribution of tree seedlings: 30.3'%1 of all tree seedlings were
found in patches with antelope dung accumulation even
though these patches only represented 9.4'1/0of the total area.
Black columns indicate the relative frequency of the specific
heterogeneity type. White columns indicate the percentage of
all seedlings that were found in the specific heterogeneity type.
Patchy seed dispersal
Earlier results of the model (see Jeltsch et al. 1996)
suggested that small-scale heterogeneities and disturbances, causing differential probabilities of tree establishment and survival, are probably a crucial factor
determining vegetation dynamics in semi-arid savannas.
Recent field investigations in the K G N P (Fig. 1) and
simulation experiments (Jeltsch et al. 1998) indicate that
among the various types of small-scale heterogeneities
and disturbances in the southern Kalahari, such as water
channels caused by topographic features, rodent burrows and colonies, diggings, areas of increased trampling. antelope tracks or termite mounds, the patchy
distribution of tree seeds in the dung of herbivores is the
most important factor producing small-scale heterogeneities that affect tree seedling establishment. Large
herbivores are known to consume A . rriolobu fruits in
the K G N P and disperse the seeds within their dung
(Leistner 1961, 1967). Thus, in the model we included
only seed patches as a source of heterogeneity in the
distribution of tree seeds. Since the actual number and
distribution of seed patches is unknown on larger scales.
we systematically investigated the impact of a range of
seed patch numbers and spatio-temporal distributions
on the dynamics of the model.
The seed patches were either distributed randomly or
with variable degrees of spatio-temporal correlation, i.e.
the location of a seed patch at timestep t + 1 depended
on the location at time t. The impact of spatio-temporal
correlation was tested because patches where herbivores
defecate are likely to show a certain degree of persistence. Three different levels of spatio-temporal autocorrelation were tested in the following way (Jeltsch et al.
1998; cf. Moloney and Levin 1996): each seed patch
was randomly positioned within an activity region of
20 x 20 grid cells. During subsequent timesteps, each of
the activity regions was repositioned at random with
probability P,,,which will be referred to as the turnover
rate, and seed patches only occurred within their associated activity regions. We investigated P,, = 0.0 (no
repositioning, i.e. high correlation), P,, = 0.2 (slight
correlation) and P,, = 1.0 (repositioning at each time
step, i.e., no correlation). At low values of P,, there was
a high probability that a grid cell that was a seed patch
during one time step would also be a seed patch during
following timesteps. Under these conditions there is a
high degree of spatio-temporal autocorrelation in the
distribution of seed patches at the level of the grid cells.
At higher values of P,, relocation of the activity regions
occurred more often. In turn. the spatio-temporal correlation in the disturbance regime was reduced.
Results and discussion of simulation
experiments
In a first set of simulation experiments we systematically varied the production rate of seed patches to test
whether the model is able to produce long-term coexistence of trees and grasses and if so, whether tree
densities are in a realistic range under these conditions.
The simulations show that at realistic, intermediate
rates of seed patch production (approx. 0.1-1% of the
modelled area per time step), trees indeed persist over
long periods of time at low densities under all three
levels of spatio-temporal correlation in seed patch formation. In fact, within this range of patch formation
rates, tree densities are in excellent agreement with
those observed at K G N P (Fig. 2), although higher
densities of trees can be found near water holes and in
the dry river beds of the Auob and Nossob river
(Leistner 1967, van Rooyen et al. 1990). The model's
ability to produce realistic pattern of tree abundance
40
correlated
--
slightly correlated
uncorrelated
patches with seed accumulations (% of total aredyear)
Fig. 2. Impact of variable numbers and spatial correlations of
seed patches on the long-term dynamics of a simulated A .
erioloba population. Top: turnover rate 0.0. i.e. the activity
regions are not relocated during the simulation run; centre:
turnover rate 0.2, i.e. in each timestep 20'%) of the activity
regions are randomly relocated; bottom: turnover rate 1.0, i.e.
the activity regions are relocated randomly in each timestep.
The horizontal lines indicate the average tree density (solid
line) and the maximum tree density (dashed line) of seven open
savanna regions in the KGNP (Bothma et al. 1992).
allowing long-term tree:grass coexistence increases the
confidence in the model.
At low rates of seed patch production ( < 0.1% of the
modelled area per time step) the model produces a
savanna grassland (Fig. 2) under all levels of spatiotemporal correlation of seed patches. With high rates of
seed patch production (approx. > li% of the modelled
area per time step), there is a permanent increase in tree
numbers and eventual development of savanna woodland (cf. Jeltsch et al. 1996). Under these circumstances.
the increase in tree abundance can be traced to two
factors, an increase in the availability of seeds and a
reduction in fire frequency due to a decrease in the
availability of grass as a fuel. Overall. these trends
support the view that patchy seed dispersal by herbivores is a key process in the production of a stable
pattern of tree:grass coexistence in the southern Kalahari (Leistner 1961, 1967, Jeltsch et al. 1996, 1998). In
the following, we will focus primarily on comparing
spatial patterns of tree distributions at K N G P with
patterns produced by model runs that result in realistic
tree abundances (i.e. seed patch production in 0.1%)to
1% of the modelled area per time step).
In general, tree distributions in the K G N P exhibit a
pattern of even spacing at small scales (lag separation
distances < 3 or 4 grid cells: Fig. 3) and a tendency
towards clustering at intermediate scales (lag distances
> 10 grid cells and < 30 grid cells; Fig. 3). Some areas
also exhibit a strong tendency towards clustering at the
broadest scales analysed (lag distances > 30 grid cells
in areas 2, 5 and 6 of Fig. 3). Clustering at the broader
scales can be traced to the existence of a marked
difference in average tree density in two or more regions of an area, which could arise from either biological causes (e.g. constrained dispersal from a seed
source) or from local environmental variability affecting tree densities (e.g. water availability).
An arbitrary sample of tree distributions from among
model runs shows that the spatial patterns produced
are often qualitatively similar to those found at K G N P
(cf. Fig. 3 and Fig. 4a-d). exhibiting a tendency towards even spacing over short distances and a tendency
towards clustering at the broader scales. What the latter
suggests is that biological processes are at least partially
responsible for some of the variability in average tree
densities observed in the K G N P study area. However,
although the patterns produced by the model are often
similar to those seen in the KGNP, the model can also
produce patterns that are qualitatively different (Fig.
4e-f). Thus the question arises as to whether pattern
variability reflects differences in seed patch number and
spatial autocorrelation in the individual model runs or
whether it arises through variability occurring at different times within a specific run. If the former is true. we
may be able to discern something about the ecological
processes producing the pattern; however. if the latter is
true. it may be more difficult to learn anything about
area I
area 2 ......... t--
O
-0 5
4I
...."........
.. lo ...................
--
L
I
.... 510
l.l
-' j
-1 5
.
0
2
-
-,
-
.........
.. . .. .20.
40
...._..I
....
I
--
Lag
Lag
area 3
area 4
0
area 5
-
~
50
100
150
area 6
200
Fig. 3. Analysis of aerial
photographs of six
representative 50 ha regions in
the KGNP (South African
Surveys and Land
Information Bureau, unpubl.
data). Spatial distribution of
trees are shown together with
their point pattern analysis
(L-value versus spatial scale
given in grid-cell units). The
L-value (Ival) is given together
with the 95% confidence
interval (max, min). L-values
within the range of the
confidence interval cannot be
distinguished from a random
pattern. L-values above the
confidence interval indicate a
significantly clumped
distribution whereas values
below the interval indicate an
even spacing of the trees.
Fig. 4. Sample tree pattern
and the analysis of the spatial
tree distributions in the range
of low-number persistence as a
result of different simulation
runs. Only adult trees are
shown. a) high correlation
(turnover rate 0.0). 105 seed
patches (approx. 0.5'%)of the
area) per timestep, timestep
900; b) slight correlation
(turnover rate 0.21, 105 seed
patches per timestep, timestep
1260; c) uncorrelated (turnover
rate 1.0). 105 seed patches per
timestep, timestep 1700; d)
highly correlated (turnover
rate 0.0), 75 seed patches per
timestep. timestep 1000, e)
slight correlation. 105 seed
patches per timestep. timestep
1500, f) uncorrelated. 200 seed
patches per timestep, timestep
2300. For the interpretation of
the point-pattern analysis see
Fig. 3.
50;
....
I..:'
0 i.,
. ,
s
. . . . .. .. .
. .. .. ... .
. .. .
'
~
*
.
. . . .
. . .
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,
.
i.
.
2 5 .~
1
;
,'I
........
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--
J
J
--
.lval
A
2 -
~
max
mn
C
-I
---
...
0
15
_>l
'
Lag
the ecological processes driving the system by applying
pattern analysis at any one point in time.
When we examined pattern production for a number
of model runs over a range of parameter settings, we
found variability in spatial patterning among time steps
within the same model run (Fig. 5). This suggests that
it would indeed be difficult to detect underlying processes from a single snapshot picture of a system with
any degree of certainty. Nevertheless, the overall pattern produced in any one individual model run does
exhibit a general tendency that is related to the underlyOIKOS 85.3 (1999)
ing ecological relationships. For example, clumping
of trees at intermediate to broad scales generally increased as spatial correlation in the production of seed
patches increased, and slightly decreased as the number
of seed patches produced each time step was increased
(Fig. 5 ) . We would expect both of these trends to occur
since spatial correlation in seed patch production will
lead to an increase in tree abundance around seed
producing trees and an increase in the rate of seed
patch production will tend to produce more widespread
dispersal.
459
Although there was variability in patterning within
and among runs, the patterns produced by the model
were. in general. consistent with the actual patterns
observed at K G N P (cf. Figs. 3 and 5): trees were evenly
spaced at the smallest scales, had a tendency towards
clustering at intermediate scales and were either clustered or randomly spaced at the broader scales. However, it should be noted that, in the real patterns, the
tendency towards even spacing at the smaller scales
occurred over a broader range of scales than those
produced by the model. This is probably due to a wider
range of root competition under field conditions than is
assumed by the model (see Model description).
Given the variability in spatial patterns observed
during the course of individual model runs, we were
interested in examining the functional relationship between the ecological state of the model system and the
spatial pattern in tree distributions produced over time.
T o do this, we examined a 500-yr period of an individual model run, comparing various ecological attributes
of the model to temporal changes in the spatial pattern
of adult tree distributions (Fig. 6). Early on (time step
1200 to 1260 in Fig. 6), the model run was characterized by a high frequency of rainfall maxima, which in
turn caused the appearance of a high number of tree
seedlings and an increasing number of trees. This
should lead to a clumped pattern at smaller spatial
scales, due to local seed dispersal near mature trees, and
a randomly distributed spatial pattern at broader
scales, due to the random distribution of seed patches
throughout the landscape. However, the relatively high
fire frequency occurring during this time period caused
the death of a relatively high number of tree seedlings
and mature trees. This was especially true of solitary
tree seedlings and trees found in the neighbourhood of
grass patches that were subject to higher fire risk. As a
consequence. the formation of tree clumps was favoured. resulting in a clumped spatial distribution
across a broad range of scales. In the following time
period (1270 to approx. 1540) there was a lower frequency of rain maxima and a lower fire frequency.
Fewer tree seedlings established and tree numbers exhibited a slight decline. Within this period, intraspecific
competition between mature trees caused a decline in
the degree of local clumping and lower fire frequency
allowed a larger proportion of seedlings to establish
and survive in randomly distributed seed patches. Both
of these factors caused a shift towards a more random
pattern in the spatial distribution of the tree population. At intermediate scales there was even a tendency
-
-
e3
€
3000
Bi SF-
0 N
~
o
~
~
o
N
~
(
D
-r---NNNNNOn"O",*
~
o
N
.
T
J-tf- r m
~
(
o
30C ZOC
IOC
m
c
I
-
m
NTwmON*Y*oN*coQ3oN
LDrnON*LDrnO
-----NNNNNmmsD<,"fPUum
-
0
high
0
.JfmmONVmmONfwmONTmmn(i
? - .
.NNNNN""?rnrnrn*.3%?SS
Spatlal scale
Spatial scale
low
correlation
none
Fig. 5. Temporal dynamics of the tree pattern under various levels of correlation and densities of seed patches. Each diagram
shows the changes of the spatial pattern over the course of time. In each time step we only distinguish whether the spatial tree
distribution is significantly clumped (black), random (white) o r significantly even-spaced (rectangles) at the different spatial scales
(given in grid-cell units).
~
~
~
~
~
Fig. 6. Detailed analysis of the
spatial and temporal dynamics
of the simulated A . erioloba
population for a 500-yr period
in a sample simulation run
(number of seed patches: 105,
turnover rate: 0.2) From left to
right: number of rainfall
maxima within the last 10 yr
(i.e. years with maximum topand subsoil moisture), number
of grid cells dominated by
perennial grasses and herbs
(average of the last 10 yr).
number of tree seedlings and
saplings (age < 10 yr), number
of mature trees (average of the
last 10 yr). spatial pattern
analysis of mature tree
distribution in 10-yr timesteps.
Arrows indicate 10-yr periods
with fire events.
r a n euenis
p grass
seeclirgs and
saplings
towards regular spacing, which is partly the result of
increasing competition for soil moisture that leads to
self-thinning of tree patches. In the final period examined (time steps 1550 to 1700), rain maxima and fire
frequency again increased slightly, causing a shift towards a more clumped distribution of trees.
The detailed examples. presented above, demonstrate
that the interaction of the pattern forming processes
works on a timescale that is relatively short compared
to the average lifespan of the trees. It also demonstrates
that different patterns emerge despite relatively small
changes in the overall abundance of trees. And. although there were differences in tree spacing patterns
over the course of a model run (Fig. 5), we found that
the general structure of the pattern was characterized
by even spacing at the smallest scale, clumping at small
to intermediate scales, and an increasing tendency towards random spacing at broader scales. Since the same
general pattern was observed in the distribution of trees
in the KGNP, we can ask a number of questions: Is this
general pattern representative of a "stable" state for an
open tree savanna? And is it, in fact, diagnostic for the
current status of the savanna under investigation? To
answer these questions we investigated pattern formation in two model runs where open savanna was developing towards savanna woodland (Fig. 7). This was
done to determine if there was a fundamental difference
in the spatial patterning of these systems as compared
to a system that would be open savanna over the long
term.
In the first model run, we increased the density of
seed patches (e.g. seed dispersal in cattle dung; cf.
O'Connor 1995) to produce an increase in tree abundance. In the second run, we changed the rainfall
regime to produce an increase in the amount of subsoil
moisture (model translation: we switched the values of
top- and subsoil moisture availability if the availability
of topsoil moisture was larger than subsoil moisture).
trees
'OO'"."'"r
frequency L
:
.
"
j
g
x
E
;
:
Spaill1 scale "
N
,7
r
"
., *
)
-
r
*
U'
We then analysed tree distributions during the transition from open savanna to savanna woodland (Fig. 7).
The patterns produced by the two models differed from
one another and also varied in the degree to which they
differed from patterns produced by model runs resulting in long term open savanna (Figs 5. 6, 7).
The transitional model produced by increasing the
production of seed patches began with a completely
random distributional pattern, but quickly shifted to a
pattern characterized by even spacing at short spatial
scales (Fig. 7). As tree abundance increased. the scales
at which there was even spacing expanded to include
increasingly broader scales. The initial (i.e.. up to year
150) tendency towards a random pattern was caused by
the random distribution of seed patches. However, with
a continual increase in tree density over time, there was
a corresponding decrease in the abundance of the grass
layer and, consequently. a decrease in fire frequency.
This led to a rapid increase in tree abundance. Strong
intraspecific competition for moisture then favoured the
establishment and survival of trees in seed patches that
were located within gaps in the tree distribution leading
to a broad scale pattern of even spacing among the
trees.
Unlike the first model, the model with improved
moisture conditions produced a pattern of clumping
across a broad range of scales during the initial increase
in tree abundance (Fig. 7). Then, as tree abundance
increased even more, the pattern developed towards
more even spacing of trees, as is typical for the simulated savanna woodland. The increased availability of
subsoil moisture facilitated the establishment and survival of trees in the neighbourhood of mature trees
despite local competition for moisture. Then. the high
seed availability in the neighbourhood of tree clumps,
together with the high rate of fire-induced mortality of
solitarily trees, led to an increase in tree clumping. This,
in turn, reduced the abundance of grass, decreasing the
fuel load, and consequently produced a decrease in fire
frequency. This allowed the canopy to close up as the
open savanna shifted to a dense savanna woodland. From these two examples, we see that the tree distribution pattern eventually becomes quite different from
the range of patterns observed in model runs producing
long-term open savanna. However, during the transitional period going from open savanna to closed woodland, the model involving an increase in soil moisture
availability passes through a phase where the tree distribution pattern looks very similar to the patterns associated with open savanna. In contrast, there is no period
N
G
P
~
~
N
m
O
~
m
N N C ) O O * - ; T l n
Spatial scale
N
m
of similarity in the transitional model being driven by an increase in seed patch production. General discussion
The diagnostic value of snap-shot pattern
Our study shows that, although different factors or
processes may lead from a state of open savanna to
savanna woodland, they may also produce markedly
different patterns during the period of transition. This
D
Spatial scale
Fig. 7. Development of an open tree savanna towards savanna woodland caused either by an increased number of seed patches
(left) or by an increased availability of sub soil moisture (right). a-g) Sample spatial distributional pattern of simulated savanna
trees after (a) 0 yr, (b, c) 100 yr. (d, e) 200 yr. and (f, g) 300 yr. h and i show the corresponding change in the spatial pattern
in the first 500 yr. The initial distribution (a) is random. Turnover rate 0.2 (slight correlation of seed patches): seed patch density:
5'%1 of the area in b, d. f, h (increased seed patch numbers) and 0.5'Ki of the area in c, e. g, i (increased availability of subsoil
moisture).
462
OIKOS 85.3 (1999)
suggests that, by analysing the distributional patterns of
trees in a savanna, we might be able to identify the
underlying ecological processes driving the system.
However, it may not be so easy. We found that one
transitional model had a pattern of tree distributions
during the open savanna phase that was indistinguishable from the patterns occurring in models producing
persistent open savanna. In both cases, trees were
evenly spaced at the smallest scales and were clumped
or randomly distributed at intermediate scales. The
same distributional pattern was also found for real tree
distributions in the Kalahari Gemsbok National Park,
suggesting that the K G N P tree population is either in a
state of low-density persistence or in a transitional state
towards savanna woodland. However. the latter is
rather unlikely due to the lack of recruitment of A.
eriolobu observed in this area (Jeltsch et al. 1996). And,
although the snapshot tree pattern is not strictly diagnostic of a "stable" tree-grass savanna, it does seem to
indicate that the tree population is not running a severe
risk of extinction. The amount of clumping of trees at
intermediate scales also indicates that the savanna is
probably not in a period of serious decline because, as
the simulations show, such a period would be characterized by a random pattern without significant
clumping.
In another field study of a savanna ecosystem conducted in the Botswanan part of the southern Kalahari,
Skarpe (1991) found a random distribution of mature
A. eriolobu trees at all scales (up to 50 m) using Ripley's
(1977) K-function for spatial pattern analysis. This
snapshot pattern was interpreted as being the result of
a trade-off between competition (leading to regular
spacing) and fire sensitivity (promoting clumping)
(Skarpe 1991). Previous results of our model contradict
this hypothesis (Jeltsch et al. 1996, 1998) indicating that
additional processes. such as seed dispersal by herbivores, are necessary to produce a random pattern. Also,
in our simulation experiments, we found that a random
pattern occurs under almost all environmental conditions examined, but tends to persist for only a limited
time period. For example. random patterns occur in
periods of lower rainfall when mechanisms leading to
clumping of the trees are weak and processes promoting random distributions dominate. Thus a random
pattern may only represent a transitory phase in a more
general pattern that is dominated by processes promoting clumping or an even distribution across a range of
scales. Under these circumstances it would be a mistake
to interpret the random pattern as being indicative of a
situation where spatially dependent processes are unimportant in determining the distribution of individual
plants, i.e., interspecific competition is weak and/or
dispersal is not a limiting factor (Skarpe 1991, Haase
1995).
Although some information can be deduced from a
single snapshot of an ecological pattern, one should be
careful not to over-interpret a single snapshot in attempting to identify the underlying processes driving
the system. In fact, even an analysis conducted over a
ten-year period would not be adequate for the savanna
system, because of the slow rate of change in the
distribution of the tree population.
What can be learned about savanna dynamics?
It is an appealing advantage of comparing pattern from
the field with pattern produced by model output that
we can examine whether or not savannas represent a
stable tree:grass equilibrium (Walker et al. 1981,
Walker and Noy-Meir 1982, Belsky 1990, Skarpe 1991,
Tainton and Walker 1992) or a disequilibrium in which
coexistence of trees and grasses is achieved through
disturbance (Scholes and Walker 1993, Belsky 1994).
The temptation is strong to interpret a given pattern on
the basis of the unspoken assumption that the vegetation is in a 'stable' equilibrium. However, the dynamic
changes in pattern observed for the modelled savanna
system indicate the non-existence of such a 'stable'
equilibrium. Instead, trees normally persist at low densities because of a set of interacting processes. During
periods of higher than average rainfall, tree seedlings
establish near mature trees or tree clumps, where more
tree seeds are available, and also in more widespread
locations due to the presence of widely scattered seed
patches in the dung of herbivores. Gone unchecked, the
open savanna would rapidly change into savanna
woodland. However, the buildup of fuel in the grassy
areas of savannas during this time period leads to an
increase in fire frequency, which controls the spread of
trees and imposes a clumped distributional pattern on
the trees. During periods of low rainfall intraspecific
competition for moisture reduces the probability of tree
seedling establishment near mature trees and additional
'establishment patches'. away from mature trees (e.g.
tree seeds dispersed in herbivore dung) are necessary for
the long-term survival of the trees. This also promotes
an even distributional pattern. Thus, grass fires in one
case. and the formation of establishment patches in the
other, work as 'ecological buffer mechanisms' that allow the system to persist when it is driven close to
critical boundary conditions (compare Walker 1989,
1992, 1993, Holling 1992).
These two buffering mechanisms, fire and the formation of establishment patches, promote the formation
of different spatial patterns, namely clumping promoted
by grass fires and a random tree distribution promoted
by the random distribution of establishment patches
(for the formation of tree clumps by fire see Menaut et
al. 1990, Jeltsch et al. 1996). As a consequence, the
system shifts between these two predominant spatial
patterns over time due to the underlying, environmentally driven non-equilibrium dynamics.
How could we learn more from pattern?
We would need several decades of spatial data to
improve our knowledge significantly regarding the natural long-term dynamics of pattern formation in savanna vegetation. Unfortunately, such a data set is not
available for the southern Kalahari. An alternative
approach, based on our simulation results, would be to
analyse spatial patterns in tree distributions along a
rainfall gradient. as is found from west to east in the
southern Kalahari (Leistner 1967. Jeltsch et al. 1997a).
According to our model, the pattern should change
from more random towards more clumped with increasing rainfall. If we found this pattern, it would
provide further field evidence backing up the validity of
our model. Similar comparative studies could be done
in areas where 'bush encroachment' takes place, i.e, the
progressive increase of trees or shrubs at the expense of
palatable herbaceous vegetation (Talbot 1961, Leistner
1967, Skarpe 1990, Archer and Smeins 1991, Perkins
and Thomas 1993, Jeltsch et al. 1997a, Weltzin et al.
1997). An increase of tree numbers may, for example.
be caused by a reduction of grass biomass by domestic
herbivores which leads to a decrease in fire frequency
and a reduction of direct competition with the herbaceous layer (Perkins and Thomas 1993, Jeltsch et al.
1 9 9 7 ~ )Alternative
.
reasons for encroachment could be
climatic changes (increased moisture availability for
trees). the formation of additional seed patches in cattle
dung (O'Connor 1995) or the reduction of a tree
seedling controlling agent (Weltzin et al. 1997). An
analysis and comparison of spatial tree patterns in
various examples of increasing tree populations may
help to detect underlying processes producing the encroachment and eventually allow for improved insights
into the general dynamics of savanna vegetation.
Clearly, a great deal can be learned through the
judicious use of pattern analysis. This is particularly
true when we couple the analysis of pattern in the field
with the analysis of pattern produced by carefully
constructed, spatially explicit simulation models. It is
under these circumstances that we can gain the most
understanding about the processes responsible for the
generation of pattern in ecological systems.
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