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A simulation model for a shrub ecosystem in the semiarid Karoo, South Africa.

Key words: automata; event-driven system; individual-based simulation model; Karoo; nonequilibrium; semiarid ecosystem; spatial and temporal dynamics.


Semiarid ecosystems are believed to be sensitive to climatic change and such forms of landuse as heavy grazing, ploughing, and strip mining (UNESCO 1980, Barrow 1991). Degradation and desertification in semiarid rangeland may be rapid, but recovery is slow because plant growth is rain limited and denudation reduces rainfall effectivity (Milton et al. 1994). The dominant species of semiarid plant ecosystems often have life-spans up to decades, so that changes in vegetation composition may take longer than human life-spans. Databases for assessing vegetation trends seldom exceed a few years. Because of the mismatch between time scales for observation and vegetation change (Scholes 1990), little is known about the dynamics of semiarid ecosystems over long temporal scales. Although models could address the problem of temporal and spatial dynamics. few have been attempted for semiarid ecosystems (Walker 1993).

Rangeland ecosystem models

The early model of rangeland dynamics, which still forms the basis of much current management, is a simple application of Clementsian theory of ecological succession (Clements 1916). This model assumes continuous and reversible transitions along a single, monotonic gradient between an overgrazed subclimax and an undisturbed climax state of vegetation and assumes that grazing and interannual variation in rainfall cause vegetation to change in the same way. Over the past decades this range succession model has been criticized because of its inability to deal with vegetation changes. especially in the arid and semiarid zones (Smith 1988, Walker 1988, Westoby et al. 1989, Friedel 1991). Instead of continuous and reversible transitions semiarid ecosystems typically show abrupt. discontinuous and irreversible transitions between discrete states. Walker 1993) pointed out, that "changes in the species composition of rangelands are commonly episodic, occurring in response to rare and extreme events, or more commonly, particular sequences of events such as a very dry year followed by a very wet year . . . In between such events, production will vary from year to year in response to variation in rainfall, but the composition of the rangeland remains essentially the same, and changes little in response to management . . . The reason for this episodic behaviour is that both successful reproduction, i.e.. establishment, and mortality in plants depend on particular conditions." He concluded that event-driven systems must be event managed and that being able to recognize the significant events is a key to successful rangeland management.

To deal with the complicated dynamics of semiarid and arid ecosystems, some range scientists (Smith 1988, Westoby et al. 1989, Milton and Hoffman 1994) suggested that these ecosystems could be described in terms of discrete states and interstate transitions. Transitions could be triggered by natural events (rainfall, drought, hail, fire) or by management actions (removal of herbivores, altered intensity or timing of herbivory, addition of fertilizer, burning). Such "state-and-transition" models are valuable tools for describing the structure of the ecosystem, but they provide little information applicable to forecasting and prediction. Some event-orientated grazing strategies make use of state-and-transition models and include information on the attributes of the most important species of the ecosystem (Hodgkinson 1992) and their response to grazing. However, additional models that improve the user's understanding of ecosystem dynamics over a long temporal scale are needed to identify significant events that drive semiarid ecosystems.

The "gap dynamics approach" to modelling plant community dynamics focuses on resource space associated with individual plants and simulates establishment, growth, and death of individuals on a small isolated plot through time (Coffin and Lauenroth 1990). This approach has been used for analyzing temporal and spatial pattern in semiarid grassland and temperate and tropical forests (Coffin and Lauenroth 1990, Belsky and Canham 1994, Shugart 1984). The model presented by Coffin and Lauenroth (1990) incorporates effects of small-scale disturbances and stochastic environmental factors, but does not consider spatially explicit processes and interactions between plots.

"Dynamic automata" models, recently developed by Jeltsch and Wissel (1993a, 1994), are appropriate for systems characterized by distinct spatial pattern and spatial interactions. They extended the method of cellular automata (Wolfram 1986) by including autonomous dynamics of a local cell so as to model the temporal and spatial dynamics of forests on a large spatial scale. Fundamental to dynamic automata models is the division of space into small subunits (Wissel 1991, 1992, Wissel and Jeltsch 1993), the cells. Each cell can exist in a variety of states, chosen depending on the aim of the model and the knowledge available for defining possible states. The states change over time according to predefined rules formulated from empirical and anecdotal knowledge. In contrast to cellular automata models, where the state of any one cell in the next time step depends only on its present state and on the state of some of its neighboring cells (Molofsky 1994), dynamic automata models assume autonomous dynamics of local cells that can be influenced by the states of neighboring cells and by such external factors as rainfall, disturbances, or management actions. The size of cells and the length of the time steps are chosen in accordance with the aim of the model and the question being investigated.

In this paper we respond to Walker's (1993) call for new rangeland models and we show that dynamic automata models can be applied successfully to semiarid ecosystems.


Aim of the model

We present a simulation model, based on long-term field-investigations, for a typical semiarid ecosystem in the southern Karoo, South Africa. The aim of the model is to capture the main events and mechanisms that determine the temporal and spatial dynamics of common plant species on a large temporal scale. Detailed information about interannual variation in rainfall (which has an important impact on the dynamics of semiarid ecosystems), and on rainfall-dependent plant attributes (including seed production, germination, recruitment, and mortality factors), are included in the model. We employ the dynamic automata method to simulate the temporal and spatial dynamics of the ecosystem in annual time steps.

Site description

The 100-ha study site. Tierberg Karoo Research Centre (TKRC), lies at the southern edge of the semiarid Great Karoo, South Africa (33 [degrees] 10' S latitude, 22 [degrees] 17' E longitude, 800) m above sea level), 26 km west of Prince Albert. Ambient temperatures ranged from -3 [degrees] C (winter minimum) to 36 [degrees] C (summer maximum). The Universities of Cape Town and Natal carried out intensive field investigations and experiments at this site over 6 yr (1988-1993). Rainfall has been recorded by an automatic weather station at TKRC since 1988, and has been manually recorded at Prince Albert jail (26 km east of the study site) since 1878. Average ([+ or -] 1 SE) annual rainfall (93 complete data sets) at Prince Albert is 167 [+ or -] 7 mm with a range of 50-400 mm (Milton et al. 1992), and can occur at any season, although monthly averages peak in autumn (Fig. 1). Winter rain is associated with cyclonic fronts and is frequent though seldom heavy. Summer rain falls during thunderstorms (Cowling 1986), often in the form of brief cloudbursts, which result in rapid run-off. The even land surface of the study site is broken by three types of topographic feature: minor drainage lines (washes), drainage lines, and heuweltjies. Neither type of drainage line carries water for more than a few hours during rain storms. Heuweltjies are low mounds of [proximate] 15 m in diameter, formed by termites (Microhodotermes viator) and maintained by the activities of various other animals (Lovegrove and Siegfried 1989). Heuweltjies are evenly scattered over the plains (Milton et al. 1992).


Vegetation, comprising perennial succulent and nonsucculent shrubs 0.2-0.4 m in height, covers 15-20% of the soil surface (Milton et al. 1992). Grasses and annuals are restricted to drainage lines and appear to play little part in the dynamics of this vegetation (Milton 1994) where succulent and nonsucculent shrubs replace one another cyclically (Yeaton and Esler 1990). The most extensive habitats are the run-off sites (plains), which cover 78% of the TKRC study site, and where plant species diversity is low (8-18 species per 25 [m.sup.2] quadrat) (Milton et al. 1992). Plant cover in plains shrubland averages 18% (range 11-13%) at the study site and is fairly evenly distributed between succulent and nonsucculent shrubs. The plains shrubland is characterized by isolated shrubs or by small, mixed-species clumps of shrubs interspersed with bare ground. Shrub density ranges from three to seven plants per square metre. Mean shrub crown diameter ranges between 200 and 500 mm, mean height from 150 to 600 mm, depending on the species. The structure and the dynamics of the vegetation of the plains are isolated from the other habitats, which have more plant-available water or nutrients (Milton et al. 1992). The vegetation at TKRC was moderately grazed by sheep from 1850 to 1987, after which time it was fenced to exclude domestic livestock. All biological data included in the model were collected in currently ungrazed vegetation.

Vegetation biological states, and transitons

Vegetation on the plains is dominated by five succulent and nonsucculent shrub species. Other plants (34 species) are too infrequent to play an Important role in the dynamics of the vegetation or show little interaction with the dominant species. In the model we consider infrequent species as a "fixed environment" of "fixed plants" that remain unchanged during the simulation and do not interact with plants of the five dominant species except that they occupy space. The five dominant species consist of three colonizer species: (1) Brownanthus ciliatus (Mesembryanthemoideae), an ascending. multistemmed, nonwoody, evergreen stemsucculent and leaf-deciduous species, (2) Ruschia.spinosa Ruschioideae), a multistemmed, spinescent shrub, and (3) Galenia fruticosa (Aizoaceae) a nonsucculent shrub. and two shrubs that replace the colonizers over time: (4) Pteronia pallens (Asteraceae), a narrow-leaved, evergreen microphyll shrub and (5) Osteospermum sinuatum (Asteraceae), a dwarf, droughtdecidous shrub. We can model individual plants, using an individual-based dynamic automata model (IBDA-model) because we have good data for the attributes of the five dominant species, and because the individuals are scattered. To do this, we divide the space into a grid of cells that represents mature plant sites.

The autonomous dynamics within a cell are given by the sequence "empty" [right arrow] "colonizer plant" "successor plant" [right arrow] "empty." A colonizer plant can be B. ciliatus, G. fruticosa, or R. spinosa, and a successor plant can be either P. pallens or O. sinuatum. Under certain circumstances this sequence can be altered to "empty" [right arrow] "colonizer plant" [right arrow] "empty." There are therefore nine possible sequences or pathways that can occur in a cell. For a given cell, the pathway followed and the duration (in time steps) of each state. is determined (1) by the variables that characterize the state of a cell (Table 1), and (2) the rule set, which determines how these variables change in the course of time with dependence on the states of neighboring cells and on external factors like rainfall, disturbances. or management actions.

Table 1. The variables that characterize the state of a cell and their range during the simulation (e) = "empty," (b) B. ciliatus," (g) = "G. fruticosa," (r) = "R. spinosa," (p) = "P. pallens," (o) = "O. sinuatum," (f) ="fixed plant."
Variable                            States

Plant species                       (e), (b), (g), (r), (p), (o),
Canopy surface area                 0-64 size units
Plant age                           0-70 yr
Number of seeds per plant           0-3000 seeds
and year

The rule set

The rules that follow determine in detail how these variables change in the course of time. They are based on published and unpublished information on the population dynamics of these five shrubs and on the spatial structure of ungrazed, plains vegetation.

Rule 1: Seed production.--Seed production of adult plants as well as germination of seeds and survival of seedlings depend on timing and amount of rainfall (Esler 1993, Milton 1995). Certain thresholds of rainfall (which differ from species to species) are required to stimulate one of these processes (Table 2). Plants of G. fruticosa, R. spinosa, B. ciliatus, and P. pallens produce seeds if rainfall exceeds a certain threshold during the growing season April to September, and the number of seeds produced depends on the total rainfall received during the growing season of the previous year. O. sinuatum differs from the other species because it flowers and seeds after any rain event exceeding 20 mm with the exception of the hottest months (December-February) summer (Milton 1992). For O. sinuatum flowers (seeds) are produced after rain that exceeds a minimum threshold during the 12 wk before anthesis. and the number of seeds increases with rain until saturation behavior occurs. Seed production of established plants that have not yet reached their maximum size is proportional to their canopy surface area. As an example, Fig. 2 shows the relation between rainfall in the 12 wk preceding anthesis and S, the mean number of seeds per adult O. sinuatum plant at TKRC, and the regression used in the model. We use the 1989 TKRC rainfall data (Milton et al. 1992) and the 1989 TKRC seed production data (Milton and Dean 1990) to calibrate the absolute numbers of seeds produced. During this year all five species produced seeds.


To calculate the seed production of individual plants of C. fruticosa, R. spinosa, B. ciliatus, and P. pallens at year t we determine the number of seeds S(t) that an adult plant (with size class 64) of each species produces in accordance with the rainfall r during the growing season of the previous year t - 1 using relations similar to that shown at Fig. 2. Then we scale the seed production linearly in accordance with the size of the individual plant. For O. sinuatum, we determine for each flowering event at year r the number of seeds S(t) in relation to rainfall in the 12 wk preceding anthesis (Fig. 2). Then we add up seed production for the whole year and scale for individual plants in accordance to their size.
Table 2. The standard parameter set for the simulation model.

Parameter                                B. ciliatus   G. fruticosa
Life-span (yr)                             10              30
Age of establishment                        1               2
Max. seed disp. distance (m)                2.5             2.5
Seed production 1989/plant               1309            2000
Minimum size of safe sites ([m.sup.2])      3.25            1
Seed viability (%)                         50               4
Seed loss (%)                              10              10

Rain thresholds(*)
  for seed production (Apr-Sep) (mm)       20              50
  for germination (Mar-Jun) (mm)           20              30
  for seedling survival (Jul-Oct) (mm)     10              15

Parameter                               O. sinuatum    P. pallens
Life-span (yr)                             50              70
Age of establishment                        1-9             1-9
Max. seed disp. distance (m)               50              30
Seed production 1989/plant                542             200
Minimum size of safe sites ([m.sup.2])    ...             ...
Seed viability (%)                          5              30
Seed loss (%)                              77              80

Rain thresholds(*)
  for seed production (Apr-Sep) (mm)       20              20
  for germination (Mar-Jun) (mm)           30              30
  for seedling survival (Jul-Oct) (mm)     20              20

Parameter                                R. spinosa
Life-span (yr)                              25
Age of establishment                         3
Max. seed disp. distance (m)                 2.5
Seed production 1989/plant                 174
Minimum size of safe sites ([m.sup.2])       1.25
Seed viability (%)                          90
Seed loss (%)                               10

Rain thresholds(*)
  for seed production (Apr-Sep) (mm)        70
  for germination (Mar-Jun) (mm)            30
  for seedling survival (Jul-Oct) (mm)      10

(*) Total rain during seeds (stipulated) critical for seed production, germination, and seedling survival.

Rule 2. Germination.--All species germinate during the germination season March to June after big rain events (Table 2), but only 50% of the viable seeds of the species G. fruticosa germinate after big rain events, and the other half of the seeds remain in a seedbank. All viable seeds of the species B. ciliatus, R. spinosa, P. pallens, and O. sinuatum germinate after big rain events. The model considers postdispersal seed loss and nonviability at time step t:

(1) G(t) = S(t) (1-l)vg(t)

where 1 and v are, respectively, the fractions of seeds lost and the fraction of viable seeds, S(t) are the number of seeds produced by an adult plant, and G(t) are the number of seed germinating. The germination function g(t) is defined as:

(2) g(t) = 0 if rain [is less than] threshold, all species

0.5 if rain [is greater than] threshold, G. fruticosa

1 if rain [is greater than] threshold, all species

except G. fruticosa.

Rule 3: Seedling survival.--To survive the critical postgermination period (July to October) young seedlings require a total rainfall that exceeds a species-specific threshold and which is fairly evenly distributed. In extremely good years, all seedlings at safe sites (see Rule 5) establish, whereas in normal years only 10% of seedlings survive. We define three rain classes for rainfall between July and October: (1) good rain: 2 mo with [is greater than] 5 mm rainfall and a total rainfall from July to October that exceeds 70 mm. (2) normal rain: 2 mo with [is greater than] 5 mm rainfall and a total rainfall from July to October that exceeds a species-specific threshold (Table 2). (3) bad year: neither good rain nor normal rain. The model calculates L(t), the number of seedlings that can survive the postgermination period as

(3) L(t) = G(t)s(t),

where the survival function s(t) is defined as

(4) s(t) 1 good rain

= 0.1 normal rain

0 bad rain.

Rule 4: Seed distribution.--The small seeds of the colonizer species B. ciliatus, R. spinosa, and G. fruticosa are dispersed by water and trapped by soil particles. Splash dispersal by raindrops expels seeds up to 0.8 m from capsules of Karoo Mesembryanthemaceae (Volk 1954). We assumed that movement in run-off water during rain events could increase dispersal distance to a maximum of 2.5 m for these small-seeded species. Based on Bond's (1988) findings that large (10-20 mm) tumbleseeds moved 10-40 m on unvegetated, stony ground, and on unquantified field observations at TKRC (Esler 1993, Milton 1995), we ruled that tumbleseeds of P. pallens and O. sinuatum moved maximum distances of 30 and 50 m, respectively (Table 2). Eventually they are trapped by pits, mat-like succulents, and shrub clumps. Fig. 3 shows the dispersal probabilities for water-dispersed seeds and for O. sinuatum tumbleseeds. An additional number of small seeds can emerge further from parent plants because some seeds are transported in dung of sheep and wild mammals or dispersed by ants. Roughly 25% of the seed production on plains at TKRC appeared to be collected by harvester ants (Messor capensis), which stored seeds weighing 0.3-28.5 mg (fresh mass) in their nest-mounds (Milton and Dean 1993). The values for postdispersal seed loss and the fraction of viable seeds are shown in Table 2 Ant nests are disturbed by aardvarks (Orycteropus afer) and foxes (Otocyon megalotis), which excavate part of the nest-mound, scattering seeds and organic matter (Dean and Yeaton 1992). This provides a pool of seeds that may germinate under suitable conditions (Rule 2). Only one species, G. fruticosa, has a considerable, long-lived seedbank: [approximate] 50% of the seeds appeared to be viable after 2 yr of field exposure (Esler 1993). The species R. spinosa has a short-lived seedbank: [approximate] 11.5% of ungerminated seeds are still viable the next year.


In the model, we distribute single seeds of each plant. We determine the direction of the seed movement randomly and choose the dispersal distance (within limits of the maximum dispersal distance) in accordance with a weighted random distribution based on field experience (Fig. 3). Seeds of colonizer species are only trapped if they are dispersed to open cells, otherwise they become deleted. Tumbleseeds of successor plants are trapped by established colonizer plants. To simulate the tumbling process we consider also the eight nearest neighbors of the cell. If one of the neighboring cells is occupied by a plant of the mound-building species B. ciliatus or R. spinosa, then the tumbleseed is always trapped. Plants of the nonmound-building species G. fruticosa trap a tumbleseed with a probability of only 60% We consider seeds that emerge in pits, in disturbed ants nests, or that are dispersed by animals only in an approximate way. To do this we disperse a certain number of seeds randomly over the cells each year in accordance with available data and field experience.

Rule 5: Safe sites.--The suitability of a cell (plant site) as a safe site depends on particular conditions. The colonizer species B. ciliatus, R. spinosa, and C. fruticosa establish in safe sites on bare ground in areas not shaded by plants (Yeaton and Esler 1990, Esler 1993). To avoid competition for water from neighboring plants, seedlings of these three species require gaps of minimum sizes (Table 2). The shaded sites within or under the edge of the canopy of established plants of the species B. ciliatus, R. spinosa, and G. fruticosa provide safe sites for seedlings of P. pallens and O. sinuatum (Yeaton and Esler 1990, Esler 1993). Seedlings of G. fruticosa cannot establish in the neighborhood of established P. pallens plants. Seedlings of P. pallens cannot establish in the neighborhood of more than four adult P. pallens plants. We define safe sites for the different species as: B. ciliatus: cell empty, gap has at least minimal size; R. spinosa: cell empty, gap has at least minimal size; G. fruticosa: cell empty, gap has at least minimal size, no P. pallens plant is nearest neighbor; P. pallens: cell with colonizer plant, not more than four nearest neighbors are P. pallens plants; O. sinuatum: cell with colonizer plant.

Rule 6: Competition.--Shortly after germination many seedlings of one or different species may co-occur within a cell, but as seedlings grow larger, competition occurs among neighboring seedlings. Only one seedling can survive within one cell. If seedlings of different species compete within one cell survival depends on the growth rates of the seedlings. Seedlings of B. ciliatus grow fast and always outcompete seedlings of the other two pioneer species, while the slow-growing seedlings of R. spinosa are outcompeted by other pioneers. The fast-growing seedlings of O. sinuatum always outcompete seedlings of P. pallens. In this way, the rule implicitly takes self-thinning into account.

Rule 7: Establishment.--We define a plant to be established if it is able to reproduce (Table 2). Establishment times (from germination to first reproduction) are assumed to be 6 mo for B. ciliatus, 1.5 yr for G. fruticosa and 3 yr for the slow-growing seedlings of R. spinosa. Seedlings of P. pallens and O. sinuatum are able to survive without growing for up to 9 yr. When their host plant becomes senescent, they establish and eventually replace it.

Rule 8: Growth.--We measure the canopy surface area C of a given plant in size classes between 0 and 64, where size class 64 corresponds with the the maximal size of an adult plant and size class I with not established plants. Canopy growth is modelled according to known plant responses to age and rainfall. We calculate the size C of the canopy surface area of an established plant at time step t + 1 as

(5) C(t + 1) = C(t) + A(a)R(r),

where the functions A and R are, respectively, the response of growth to a given age a and to the rainfall r during the growing season April to September. For example, Fig. 4 shows the functions A and R for the species G. fruticosa. Old plants of the species B. ciliatus and R. spinosa show negative growth because parts of the plant mat that are affected by parasites die and disintegrate (S. J. Milton, personal observation). Rates of growth were found to have little influence on the dynamics of ungrazed vegetation, because ungrazed plants reach their maximum size within a decade. However, growth rates have a large influence on a grazed plant assemblage.


Rule 9: Mortality.--One characteristic of the plant community is that considerable mortality occurs only during the seedling stage (see Rules 5 and 6) and when plants have reached their maximal age. Occasionally large proportions of the B. ciliatus, R. spinosa, and P. pallens population die after such catastrophic events as hail (Milton and Collins 1989, Powrie 1993) or extreme drought (S. J. Milton, personal observation). Catastrophic events are not yet included into the model. In our model a plant dies deterministically after reaching its maximum age (Table 2).

The simulation

In this section we demonstrate how we transform the rule set into an individual-based dynamic automata (IBDA) simulation model. Before starting a simulation we have to fix: (1) a parameter set, (2) an initial plant distribution, and (3) a weather scenario that delivers monthly rainfall data for each time step (year). We divided the space into a 77 x 53 grid where each cell (0.5 x 0.5 m) can contain one established plant and one or more nonestablished seedlings. We simulate the spatial and temporal dynamics of our (1020.25 [m.sup.2]) system in time steps of 1 yr. Because important processes like seed production, germination, and seedling survival are dependent on the timing of rainfall, and because we have monthly rainfall data, the model works internally with time steps of 1 mo, although the output shows only annual time steps. The structure and sequence of an actual simulation follow the phenology of the five study species. At the beginning of each time step (year) t, we employ the submodel "SEED" to calculate for each species: (1) the number of seeds S(t) that are produced by an adult plant (Rule 1), (2) the number of seeds G(t) that germinate between March and July (Rule 2), and (3) the number of seedlings L(t) that can survive the postgermination period (Rule 3), all of which variables depend on the timing and the amount of rainfall. Using this information we can simulate for each plant the distribution of individual seeds (see Rule 4). To proceed efficiently we distribute only seeds that have a chance to survive. For plant species without seedbanks we distribute only the number

(6) [S.sub.distr] = L(t)C(t)/64

(7) = S(t)(1 - 1)vg(t)s(t) C(t)/64

of seeds that can survive the postgermination period (if they are trapped at unoccupied safe sites) considering the actual size C(t) of the plant, postdispersal seed loss 1, viability v, germination g(t), and seedling survival s(t). For species with a seedbank (R. spinosa and G. fruticosa) we distribute the number

(8) [S.sub.distr](t) = S(t)(1 - 1)vC(t)/64

of seeds, considering at the moment only the size C(t) of the plant, postdispersal seed loss 1, and viability v. After adding for each cell the seeds newly distributed to the seeds already in the seedbank we apply the germination function g(t) and the survival function s(t) in order to determine for each cell the number of seedlings that can survive the postgermination period (if they are trapped at unoccupied safe sites). Next we decide, for each cell, whether these seedlings are within a safe site or not. To do this we use information about the spatial arrangement of the established plants and employ Rule 5. In the case where several seedlings are trapped in a safe site we determine the surviving seedling in a cell in accordance with Rule 6. Surviving seed(ling)s eventually establish (Rule 7) and all the other seeds (with exception of 70% of the seeds of G. fruticosa and 11.5% of the seeds R. spinosa, which have a seedbank) are deleted. In the last step we simulate growth and death of established plants in accordance with Rules 8 and 9. The cycle for 1 yr is now complete, and we can proceed with simulating the cycle for the next year.


A simulation

The spatial and temporal dynamics of the shrub community were simulated for 1000 yr. Parameters were chosen in accordance with the reference parameter set (Table 2). The simulation started with a plant distribution in which single plants of the successor species P. pallens and O. sinuatum were distributed randomly over the grid and where single plants of the colonizer species B. ciliatus, R. spinosa, and G. fruticosa were distributed randomly over the remaining safe sites. The plants were aged at random. We consider uncommon species to be a "fixed environment" (see above, The model: Vegetation, biological states, and transitions). To create the "fixed environment," we distributed an additional number of "fixed plants" over the grid. The starting number of plants of each species (including the fixed plants) was chosen in accordance with absolute abundance of that species at the study site. Because seed production, germination, and seedling survival depend on timing and amount of rainfall, we need a realistic, long-term rainfall scenario that we can use as input for the submodel "SEED." The 93 complete monthly data sets of the rainfall records kept since 1878 at Prince Albert were used as a rainfall scenario for the simulation. To provide rainfall data for longer periods we repeat this scenario.

Fig. 5 shows the outcome of a simulation run for the first 240 time steps. All five species were able to coexist for the simulated time period of 1000 yr. Instead of showing an equilibrium state, the dynamics of the simulated system are characterized by long quasistable periods where the abundances of the species change very slowly. These quasistable periods are interrupted by sudden, discontinuous changes in the species composition. In such cases, many plants of one species establish in the same year or many plants of one species die simultaneously. Furthermore, without any such extrinsic pressures as disturbances or grazing, we observe a nearly continuous decrease in plant density between the simulation-years 22 and 80. This indicates that the dynamics of the system are complex, unlike equilibrium behavior. Because plant species compete for safe sites in our model, our finding that the species can coexist for long periods is not trivial. Fig. 6 shows the spatial distribution of the plants at different time steps (t = 1, 85) during the simulation. The initial distribution (Fig. 6, top) (t = 1) is an essentially random "salt and pepper" distribution. Fig. 6 (bottom) shows the spatial distribution of the plants at t = 85, after a sequence of 58 yr (time step t = 22 to t = 80) where overall plant density decreased continuously followed by 3 yr with recruitment events of all five species (Fig. 5). The distribution of the plants now is more clumped.


To test how sensitively the model predictions at any time depend on the initial plant distribution we have varied the initial number of the plants as well as their age structure. We find that changes, as long as the relative abundances of the species do not differ more than say 10% from the relative abundances observed at the study site, do not influence the dynamics of the plant community qualitatively, because the "recruitment pattern" (Fig. 7) generated from the rainfall data is of such dominant importance that slight differences in the initial number of the plants will be smoothed away within at least 80 yr (one generation). But clearly, if the relative species abundances differ too much from the relative species abundances at the study site (e.g., if we use the relative species abundances observed at a heavily grazed adjoining farm [Milton and Dean 1990] where overall plant density is lower and the unpalatable species P. pallens and B. ciliatus are dominating), the resulting dynamic behavior is different in a way that the system needs much longer to "regenerate. "


Densities generated for the five species by the simulation during 240 time steps were generally lower than densities of these species measured in ungrazed vegetation at the TKRC study site (Table 3). This is because the plant densities plotted by the model exclude seedlings. The simulation produced densities of G. fruticosa that were 2-3 times greater than densities of other species. Experimental removal of established plants from areas of 25 [m.sup.2] in ungrazed vegetation at TKRC (Milton 1995) resulted in establishment of C. fruticosa at an average density of 14 Seedlings/[m.sup.2] (14700 seedlings per 1020.25 [m.sup.2] simulation grid). We therefore maintain that high densities of C. fruticosa, relative to other species, could realistically be expected to occur following gap formation, resulting from drought- or age-induced mortality of these species.

Table 3. Densities of the five study plant species and "fixed" species (see The model: Vegetation, biological states, and transitions) in ungrazed vegetation sampled in 18 plots (each 25 [m.sup.2]) at the Tierberg Karoo Research Centre in 1989, densities of these species linearly scaled to the 1020.25 [m.sup.2] surface used in the simulation, the absolute extremes for densities of these species in the 18 plots scaled to the 1020.25 [m.sup.2] surface used in the simulation, and the range of these species during the simulation.
                 25 [m.sup.2]    1020.25
                 plot            [m.sup.2]    1020.25
                 (aver-          (aver-       [m.sup.2]   Simulation
                  age)            age)        (range)     range

B. ciliatus        3.3            136           O-489        0-100
G. fruticosa      15.0            613         163-1102     172-1300
O. sinuatum       16.4  672        41-1795     50-450
P. pallens         6.5  265         0-653     130-440
R. spinosa        14.8  606        41-2040     23-350
Fixed species     33.3           1362         205-2611     345

Analyzing the simulation

We now analyze the reasons for the sudden and discontinuous dynamics in more detail. It is clear that, in our model, most of the conspecifics that establish in the same year will die simultaneously because we have given each species a fixed age. This is an idealization: if variable life-spans are taken into account, these effects will be smoothed. Clearly, if recruitment was poor for a species in years that followed a "big" recruitment event, then the population of this species will break down when the maximum age of these plants is reached. Things become more complicated if we ask why such big recruitment events occur.

Fig. 7 shows the time series of the three colonizer species together with L(t), the number of seedlings that can survive the postgermination period. We see that two conditions have to be fulfilled simultaneously to facilitate a big recruitment event: (1) timing and amount of rainfall over the year has to fulfill all conditions that guarantee that seeds are produced, are able to germinate, and that seedlings can survive, and (2) vegetation density has to be low so that a high number of unoccupied safe sites is available. A combination of these conditions is a coincidence of independent events, because single rainfall events are uncorrelated and unpredictable in the southern Karoo (see also Fig. 5). In accordance with this rainfall sequence, big recruitment events always have the following temporal pattern. During prolonged, unfavorable rainfall conditions, plant density decreases continuously and the number of gaps within the vegetation (safe sites for colonizer plants) increases. When favorable rainfall conditions occur, many young plants of one or more colonizer species are able to establish in the gaps. If these plants senesce simultaneously, they offer numerous safe sites for the successor plants. Many successor plants can then establish if rainfall conditions are favorable for them. We see that a combination and particular sequence of different, independent events such as a long period with unfavorable rainfall followed by a year with favorable rainfall and lag effects ("memory") are the events that drive the spatial and temporal dynamics of this plant community.

Recruitment: rainfall conditions

The rules and the parameter set of the submodel "SEED," which determine seed production, germination, and seedling survival, were drawn up and calibrated in accordance with the monthly rainfall data from TKRC weather station and phenological data collected from 1988 to 1991. Using complete sets of monthly rainfall data from Prince Albert for 93 yr we investigate how the probabilities for seed production, germination, and seedling survival for the five species depend on the different parameters included in "SEED." As previously explained, seed production, germination, and seedling survival can be expected to be uncorrelated.

To illustrate this point we show the outcome of the submodel SEED for B. ciliatus. Fig. 8 shows the number [S.sub.distr] = S(t)(1 - l)v of viable seed produced per plant, which are not lost to ants, the number G(t) of seeds able to germinate, and the number L(t) of seedlings that can survive the postgermination period, calculated for the 93 yr of the rainfall scenario from Prince Albert. The output of the submodel SEED shows (1) that B. ciliatus produces seeds in 98% of all years of the the rainfall scenario, that (2) rainfall is sufficient for germination of B. ciliatus seeds in 78% of all years, while (3) rainfall is sufficient for seedling survival only in 54% of all years. Putting together all three processes we see that recruitment of B. ciliatus can occur in only 44% of all years (Fig. 8). Furthermore SEED shows that rainfall is sufficient for recruitment of G. fruticosa in 33% of years. for R. spinosa in 22% of years, and for P. pallens and O. sinuatum in 28% of all years. The major components of the submodel "SEED" are the different thresholds for seed production, germination, and seedling survival. Thresholds for seed production, germination and seedling survival differ between species (Table 2), although the rules (Rules 1, 2, and 3) are identical for all five species, (an exception being the opportunistic seeding of O. sinuatum). The values for these thresholds can be fixed in accordance with the rainfall data and the phenological data from the TKRC. Nevertheless, it is important to check how sensitively the submodel "SEED" (Rules 1, 2, and 3), and thus the whole simulation model, responds if we vary the parameters of our model. The percentage of years without seed production, germination, and seedling survival (calculated with SEED) depends on the species' rainfall thresholds (Fig. 9). Restrictions are strongest in case of seedling survival. To survive the critical postgermination period (July to September) young seedlings require a certain total rainfall (the threshold). Additional rainfall has to be distributed so that 2 mo of the postgermination period have [greater than] 5 mm rainfall. The latter condition was only fulfilled in 41 of the 93 yr. Therefore seedling survival fails in 44% of all years, even if the threshold is low (Fig. 9, curve 1). While thresholds for germination and seedling survival differ only slightly for the five species (Table 2), differences are much higher for seed production, which reflects differences in survival strategies among the species.


For plants in semiarid climates, seed production represents a large investment in energy and water. Given the low percentage of years where seedlings can establish, seed production wastes energy when seeds do not lead to established seedlings. On the other hand, a plant has a high probability of missing favorable opportunities for reproduction if it does not produce seeds every year. Plants therefore have to minimize physiological costs but optimize reproductive success. The species B. ciliatus and P. pallens have risk-spreading strategies and produce seeds nearly every year, while G. fruticosa, R. spinosa, and O. sinuatum produce seeds only in years when rainfall is abundant during the previous growing season. We suggest that rainfall thresholds for seed production represent species-specific trade-offs between energy conservation and reproduction.

Recruitment: spatial conditions

Apart from rainfall conditions, spatial relationships are important for seedling survival. We can understand spatial relationships using simple geometric arguments. A colonizer plant requires a gap of a certain size as a safe site within the vegetation. Fig. 10a shows the number of safe sites K(p), which varies with the density of adult plants p. There are sufficient safe sites for B. ciliatus only at low plant densities (p [is less than 2000 plants), while for G. fruticosa and R. spinosa safe sites are still available at higher plant densities. The spreading ability of a colonizer plant does not depend only on the density K(p) of available safe sites, but also on its seed dispersal mechanism. If its seed dispersal radius is small, a single cell can trap numerous seeds of one plant. Thus. the number of cells O(L) that are occupied by seeds of a single plant shows a saturation behavior as seed number increases (Fig. 10b).


Contradiction between episodic behavior and cyclic succession?

We have shown that rainfall data from the Prince Albert station generates episodic behavior in the local plant community. To demonstrate the sensitivity of event-driven behavior to rainfall scenarios, we use two rainfall scenarios. The first sequence is the one for Prince Albert data (Fig. 1), and the second. the hypothetical sequence (Fig. 1) has a higher annual rainfall (5-10 mm more rain per month) and a bimodal rainfall distribution with similar peaks during the germination and recruitment periods. The Prince Albert rainfall scenario is used during the first 100 yr of the simulation (thus the first 100 yr of the simulations in Figs. 5 and 11 are identical) and the second rainfall scenario is used for the next years (Fig. 11). When the second rainfall scenario begins, behavior immediately switches from that of an event-driven system to that of cyclic succession. During the first scenario, recruitment success of the species varies greatly between years, whereas with the second scenario, recruitment occurs in 66% of all years. The second scenario therefore results in rapid gap filling by the colonizers (R. spinosa and G. fruticosa), their replacement by successors (O. sinuatum and P. pallens) and efficient filling of gaps created by their subsequent mortality. P. pallens occurs only at a low density because our rules stipulate that faster growing O. sinuatum seedlings will win all competitions in which seedlings of these species are distributed to the same cells. B. ciliatus, a colonizer of large gaps, is rare because long-lived shrubs that colonize small gaps remain at high densities in the absence of natural disturbances. Disturbances that cause large gaps have not been included in this version of the model. Both parts of the simulation are rain driven through the effects of weather on the submodel "SEED." However, in the first part of the simulation, availability of safe sites and rare occurrence of rainfall suitable for recruitment lead to episodic behavior. In the second part of the simulation, recruitment occurs in most years but community composition is driven by plant life-spans, competitive interactions, and the availability of space, giving rise to regular cyclic changes in composition.


Further development of this model will require consideration of the effects of major disturbance events (hail, drought, and grazing) on plant communities developing under various rainfall scenarios. At this stage we suggest that the occurrence of episodic or cyclic behavior in plant communities is largely dependent on the rainfall input, thus episodic and cyclic behavior are not necessarily contradictory, but can be exhibited by the same system.


What we have learned

Bookkeeping.-- The technique of individual-based modelling requires a detailed bookkeeping of all processes that are relevant for the temporal and spatial dynamics of the plant community. By compiling the rule set we were forced to organize the knowledge already available, and we had to give this knowledge a hierachical structure. The main guidelines for structuring available knowledge were the aim of the model and the level of detail available. As a first result, we are able to present the current knowledge about the plant community in a clear and reasonable way.

Secondly, the transformation of the rule set into a qualitative simulation model and the comparison of the model output with the field situation was a strong test of the self consistency of the rules (and knowledge). At this stage we used the model to detect gaps in our knowledge. For example, in Rule 5 we determine a minimal gap size necessary for the survival of seedlings of different colonizer plants. Then we calculated the number of suitable gaps available at different plant densities (Fig. 10a) and we found that the number of gaps suitable for the species B. ciliatus was low for plant densities typically produced by our simulations. Also, B. ciliatus became extinct quickly during our first simulations. Going back to the study site we found that densities of B. ciliatus are indeed low in undisturbed vegetation, but that B. ciliatus plants occur more frequently in disturbed areas (for example after small-scale disturbances of digging animals or at an abandoned parking lot near the gate of the study site). The conclusion we drew from these observations was that an additional factor, the small-scale disturbances (see end of Rule 4), had to be considered. After adding this factor we found that B. ciliatus is present in most of the simulation years (Fig. 5) only at a low density, but that this species plays a very important role in years where plant density is low (e.g., years 168 to 191 of Fig. 5). Its ability to colonize very quickly guarantees that large gaps in open vegetation are closed immediately and averts rapid soil degradation. Thus, besides detecting a missing process in the rule set we now understand the specific function that this process fulfills within the structure of the plant community.

The logical framework of the model gives each process a certain importance that is not always obvious without the model. For example, the results of the submodel "SEED" presented above in Results: Recruitment: rainfall conditions are basically a new arrangement of knowledge already available and simply accessible without using our model. But within the logical structure of the model the submodel "SEED" becomes central to understanding the effect and the importance of the rainfall events. On the other hand, the model gives new insight by linking different aspects that before seemed to be unconnected. For example, it was not obvious that the interplay between temporal aspects and spatial effects defines the driving events of this plant community.

Range succession and state-and-transition.--Applied ecology disciplines such as range management necessarily are based on a model that shows how their ecosystem functions (Westoby et al., 1989). This model is a philosophic system of concepts, generalizations, or assumptions rather than a qualitative model that guides management strategies or what data are collected. With our quantitative simulation model we were able to test the two main concepts existing for rangeland ecology, the "range succession model" and the "state-and-transition" model. The results of our model do not support the range succession concept and we suggest that it should be replaced with the state-and-transition concept. Our model is an good example that demonstrates the viability of this concept. The plains rangeland plant community at the TKRC showed essential properties described by the state-and-transition concept. (1) Event-driven change (Walker 1993): Changes in species composition are episodic, occurring in response to rare or extreme events. (2) Time scale of changes (Wilson and Hodgkinson 1991): Because of the episodic nature of deaths and recruitment population changes may take three or more recruitment events and this may take 10, 20, or more years. (3) Demographic inertia (Westoby et al. 1989) and lag effects: A plant population requires a rare event for establishment to occur, but once this has occurred, the resulting cohort persists for a long time (Williams and Roe 1975, Crisp 1978, Griffin and Friedel 1985, Austin and Williams 1988), and the plants of the cohort die simultaneously. Long periods with unfavorable rainfall lead to increasing numbers of safe sites so that big recruitment events can take place when rainfall conditions are favorable. (4) Unpredictability of vegetation change (Harrington et al. 1984): The sequence of rainfall events and the magnitude of their effects on plant populations cannot be predicted. (5) Variation in fodder production (Walker 1993): The abundance of palatable species (in our model O. sinuatum, R. spinosa, and G. fruticosa) varies greatly and unpredictably between years.

Our model uncovers a problem of the state-and-transition concept concerning the definition of events, possible states, and transitions. Obviously the dynamics of the plant community are event driven, but do such events drive the system toward a different state? It seems that the term "event" needs to be stated more precisely. The main events included in our model (recruitment events due to a rare rainfall pattern) are integral parts of the community dynamics and do (as long as their frequency does not change) allow the persistence of the system. Therefore the state of the plant community consists of the episodic and fluctuating behavior where the abundances of the species change in an episodic and unpredictable manner. In the following we will call this type of event an "integral" event because such events have an outstanding importance by constituting the dynamics of the plant community of which they are an integral part. Although "integral" events change the species composition of the plant community temporally (for years) dramatically they do not change the dynamic state of the system on a larger time scale. In contrast, for example, catastrophic events (that also change the species composition of the plant community dramatically) may drive the system to a different state because they bring about irreversible changes in species composition over a large time scale of, say, 100 or 200 yr. To distinguish "integral" events from this latter type of events we propose the definition of the term "transition trigger." We define a "transition trigger" to trigger a transition towards a different (dynamic) state of the system. Our definition includes events like drought. hail, or fire or continuous changes of environmental conditions due to management actions like altered intensity or timing of herbivory or a changed rainfall regime. This definition is consistent with definitions given in the state-and-transition concept (see Westoby et al. 1989 or Walker 1993).

We have shown that in our model a changed rainfall regime is such a "transition trigger" that drives community dynamics from the event-driven state to a state consisting of a regular cyclic succession. This is because the frequency of "integral" events had changed qualitatively. The state-and-transition concept covers a wider range of "transition trigger" than the version of the model presented in this paper. We have not yet included grazing or large-scale disturbances like hail or drought. Field observations (Milton and Dean 1990) and our own experience with the model suggest that, for example, heavy grazing will drive the system toward a degraded state where plant density is low and the pioneer species B. ciliatus becomes important.

Persistence of the plant community.--A characteristic feature of arid or semiarid rangelands is their changeability (Harrington et al. 1984). Because of their lack of "persistence" they are often called "fragile" or "unstable." However, the results of our simulations support the viewpoint of Harrington et al. (1984) that these ecosystems are "resilient" to the widely fluctuating stresses that they experience (Holling 1973, 1980). They persist within their state as long as responses to events do not involve a fundamental alteration to the way the system functions, i.e. recruitment in most years (see above, Results: Contradiction between episodic behavior and cyclic succession). We found that the episodic, event-driven behavior is an optimal adaption of the plant community to the fluctuating environment. The main processes responsible for the "persistence" of the plant community follow. (1) Species respond to the fluctuating environment (rainfall) in different ways. This process has long been known to support coexistence in ecological systems (Westoby 1980). (2) Life-spans of the component species differ from one another. This is a desynchronizing factor that works against the synchronizing power of the events. Guilds of species that recruit following the same event die at different times so that new establishment sites are available at all times. (3) Spatial aggregation: the small seed dispersal radius of colonizer plants leads to a patchy spatial distribution of colonizer plants. The single patches of this mosaic oscillate in a desynchronous way because different pathways of the autonomous dynamics are realized. We found that the right balance between these factors, especially between events (synchronizing factors) and desynchronizing factors, guarantees the diversity of the plant community. In cases where the balance is disturbed the system becomes impoverished. This is shown by the simulation with the second rainfall scenario (Fig. 11). In this case condition (1) is not fulfilled and recruitment is possible for all species in most of the years. Consequently, the species B. ciliatus and P. pallens nearly disappear, and R. spinosa can only survive because of their spatial aggregation.

Comparison with other rangelands

Many of the biological processes incorporated in our model are known to he important in structuring or driving plant communities in other arid and semiarid regions. These include intraspecific competition among long-lived dominants (Yeaton and Cody 1976), facilitation of species that require "nurse" plants by species that establish on bare ground (Connell and Slatyer 1977, McAuliffe 1988), interspecific in seed dispersal and dormancy permitting both disturbance-coupled and -uncoupled recruitment (Grubb 1988), differential response of component species to rainfall seasonality and drought (Westoby 1980), and episodic recruitment followed by lag effects (Goldberg and Turner 1986, Friedel et al. 1993).

The simulation also produced rainfall-related temporal fluctuations in the densities of component plant species. Long periods of community stability were terminated by episodic recruitment or mortality, subsequent changes in species composition, and new stable states. Although there is no long-term database that can be used to verify the results of the model in succulent Karoo plant communities, comparable dynamics have been reported in other arid ecosystems. On the basis of observations in Australian shrublands, Westoby (1980) predicted that plant communities in arid regions undergo lagged temporal variation in growth-form mixes because plant species respond differently to given sequences of rainfall and drought. Long-term ([is greater than] 70 yr) monitoring data from Sonoran Desert plant communities of Arizona and Mexico supported Westoby's (1980) conceptual model by demonstrating that large changes in cover, density, and relative abundance of species occurred and were related to rainfall (Goldberg and Turner 1986, Turner 1990). Shorter term studies in semiarid South Africa (Hoffman and Cowling 1990) and arid central Australia (Friedel et al. 1993) provide further support for the notion that species composition in arid and semiarid areas is temporally dynamic.

Implications for management

The results of our model confirm existing doubts about the application of Clementsian ecological succession theory to semiarid ecosystems and support the state-and-transition concept. Of course, the first (trivial) implication arising is that management should be guided by the state-and-transition philosophy if mechanisms mentioned above (see Range succession and state-and-transition above) are important in the rangeland considered. Nevertheless this point is discussed ardently in literature (Westoby et al. 1989. Friedel 1991, Wilson and Hodgkinson 1991, Hodgkinson 1992, Walker 1993). The results of our model have the following specific implications for management. (1) Event-driven change: We identified the rare recruitment events to be the key processes driving the dynamics of the plant community. Such events are opportunities (Westoby et al. 1989) for management to influence vegetation change in a desirable direction (e.g., maintain a viable population of palatable species within the plant community [Wilson and Hodgkinson 1991]), which may occur once in 10 or 20 yr. To recognize these rare opportunities managers should monitor the vegetation and estimate the densities of safe sites for the different species or functional groups of the rangeland to assess if a large recruitment event may be possible. On the other hand, all reproductive processes of desirable species should be promoted. If, for example, seed production fails because sheep have eaten all the flowers, excellent rainfall conditions following for germination or seedling survival are "wasted." A specific management goal requires a good knowledge of the phenology of the species present at the rangeland. (2) Long time scale of changes: This property has two implications. First, changes because of bad management are only visible after decades, and good management will take many years to show a positive response. For that reason management should be long-term management. Of course, this requires a good memory for the history (detailed record-keeping) of the rangeland. (3) Demographic inertia and lag-effects: Knowing the age structure of the plant community is important for improving the rangeland and for the rehabilitation of degraded plant communities. For example, in cases where rangeland is dominated by a young cohort of nonpalatable plants withdrawal of sheep may not be sufficient to improve the quality of the pasture: management actions like bush-clearing may be more effective in this case. (4) Unpredictability of vegetation change: The unpredictability of the events and an uncertainty and lack of precise knowledge snake management necessarily risk based (Walker 1993). Flexibility to respond rapidly to an event is needed ("event-orientated management"). Therefore, a flexible herd size and herd structure should be an integral part of the management strategy. (5) Variation in fodder production: Our simulations show huge fluctuations in the densities of some species in some years. To avoid overgrazing after "vegetation breakdowns" a flexible stocking rate is required (Walker 1993). We conclude that a successful management of arid or semiarid rangelands that show episodic event-driven dynamics must be flexible and centered around a detailed knowledge of the system.

Applications and extensions of the model.--The basic IBDA model for the Karoo can be extended potentially for investigations of effects of grazing intensity, frequency, and seasonality on the spatial and temporal dynamics of semiarid ecosystems. It could also be used to compare the effectiveness of active interventions (seed addition, microhabitat modification, bush clearing) for rehabilitation of event-driven rangelands. However, in this paper we have not only presented a particular model for a particular plant community, we have also introduced a new modelling technique. This technique, the method of individual-based dynamic automata modelling, is a powerful tool to model arid and semiarid plant communities and gives way to a great variety of applications. If data and knowledge about the life history of the dominant species and the impact of environmental factors are available, other ecosystems could be modelled as well using this technique.


Field studies by S. J. Milton were supported by the Foundation for Research Development and the Department of Environmental Affairs, South Africa, the Southern African Nature Foundation and the FitzPatrick Institute, University of Cape Town. Funding provided by the UFZ-Centre for Environmental Research, Leipzig enabled all authors to travel between Germany and South Africa for co-operative work. The authors thank W. R. J. Dean, W. Bond. M. A. du Plessis, M. T Hoffman, F Jeltsch, I. A. W. Macdonald, W. R. Siegfried, T. Stephan, B. H. Walker, and R. I. Yeaton for assistance during the development of ideas or for comments on drafts of this manuscript, and especially G. R. Shaver and two anonymous reviewers whose critical and thorough comments markedly improved the content and tone of this manuscript.

(1) Manuscript received 24 June 1994; revised 6 February 1995; accepted 9 February 1995; final version received 17 March 1995.


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THORSTEN WIEGAND Department of Ecological Modelling, UFZ-Centre for Environmental Research, Leipzig-Halle, Permosertrasse 15, 04318 Leipzig, Germany

SUZANNE J. MILTON FitzPatrick Institute, University of Cape Town, Rondebosch 7700, South Africa

CHRISTIAN WISSEL Department of Ecological Modelling, UFZ-Centre for Environmental Research, Leipzig-Halle, Permosertrasse 15, 04318 Leipzig, Germany
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Author:Wiegand, Thorsten; Milton, Suzanne J.; Wissel, Christian
Date:Oct 1, 1995
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