An increase in the frequency of high rainfall years resulted in an increase of the z-value (Figure 3 and Figure 4). For example, a 20% increase in high rainfall years yielded a mean z-value of 0.01 (Figure 4). With an increase of 50%, the maximum z-value was 0.02. This upper limit could be clearly assigned, because the carrying capacity of cell type T had been reached, i.e. no further recruits were able to establish in the sub-canopy of trees. Even though additional juveniles may still emerge in cell type M, they are prevented from reaching the open due to a low probability of P_matrix. Additionally, fire mortality is higher in cell type M.

A decrease in mean precipitation resulted in a negative population trend (scenario 2, Figure 3) with a lower limit of z at -0.027 (Figure 4). A 10% increase of P_low, corresponding to a probability of occurrence of low rainfall years of P_low = 0.165, yielded a mean z value of -0.008. This continued to z = -0.02 for a 40% increase of P_low. Above a 60% increase of P_low 30% of all replicate runs resulted an extinction of the Grewia flava population within the 500 year time frame (not shown, however compare Figure 3). Comparing the upper and lower limits of the z-value for scenario (1) and (2), negative effects associated with an increase in drought events were more effective than the positive effects related to higher recruitment rates associated with comparable rainfall excess (0.02 as opposed to -0.027 see Figure 4).

An increase in inter-annual variation between low and high rainfall years below 100% did not cause any significant departure from stable population dynamics. However, above a threshold value of 100%, corresponding to a reduction of P_average from 0.72 to less than 0.44 (see Table 1), we observed a decrease in the mean z-value. For example, for an increase in variation of 150%, z was -0.02 (Figure 4). Here, rainfall almost exclusively occurred as either high or low (P_low = 0.375, P_high = 0.325). Considering the negative and positive effects of low and high rainfall years, negative effects, such as drought, outweighed positive effects associated with enforced recruitment in high rainfall years.

Despite constant mean and inter-annual variation in precipitation, increasing period length (PL) and intra-cycle variation (ICV) led to negative population trends and a decrease in z-value (Figure 5). Here, we found varying ICV thresholds for the PL scenarios tested, i.e. an abrupt departure from stable population dynamics. For example, a 10-year alternating rainfall cycle with an increase of 80 % in ICV (corresponding to a 5-year 'poor' phase with P_low = 0.27 / P_high = 0.026 and a 'good' phase with P_low = 0.03 / P_high = 0.234) yielded a z-value of -0.003, whereas population dynamics were stable for an ICV of 70%. Further thresholds were found above 60% ICV for a PL value of 20 years, 50% for 50 years, 40% for 100 years and 15% for 250 years. In the latter case, a large proportion of the population became extinct below 50% ICV where z was close to its lower limit. It is noteworthy, that we found no significant difference between similar PL scenarios that began with different phase conditions.

In order to assess the predictive power of SGM it is important to discuss the inference of actual precipitation amounts on probability values. Threshold values for extreme rainfall years are based on expert knowledge and may vary depending on the plant species considered. Moreover, percentage changes in extreme rainfall years, as applied in our model, are not necessarily equivalent to percentages changes in mean annual rainfall. For example, an increase in 'high' rainfall years by 10% represents an increase in mean annual rainfall of approximately 10%, depending on actual precipitation amounts in each 'high' rainfall year. Thus, it is important to derive principal population trends from the SGM model results rather than absolute predictions. Another issue regards the temporal dimension: long-term SGM simulation periods deviate from short-term predictions of climate change. However, population trends applied for only a few decades may yield biased results due to the cyclicity of Grewia population dynamics (see 17). Finally, we stress that potential deviations in the annual probability of drought mortality may modify the output of the model. For example, a sensitivity analysis showed that a 100% increase in drought mortality for adult shrubs decreased the model output by 10% (see 18).

In the open savannas of the southern Kalahari, Grewia flava typically grows beneath the dominant tree species Acacia erioloba 34, because bird-mediated seed dispersal predominately confines new establishments to woody plant microsites. However, occasionally large individuals may be found in the open grassland matrix at former tree sites, suggesting high longevity of Grewia flava. Under high cattle grazing a substantial proportion of seeds may be distributed into the open matrix vegetation, since cattle feed on the foliage and fleshy fruits of Grewia flava 18. This may result in a substantial increase in Grewia cover, particularly around boreholes (e.g. 10). Grewia flava has excellent resprouting capabilities after fire 35 and low drought mortality rates 36. Size class distributions suggest a demographic bottleneck in early life stages due to low rates of emergence and high juvenile mortality 36.

The computer model SGM (see 1718) represents a grid-based approach iteratively simulating population dynamics of Grewia flava in annual time steps for a period of 500 years. Based on empirical demographic and spatial data from the Kimberley region of the southern Kalahari (see 36), an initial population of 15 shrubs ha^-1 was distributed on a 200 × 200 cell grid, with each cell representing 5 m × 5 m of savanna vegetation. The SGM grid is developed in two layers: a landscape and a population layer. In the first layer micro-site types of the savanna vegetation may change in the course of time. In the second layer SGM simulates population dynamics of Grewia flava. A cell type was classified as either tree (T) (i.e. occupied by Acacia erioloba) or matrix type (M) (i.e. grassland vegetation). However, it switched from T to M status when a tree died after a mean life span of 200 years. The converse occurred where a new tree establishes. Initial tree distribution and spatial recruitment pattern was random with a constant density of 5 trees ha^-1 for the entire simulation period. In each time step, SGM simulated important life history stages, environmental conditions and the key ecological processes of Grewia flava, the majority of which are governed by annual rainfall patterns (see Figure 1). For example in the model, rainfall directly determines the likelihood of a fire, drought and fire mortality, fruit crop size, emergence and annual shrub growth rate.

Annual rainfall in the southern Kalahari is highly variable, with precipitation ranging between 200 and 700 mm yr^-1. Rain falls almost exclusively during summer (November to April) with an inter-annual mean of 417 mm for the period 1940 – 2000 (Kimberley airport, South African Weather Service 2001, unpublished data). For this period we defined a threshold value of 150 mm below and above the long-term mean to classify years into 'low', 'average' and 'high' rainfall years (thresholds of 267 mm yr^-1and 567 mm yr^-1, respectively) (see 18). Accordingly, frequency of extreme rainfall years resulted in an annual probability of P_low = 0.15 for low and P_high = 0.13 for high rainfall years (P_average = 0.72). Similar classifications have been applied in other studies, e.g. in a spatial simulation model of Acacia raddiana in the Negev Desert 37.

Fruit production rates are based on mean crop size of Grewia size classes and may vary depending on the annual rainfall (see 18). Seed dispersal is mostly zoochorous and spatially aggregated with a large proportion of seeds deposited in woody plant microsites. It is a crucial factor for the population dynamics and long-term viability of Grewia flava 36 and represented by two parameters: probability of seeds removed from a shrub and deposited in cell type M, P_matrix, and cell type T, P_acacia. P_matrix varies randomly per year between 0.1–0.01% and represents occasional seed distribution through, e.g. small mammals. P_acacia refers to bird-mediated seed dispersal with a range of 1–5% in high rainfall years, 2.5–7.5% in average and 5–10% in low rainfall years. Based on estimates from empirical data 36, the assumption of relatively constant seed dispersal rates may be reasonable as the proportion of seeds removed is most likely to be higher in years of lower fruit set. Fruit production typically varies more than bird abundances, and unless birds switch in exactly compensatory fashion to other fruits to the degree that Grewia flava becomes less abundant, fruit removal may be higher in years of low population size.

Even though microsite types differ markedly in micro-environmental conditions, they do not differ in emergence rates of Grewia flava seeds 36 suggesting a similar emergence probability for cell type M and T. However, depending on annual rainfall conditions, the probability of emergence in the model may vary between 1–2% in average and 2–4% in high rainfall years (no emergence occurs in low rainfall years). As survival of Grewia flava seeds in the soil is very low, seeds that do not emerge are assumed to die.

The occurrence and intensity of a fire depends on the amount of rainfall, since precipitation determines annual grass biomass production and thus fuel load 38. In the SGM, fire was simulated probabilistically for the entire grid and may only occur in high and average rainfall years. Average frequency of 7.9 years in the model is supported by fire intervals reported from similar savannah types 39. The impact of fire varies spatially and demographically: a seedling in the matrix vegetation has a 95% chance of being killed in a fire, compared with 4% for adult shrubs which largely resprout in the following year. For T cells, fire mortality probability is 0% for adults and 75% for seedlings. These model assumptions are realistic, since grass fuel accumulation and fire severity beneath trees is less than in tree inter-spaces (e.g. 4041). In the SGM, the probability of drought mortality varies between life stages and the annual rainfall conditions (see also model description in 18). Annual drought mortality for adults is restricted to low rainfall years with a probability of 3% for both cell types. This assumption is based on data from O'Connor 42 and Schurr 36. For seedlings the annual drought mortality probability is 90% for average and 50% for high rainfall years (see 18). All seedlings are assumed to die in years with low rainfall conditions.

In the model, each shrub was assigned a size class with an average canopy cover. For definition of shrub size classes we used the canopy volume to group Grewia flava individuals into categories of small (Grewia_S; <1 m³), medium-sized (Grewia_M; 1–10 m³) and large plants (Grewia_L; >10 m³) as well as seedlings (Grewia_seedling; temporary state after emergence, transforming to Grewia_S with the following year). We assumed a maximum age for each size class member with MaxAge_S = 5 years for small and MaxAge_M = 25 years for medium-sized shrubs. These estimates are based on annual shoot growth rates and aerial photograph analysis of the study area (see 18). In each annual time step an individual can accumulate a growth year and, if MaxAge is reached, attain the next size class with P_{transition(S)} = 0.2 for small and P_{transition(M)} = 0.1 for medium-sized shrubs. No growth occurs in low rainfall years.

To incorporate intra-specific competition we used a simple causal approach that incorporates cell-based shrub cover. Therefore, we defined a carrying capacity of K = 25 m² as maximum total cell shrub cover. If shrub cover exceeded K, individuals in the cell died. Density-dependent mortality was simulated annually by removing the smallest individuals first, i.e. in descending order of size class, until shrub cover <K (for further details see 18). Inter-specific competition between Grewia flava seedlings and the grass layer were neglected: empirical tests showed that emergence rates were similar within and outside of the grassy vegetation matrix (see 36). For the adult stage, inter-specific competition with other woody plants is of low importance because Grewia flava often occurs as a mono-dominant species, particularly on bush encroached rangelands.

The standard scenario of the model, based on rainfall for the period 1940 – 2000, led to stable population dynamics (Figure 2). As standard deviation was generally high we performed 100 replicate simulation runs (± 0.0173 for the standard scenario, see Figure 2). To determine and compare population trends between the climate change scenarios we then used a simple linear regression of years vs. shrub density to calculate the z-value, i.e. the slope of the year – shrub density relationship. This is a reasonable approach for analyzing time-abundance relationships (e.g. 43). When population dynamics were stable for 500 years, i.e. z was ≅ 0, the number of new recruits more or less resembled the number of fire- and drought-killed shrubs (Figure 2). Significant recruitment events mostly occurred in high rainfall years without fire.

Based on the standard scenario we performed simulation experiments for each of the four different scenarios of climate change with variation in inter-annual mean of precipitation, coefficient of inter-annual variation, and temporal auto-correlation (see Table 1):

(1) increase in precipitation, i.e. stepwise increase of P_high resulting in a higher probability of high rainfall years (increase in recruitment events). Probability of low rainfall years is constant whereas frequency of average years is reduced, accordingly.

(2) decrease in precipitation, i.e. stepwise increase of P_low resulting in a higher probability of low rainfall years (increase in drought events),

(3) increase in inter-annual variation of precipitation, i.e. stepwise increase of P_low and P_high resulting in a higher probability of low and high rainfall years at the cost of average years (unaltered inter-annual mean),

(4) increase in temporal auto-correlation of rainfall with 'good' and 'poor' phases.

For the scenario (4) we varied two parameters:

(4a) introduction of rain cycles with increasing period length PL, i.e. longer phases of favorable and unfavorable rain conditions and,

(4b) increase in intra-cycle variation ICV, i.e. increasing variation between alternating favorable and unfavorable periods within one rain cycle.

For PL we applied a period length of 10, 20, 50, 100 and 250 years, respectively, with each period subdivided into a similarly long 'good' and 'poor' phase. For example, a 10-year period length was subdivided into a 5-year period of superior and a 5-year period of poor conditions. For an increase in ICV we alternately increased and decreased high and low rainfall probabilities in each period, respectively (see 4_a and 4_b in Table 1). For example, for an ICV value of 100% P_low equaled 0.00 and P_high 0.26 in a 'good' phase, whereas values were 0.30 and 0.00, respectively for a 'poor' phase. Through this procedure inter-annual mean and variation of rainfall probabilities were identical for each scenario and the default set of rainfall probabilities. Combined increase in PL and ICV resulted in an increase of positive auto-correlation. Near-decadal epochs of above- and below-normal rainfall have been identified for the period 1955–1991 30 and may increase in ICV under predicted climate change.