N. Georgiadis, Mpala Research Center, PO Box 555, Nanyuki, Kenya. Fax: 254 176 32752. E-mail: firstname.lastname@example.org
1Where wildlife is not formally protected in Africa, abundant species are often consumptively managed, for benefit, control, or both. Such management should preferably be based on some understanding of the ecological processes regulating populations, the extent to which wildlife conflicts with livestock, and the rates at which wildlife can be harvested on a sustainable basis.
2We report the results of a simulation modelling analysis of the dynamics of a plains zebra population in Laikipia District, Kenya. This largely unfenced, non-protected area of 9666 km2 is comprised of privately, publicly, and communally owned properties supporting livestock and wildlife. We examined the influence of rainfall on zebra abundance, and evaluated the sustainability of alternative harvesting regimes.
3The model was stage- and sex-structured, corresponding with age/sex classes recognizable in the field. Vital rates and age at first reproduction were adjusted annually using two alternative approaches, both involving annual re-calculation of carrying capacity as a function of rainfall. The first adjustment factor was simply a function of carrying capacity – a solely ‘Rainfall-Dependent’ (RD) mechanism. The second adjustment factor was proportional to the ratio of carrying capacity to population size – a Rainfall-Mediated Density-Dependent (RMDD) mechanism.
4Both versions of the model reconstructed known zebra dynamics over the last 16 years within the precision afforded by sample counting. However, RMDD gave a better fit, yielded parameter settings that were meaningful ecologically, and better predicted the impact of a severe drought than did RD. We infer that rainfall strongly influences the abundance of Laikipia's zebras by a mechanism involving adjustments in the strength of density dependence.
5Simulations using RMDD suggested that previously allocated annual harvest quotas in Laikipia (up to 15%) might not have been sustainable, but a 6% annual harvest would be a sustainable harvest.
6Synthesis and applications: The model captures a fundamental feature of the mechanism by which rainfall limits large-bodied savanna ungulates: populations typically decline faster during dry phases than they can increase during wet phases. As a result, the greater the variation in annual rainfall, the greater the proportion of time the population spends below carrying capacity. The model has been adopted as the basis for zebra management in Laikipia. To our knowledge, this is the first population-specific model to be used to guide the consumptive management of a wild ungulate in Kenya.
Protected areas in Africa rarely provide sufficient space to support the long-term viability of large mammals (e.g. Woodroffe & Ginsberg 1998; Caro 1999), but few opportunities remain to expand protected areas further (Musters, Graaf & ger Keurs 2000). The remaining option is for wildlife to share landscapes with humans and their livestock (Du Toit & Cumming 1999). In this context, active management is required, preferably based on some understanding of the ecological processes regulating populations, the extent to which wildlife competes and conflicts with livestock, and the rates at which wild herbivores can be harvested on a sustainable basis.
Scrutiny of factors affecting the density of herbivores in Africa has focused largely on comparisons between wild species in protected areas, and, to a lesser extent, livestock in non-protected areas. Such studies have shown that the biomass of wild ungulates in African savannas (Coe, Cummings & Phillipson 1976), and of individual herbivore species (East 1984), increases with mean annual rainfall at greater than a linear rate, with soil nutrient status potentially playing a covariate role (Fritz & Duncan 1994). While this implies that savanna herbivores are fundamentally resource-limited, the mechanisms are poorly understood. Several studies of wild ungulates have identified food as a key limiting factor (Sinclair 1974; Sinclair, Dublin & Borner 1985), but others have implicated predation or disease (Sinclair & Norton-Griffiths 1982; Sinclair 1985; Gasaway, Gasaway & Berry 1996), and illegal hunting (Dublin et al. 1990) as playing key roles. More recently, simulation models strongly suggest that dry season rainfall has influenced Serengeti wildebeest Connochaetes taurinus dynamics since the late seventies (Pascual & Hilborn 1995; Pascual, Kareiva & Hilborn 1997; Mduma, Sinclair & Hilborn 1999), but apparently not the Serengeti plains zebra Equus burchelli population, for which predation might be the dominant factor (Senzota 1988). In Kruger National Park, greater kudu Tragelaphus strepsiceros dynamics are largely rainfall-dependent, with cold periods and predation featuring intermittently (Owen-Smith 2000). Studies of livestock dynamics in savannas have depended upon simulation models, largely because their numbers are influenced by human management practices (e.g. Illius, Derry & Gordon 1998; Fynn & O’Connor 2000). Even so, rainfall strongly affects livestock abundance, especially through the occurrence of drought in communal areas (e.g. Campbell et al. 2000).
While these studies provide an essential framework for comparison, little is known about the dynamics of wildlife in non-protected areas, where predators are reduced or eliminated, and livestock often dominate the herbivore biomass. Models featuring more realistic assumptions about fluctuating resources are needed to understand the dynamics of wild herbivores in non-protected areas, and thereby to establish sustainable harvest rates. To meet these needs for plains zebras in Laikipia District, central Kenya, we used simulation modelling to assess the influence of rainfall on zebra abundance, the impact and sustainability of different harvest rates and regimes, and implications for the sufficiency of sample counting for censusing zebras.
Consumptive wildlife management in Kenya
After a hiatus of 15 years, consumptive management of wildlife resumed in Kenya in 1992 on an experimental basis. Permits to harvest locally common wild herbivores were issued by the national conservation authority, the Kenya Wildlife Service (KWS), to residents of five non-protected areas (including Laikipia District, the focal area of this paper) who could thereby offset the costs of hosting wildlife on their lands through the sale of game skins and meat, or simply benefit from local consumption. Annual quotas were set by KWS at up to 15% of the population size of each species, which were to be esti-mated by annual censuses. Quotas were prescribed with reference to a similar exercise carried out in Zimbabwe (Martin & Thomas 1991), and the principles used to derive harvesting strategies were summarized in an appendix of the KWS Wildlife Utilization Study (1995a; Report 3, Annex D).
Assumptions about population growth and regulation employed in the KWS prescription are those often used to illustrate the principles of harvesting natural populations: logistic (or related) population growth, with density dependence based on a fixed carrying capacity. While useful for illustration, the assumption of constant carrying capacity is not warranted in savannas: a herbivore population can be well above carrying capacity at the end of the dry season, and well below carrying capacity a few weeks after rain, with no change in overall density. Because changes in population size are not symmetric around a fluctuating carrying capacity, which itself increases faster than it declines, the assumption of a fixed (mean) carrying capacity can be misleading (Caughley 1987; Caughley & Gunn 1993; McLeod 1997; Saether 1997; see below). Accordingly, we employed a carrying capacity that fluctuates with annual rainfall. This usage does not address rapid seasonal changes within years, but it does capture marked changes in carrying capacity between years.
study area and field methods
Laikipia District (9666 km2) is comprised mostly of semi-arid rangelands, divided into a mosaic of privately, publicly, or communally owned properties (Fig. 1). Wildlife has been eliminated from the wetter southern and south-western periphery of the District, much of which is cultivated, but zebras are scattered at varying densities across the remaining 7000 km2, which they share with livestock and many other wild herbivore species. Long-term residents agree that, prior to the 1960s, wildlife was almost eradicated from Laikipia, largely by shooting for rations, a practice that began during World War II. In the 1960s zebras were uncommon on many properties, considered a pest, and shot to reduce what was (and generally still is) believed to be intense competition with livestock. In 1977 a national ban on consumptive use of wildlife was imposed in Kenya. Thereafter, consumption of zebras declined in Laikipia, local interest in wildlife conservation grew, and wildlife numbers increased. A controlled harvesting strategy was introduced in 1992 upon the creation of the Laikipia Wildlife Forum, whose members are local land owners with a shared interest in conservation by managing wildlife for profit.
The Department of Range Surveys and Remote Sensing (DRSRS, Ministry of Environment, Government of Kenya) first counted large herbivores in the entirety of Laikipia District in 1985, by systematic aerial sampling (Norton-Griffiths 1978). Over the following 16 years a further nine sample counts were conducted, the last four at high resolution (2·5 km spacing between transects), in collaboration with the Mpala Research Centre (N. Georgiadis & G. Ojwang’ 2001, unpublished report). Over this 16-year period, numbers of wild herbivores in Laikipia weighing over 10 kg fluctuated between 53 000 and 70 000, almost half of them plains zebras. During a severe drought in 1984, zebra die-offs were observed in Laikipia. Partly for this reason, zebra numbers were low when counting began in 1985. Thereafter, the population increased over a succession of average to wet years and stabilized at 30 000–35 000, until 2001 when the population declined again following the worst drought on record (Fig. 2). In May 2000 zebras were again observed to die from drought-related causes in northern Laikipia. We used these figures as the basis for modelling.
Zebras in Laikipia do not migrate seasonally, although they do shift habitats locally and in response to patchy rainfall. Laikipia is well-watered, with two perennial rivers (Ewaso N’yiro and Ewaso Narok) and many run-off catchment dams, particularly on commercial properties. Most of these hold water well into ‘normal’ dry seasons, and all dry out only in droughts.
Zebra demography was monitored on three properties in Laikipia during 1999–2000 (Mpala, Ol Jogi and El Karama ranches). Age structure was recorded by classifying at least 200 individuals into the age and sex classes defined below. Population structure was assessed on each property by driving a predetermined set of roads and recording the number, group composition, sex, age class, and reproductive status (i.e. bachelor, stallion, lactating or non-lactating female, when possible) of all plains zebra within 100 m of the road. Several factors ensured that animals were counted only once per survey, including: (i) each survey was conducted during a single, continuous period of 6–12 h; (ii) roads were selected to cover the region without passing through areas already surveyed that day; and (iii) the timing of the surveys minimized sampling during times of the day when animals were most active. Terrain, vegetation cover, and the location of roads determined the likelihood of observing zebra from the vehicle. Because these features differed among properties, the estimated percentage of the population sampled on each occasion varied from 75 to 90%.
Model description and parameter estimation
Using the Serengeti wildebeest population as a case study, Pascual et al. (1997) showed that a wide variety of qualitatively different models, structured or unstructured, could be made to reproduce observed fluctuations in total numbers, provided that rainfall-dependent processes in all models were based on the amount of rain falling in the dry season. This feature so dominated how well models fit observed data that different models could be made to fit equally well, each with rainfall dependence affecting different components of the model population. Because each model resulted in profoundly contrasting management implications, however, Pascual et al. (1997) cautioned that deriving the appropriate management strategy depends not only on using a model with the correct limiting factors, but also on choosing a model that mimics how those factors affect real populations. Given these caveats, we designed an annual time step simulation model such that, by monitoring rainfall-dependent responses by known individuals in different age/sex classes over time, we will ultimately be able to quantify the rainfall-dependent mechanisms operating on this population.
Our model is stage- and sex-structured into three age classes identifiable in the field: foals (< 1 year old), sub-adults (1–3·4 years; 3·4 years, or 41 months, is the youngest age of first conception in plains zebra, M. Hack, unpublished data), and adults (3·4 years +; see Table 1). We assumed foal sex ratio to be unity, but field observations suggest sex ratios at older ages are biased in favour of females (Table 2). We also wished to simulate harvesting regimes in which the sex ratio is biased. Accordingly, sexes in sub-adult and adult (but not foal) classes are treated separately. The initial stage- and sex-structures of the model population were set to means from observations in Laikipia during 1999–2000 (Table 2).
Table 1. (a) Demographic data from plains zebra populations in eastern and southern Africa. Minimum values (italics) were used as resource-independent parameters in the Laikipia zebra model (maximum values in bold type). (b) Mean, minimum and maximum values exhibited by the best fit versions of the models under Rainfall-Mediated Density Dependence (RMDD), and Rainfall Dependence (RD)
Table 2. Age and sex structure of the plains zebra population at different times and locations in Laikipia, Kenya, compared to best fit model values under Rainfall-Mediated Density Dependence (RMDD)
Location or Model
Adult females (%)
Adult males (%)
Adult sex ratio (M/F)
Ol Jogi Game Sanctuary 6/1999
El Karama 6/1999
Observed Mean 1999
Ol Jogi Game Sanctuary 7/2000
El Karama 7/2000
Observed Mean 2000
RMDD Model Mean 1985–99
RMDD Model Mimimum 1985–99
RMDD Model Maximum 1985–99
In successive years, numbers within each stage and sex class were calculated using difference equations given in Table 3 (eqns 1–3). Foal numbers were calculated as equal to the number of adult females surviving from the previous year, less the number harvested, and less the number having given birth in the previous year (females only rarely foal 2 years in succession; M. Hack, personal observation). The sub-adult age class (1–3·4 + years) for each sex consisted of the foals surviving from the previous year (half of them male, half female), added to those sub-adults surviving from the preceding year that did not mature to adulthood. The adult population of each sex was derived as those adults surviving from the previous year, subtracting harvested animals, and adding the number of surviving sub-adults that matured from the previous year.
Table 3. Equations, and best fit parameter settings
Eqn 1. Foals
Yt = (Ft−1 − (q · p) − Yt−1) · (sF + δF) · 1 − δβ
Eqn 2. Sub-adult females
For males: substitute sub-adult male survivorship
Eqn 3. Adult females
For males: substitute sub-adult and adult male survivorship
Best fit values for RMDD and RD, respectively: foal survival 0·164, 0·001; sub-adult survival 0·173, 100; adult survival 0·083, 0·001; birth rate 0, 100; age at first conception 0·625, 12·5
In the absence of data from zebras in Laikipia, rates of birth, death, and age at first conception from plains zebra populations elsewhere in Africa were used as model parameters (Table 1), but these were further refined using the following rationale. Rates of birth, mortality, and age at first conception were assumed to be the sum of density-independent factors (excluding rainfall), and rainfall-dependent factors (after Owen-Smith 2000). Density-independent values were assumed not to vary, and to equal the minimum (or for birth rate, maximum) values observed in wild populations (Table 1). Sub-adult mortality cannot be estimated as accurately as foal and adult mortalities, being confounded with dispersal from the study area, following dispersal from the natal group. Transitions involving dispersal from the natal group involve greater risks compared to the relative security afforded to adults by harem membership, but are not as risky as the first 2 months of life. We therefore assumed density-independent mortality in sub-adults to be intermediate between that of foals and adults.
Given a sex ratio bias at older ages in favour of females, density-independent survivorship rates for sub-adults and adults were set higher in females than in males (s in Table 3). Density-independent age at first conception was set at 3·4 years. Each vital rate, v, was then adjusted annually by adding (or, in the case of birth rate, subtracting) a rainfall-dependent factor, δv, defined in one of two alternative ways.
1. Rainfall-Dependent, derived as an inverse function of carrying capacity (RD; eqn 5 in Table 3).
2. Rainfall-Mediated Density-Dependent (RMDD), derived as a function of the ratio of population size to carrying capacity (eqn 6 in Table 3).
Each adjustment factor responded to fluctuating rainfall with different sensitivities, defined by the constant cv in eqns 5 and 6. While sensitivities of vital processes to resource availability are not known for any zebra population, observed variation within and among other populations again provided preliminary guides. For example, mortality can be far higher in foals than adults (Table 1), suggesting greater sens-itivity to resource availability in foals than adults (a pattern typical of most ungulates, e.g. Gaillard et al. 2000). To mimic these patterns, we assumed that observed variation in vital rates was solely rainfall-dependent, and adjusted sensitivities in an iterative manner until variation in each rate most closely matched the range of variation observed in wild popu-lations (Table 1).
We define carrying capacity as the maximum number of individuals that could be supported in a given year, at maximum reproductive output. Lacking independent information about zebra carrying capa-city in Laikipia, a power equation was used to calculate zebra carrying capacity from rainfall each year (Table 3, eqn 4), allowing carrying capacity to increase with rainfall at greater or less than a linear rate. Values of the coefficient and the exponent in this model (h and i, respectively, eqn 4 in Table 3) were obtained empir-ically, by an iterative best fit process described below. Thus, carrying capacity was a derived quantity rather than a fixed input.
Rainfall was taken from five long-term recording stations throughout the zebra range (El Karama, Kamwaki, Mugie, Ol Maisor, and Ol Pejeta ranches). These yielded 36 years of rainfall data, from 1965 to 2000, with an overall annual mean of 639 mm (± 191 mm SD), and with maximum and minimum values of 1021 mm in 1997 and 302 mm in 2000, respectively. The ability of two versions of the rainfall series to account for observed fluctuations in zebra numbers was compared. The first was total annual rainfall (January–December), the second was the sum of rain falling in dry season months (September, December, January and February), which comprised an average of 17% of the annual total (March–February).
Available information about the history of zebra harvesting in Laikipia is qualitative: long-term resid-ents agree that wildlife was scarce prior to the 1960s, due to intensive harvesting for rations. Offtakes declined (but did not cease entirely) after 1977, when a national ban on wildlife consumption was imposed. In the years 1992–96 the harvest quota was 15%. In 1997 and 1998 quotas were reduced to 10% and 8%, respect-ively, with a 6% quota thereafter. Records of numbers of zebras actually harvested were not kept until 1996. Annual harvests for the years 1996–2000 were 1350, 1777, 1523, 1568, and 635 zebras, respectively, and these actual harvest rates were used in the model. Based on model estimates of population size in each of these years, the mean proportion of the population harv-ested between 1996 and 2000 was 4% (that is, less than the quotas). Given that harvesting practices have not changed greatly, this value was used in the model to estimate offtakes in the years 1992–95. Except where otherwise specified, we set the proportion of females in the harvested population to be equal to the observed mean for the period 1998–2000, which was 0·37. For model simplicity, all harvesting occurred prior to annual reproduction.
The RMDD version of the model, fitted to the population series from 1985 to 1999, was used to derive a ‘best estimate’ for the zebra population size in 1977, when consumptive management ceased. The best estimates was obtained iteratively, with starting population size increased from 5000 in increments of 1000.
The RMDD model was also used to assess the dynamics of the Laikipia zebra population under a variety of ‘future’ harvesting regimes. Starting with the zebra numbers set to the 2001 census (26 095), simulations were performed with the population under different harvesting regimes, projected over the next 35 years (other factors potentially affecting zebra numbers, e.g. predation, competition, land use and disease, were assumed to remain constant over this period). The annual rainfall series for the last 35 years was used for the next 35 years, but shuffled randomly 50 times without replacement. Thus, total rain falling over the next 35 years, as well as rainfall variability, were identical between model runs, but each rainfall series was unique. Means and standard deviations resulting from these simulations for population size, cumulative harvest, and adult sex ratio were calculated for a range of annual harvesting rates (0%−10%), and adult sex ratios in the harvest (50% females, 37% females).
An iterative process featuring maximum likelihood estimation (MLE) was used to obtain the best fit of the model to observed zebra dynamics between 1985 and 1999, following Pascual et al. (1997) (citing Green 1984; Arnold 1990). Model predictions regarding population size (N) over time, and ranges of variation in foal survival (sF), adult survival (sA), age at first conception (α), and birth rate (β), were contrasted to actual observations from other populations (Table 1). Added together, these contrasts yielded an index, M, of how well the model fits not only observed population fluctuations, but also a ‘generalized’ zebra demographic profile:
In this equation, j= min refers to the minimum (density-independent) observed rate or age, and j= max refers to the maximum observed rate or age. The maximum likelihood estimates for all parameters are those values that minimize M. For any given set of model parameters, carrying capacity was adjusted by varying h and i in eqn 4 until M was minimized.
Model runs using dry season rainfall (as opposed to annual rainfall) were unable to reconstruct the known dynamics of Laikipia's zebras (M > 11). By contrast, both RD and RMDD versions of the model, using total annual rainfall, reconstructed known dynamics well within the precision afforded by sample counting (Fig. 2, and Table 1). Both reproduced a 25% increase from low numbers after a severe drought in 1984, to high but relatively stable numbers in the 1990s, while offtake was moderate, and a decline in numbers following the drought of 1999–2000. However, the RMDD model gave a superior fit, both to the population time series between 1985 and 1999, and to the demographic profile (MRMDD = 1·05 compared to MRD = 1·87; Table 1). Moreover, for the RMDD model, values of the constants adjusting sensitivities of vital rates to rainfall fluctuation (cv in eqn 6,Table 3) were ranked consistently with expectations based on typical dynamics of large ungulates (e.g. Gaillard et al. 2000): age at first conception was relatively sensitive to varying resources, mortality of foals was more sensitive than adult mort-ality, and birth rate was relatively insensitive. By contrast, in the RD model, the constants adjusting sensitivities of vital rates to rainfall fluctuation (cv in eqn 5, Table 3) displayed implausible values, and were not ranked consistently with expectations based on typical dynamics of large ungulates.
To test the strength of the rainfall ‘signal’ in the population series, we estimated the probability of obtaining a better model fit using 100 randomized rainfall series, and found this to be << 0·01 (with observed rainfall, MRMDD= 1·05, and no better than 1·94 among 100 randomized rainfall series). Thus, there is a strong rainfall signal in this zebra population time series.
Since models were fitted to the census time series between 1985 and 1999, omitting the census in 2001 following the severe drought in 1999–2000, we could test the ability of each to predict the impact of the drought on the zebra population. The census conducted at the end of the drought in February 2001 yielded an estimate of 26 095 zebras. The RMDD model prediction for 2001 was 28 008 zebras, while the RD model prediction for 2001 was 32 477 zebras. In all comparisons therefore the RMDD model performed better than the RD model, and all further results focus only on the former.
Values of observed demographic features not directly used in the model fitting process (age structure, adult sex ratio, and number of foals per 100 adult females) differed from modelled values in 1999 and 2000, both in absolute terms, and in degree of variation (Table 2). However, the model predictions did not measure the same quantity as observed values: model values were accounted over the entire year, whereas observed values were instantaneous estimates made at different times during the year. Accordingly, biases are expected between observed and modelled values, especially for demographic variables most sensitive to fluctuating resources, such as the proportion of foals, and ratios that feature these variables, such as the number of foals per 100 adult females (Table 2).
Past zebra management in laikipia
The RMDD model fit to the population series between 1985 and 1999 yielded a ‘best estimate’ starting population size in 1977 of 11 000 zebras. This agrees with the reported history of far fewer zebras in the 1960s and 1970s than in the 1980s and 1990s. The model trajectory using this setting (Fig. 3, black line) shows the model population growing until the 1984 drought, when zebra numbers declined. Following the 1984 drought the model population continued to increase until harvesting was resumed in 1992, after which numbers fluctuated between 30 000 and 35 000 until the drought of 1999–2000, when numbers again declined.
Overlaying this ‘best estimate’ model trajectory of total zebra numbers on the pattern of fluctuating carrying capacity (Fig. 3; dotted line) showed the model population exceeded carrying capacity only during the most severe droughts. Carrying capacity was 19 754 in 1984, and 19 558 in 2000. These are the only periods when zebra die-offs have been recorded in Laikipia. Mean carrying capacity between 1965 and 1999, calculated using ‘best fit’ settings for parameters in Table 3, was 73 075 zebras, but varied greatly between a minimum of 19 558 and a maximum of 155 197 (SD 32 510).
Comparison of this reconstructed history to two hypothetical management scenarios is instructive. First, no harvesting after 1977 (Fig. 3; open circles), when compared to the actual record (black line), shows the impact on the population of harvesting in the 1990s. A second harvesting scenario depicts the population trajectory assuming the full quota had been taken each year since harvesting resumed in 1992 (in fact, the full quota has never been taken in any year). Had all of these quotas been met, the population would have declined sharply (Fig. 3; filled circles), showing that the management policy at the time was far from sustainable.
Sustainability of zebra management
Simulations of population dynamics under no-harvest regimes yielded an important contrast between those employing constant rainfall (equal to the series mean; curve 1 in Fig. 4) vs. naturally varying rainfall (curve 2 in Fig. 4). Although the total amount of rain falling over 35 years was the same for both, the mean model population under variable rainfall remained below that under constant rainfall, which asymptotically approached mean carrying capacity. The difference arose because a population experiencing variable rainfall would decline faster during droughts than it could increase during wet periods: in no-harvest simulations, the maximum rate of population decline (0·69) was much further below zero growth than the maximum rate of population increase (1·13) was above zero growth. Therefore, simulations employing an assumption of constant rainfall would be misleading.
Subjected to variable rainfall, population trajector-ies under different harvesting regimes were consistent with expectations, and have implications for zebra management. For example, with increasing harvest rate, not only would the population size decline, but variation in population size would also decline (Fig. 4, curves 4 and 5). This is because fluctuations in rainfall exert progressively less influence via rainfall dependence the further the population is depressed below carrying capacity. Finally, if more males than females are harvested, a given harvest rate would result in larger population sizes (Fig. 4, curve 3) compared to a regime in which equal numbers of males and females are taken (Fig. 4, curve 4; harvest rate in both simulations was 6%).
Summaries of these results are plotted in the form of mean values of key population variables after 35 years, under different harvesting regimes (Fig. 5). Cumulative offtake peaks at 8% annual harvest if equal numbers of males and females are taken, but continues to increase beyond 8% if more males are taken than females (Fig. 5a). Final population size is also greater with more males taken than females (Fig. 5b). However, the effect on the living population of biasing the harvest sex ratio is extreme at greater than 6% annual offtake (Fig. 5c). While these values should be treated with caution, because observed adult sex ratios (Table 2) are nowhere as extreme as indicated by the simulations at higher harvesting rates, we suggest an annual harvest of 6% to be appropriate for the Laikipia zebras.
Plains zebra dynamics in Laikipia appear to be strongly affected by annually fluctuating rainfall, evid-ently by modifying the strength of density dependence, rather than by a direct effect that is independent of density. This is consistent with little apparent influence by parasitism, disease or unsanctioned hunting, and the fact that predators, while present, may occur at lower than natural densities, due to their removal from some properties. While competition with other grazers is not featured explicitly in the model, this does not rule out a limiting role for competition. Over the period for which census data are available, competition may have had a constant effect, or more likely an effect varying systematically with rainfall. Accordingly, best fit carrying capacities derived from eqn 4 may already ‘account’ for competition. Testing this hypothesis is difficult, however, because the parameter h in eqn 4 is deceptively simple: it is the only feature of this model by which other ecologically important factors, such as habitat suitability, soil fertility, and geographical area, might (implicitly) exert limiting effects on the population.
implications for zebra management in laikipia
This modelling exercise arose from the need for zebra management in Laikipia to be based on quantitative dynamics. We have begun this process by providing a tool (the RMDD model) which, given rainfall and harvest rates, yields an expectation for the population size each year, and suggests a sustainable harvesting rate (6%). If the population consistently falls below model predictions based on rainfall and harvesting, this should prompt a search for additional factors, such as illegal harvesting, loss of habitat, increased competition with livestock, disease, or increasing predation.
For a rainfall ‘signal’ to be detectable in the population series, both variables – rainfall and zebra numbers – must be measured with sufficient precision. Therefore, a corollary of the likelihood that zebra abundance is influenced by rainfall is that our confidence is increased in the sufficiency of sample counting for censusing zebras (at least in its precision). While sample surveys are the least expensive monitoring option, they are costly (currently about US$12 000 to survey the whole of Laikipia District at 2·5 km transect spacing). Modelling was also prompted by the need for a less expensive alternative for setting harvest quotas that would be independent of, but complementary to, censuses. Lack of funds for counting in 2000 gave conservation authorities little choice but to use the model to estimate total numbers, and thereby set the harvest quota. The close agreement between predicted and observed estimates in 2001 appears to validate the decision in 2000 to base the harvest quota on the model, instead of on a census. We propose that harvest quotas be based on model predictions in place of counts at least in alternating years, preferably 2 years out of 3 (saving $24 000 every 3 years). The effects of different harvesting regimes are unlikely to be detected by sample counting at less than 3 years intervals between counts.
To our knowledge, this is the first time in Kenya that a population-specific model has been used as the basis for the consumptive management of a wild ungulate species. We emphasize that modelling is a valuable but grossly underused tool for conservation and management of wildlife in eastern Africa. Because funds for monitoring populations are increasingly scarce, modelling should be a fundamental component of any management policy, helping not only to improve our understanding of observed dynamics and therefore management, but also to make management affordable.
implications for savanna herbivore dynamics
Our estimate of carrying capacity remains hypothetical. We believe current model settings result in an overestimate of carrying capacity, partly because the demographic data used in model fitting were gleaned from zebra populations elsewhere, and most probably do not span the extremes observed in nature. Even so, model predictions of total population size are relatively insensitive to a wide range of mortality rates. For example, a 20% reduction in minimum foal mortality (from 0·19 to 0·15) yields a 2% decline in the model prediction for total zebra numbers in 2001 (also, an 8% decrease in mean carrying capacity, and a slightly improved model fit). Tests of model predictions await independent estimates of vital rates, and their variation with rainfall, for the Laikipia population. Yet this approach captures features we believe to be characteristic of the behaviour of true carrying capacity: annual fluctuations of far greater amplitude than those of the population itself, but with little ‘carry-over’ effect from one year to the next. Irrespective of how other model parameters were set – and dozens of different combinations were tried – best fit values of the rainfall exponent i in eqn 4 were invariably greater than unity (for RMDD, best fit i was 1·7, M= 1·05; an inferior model fit was obtained with i fixed at unity, M = 1·40). Although both settings yielded similar estimates of population size in 1999 (34 765 when i= 1·7; 33 967 when i= 1), predictions of population size in 2001 following the drought were closer to the census estimate (26 095) with i = 1·7 (28 008) than with i= 1 (31 343). These comparisons support a value of i > 1.
If this result is not a model artefact, the mechanism whereby carrying capacity should increase with rainfall in a greater than linear fashion has not been established. Relationships between savanna grass production and rainfall (R) are typically reported as linear (e.g. Rutherford 1980; Deshmukh 1984; Le Houérou, Bingham & Skerbeck 1988; Illius & O’Connor 1999; assumed by Pascual & Hilborn 1995), or less than linear (Dye & Spear 1982), with grass biomass (or available forage density, B) per mm of rainfall declining at higher rainfall values (i.e. B∝Rk, where 0 < k ≤ 1). Thus, factors in addition to forage density must account for an exponent of i > 1 in our derived relationship between carrying capacity and rainfall. One possibility begins by recognizing that carrying capacity is not solely a function of available forage density, but also the cost of acquiring energy to meet production needs (C. Craig, personal communication). Thus, if the per capita cost of foraging (F) declines with increasing rainfall in a non-linear fashion (say F ∝ Rm, where m < 0), then carrying capacity, C, is a function of B/F, or ∝Rk–m, where i∝k–m > 1. To our knowledge, data permitting a test of this hypothesis are not available for zebras. A related possibility arises as a result of rainfall affecting the spatial distribution of surface water for drinking, which in turn limits the foraging area avail-able to water-dependent herbivores during the dry season (Illius & O’Connor 2000). The cost of meeting production requirements increases as the dry season progresses because individuals must travel ever further between drinking and foraging locations.
the importance of rainfall variability
The exercise also captured a fundamental feature of the mechanism by which rainfall influences abundance: the difference in rates at which naturally dynamic ungulate populations decline during dry phases, and increase during wet phases. For any large-bodied grazer population limited by rainfall, rates of decline in drought years are typically higher than rates of increase in wet years (Illius & O’Connor 2000). Thus, the greater the variation in annual rainfall, the lower the ratio of mean rate of increase to mean rate of decline, and the greater the proportion of time the population spends below carrying capacity. This is the principal reason why the ratio of mean population size to mean carrying capa-city was less than unity; in the absence of harvesting, this ratio for the Laikipia population was approximately 0·73 (Fig. 4). In addition to the absolute quantity of rainfall therefore the variability of rainfall appears to be an important factor limiting savanna herbivore populations. An implication of this effect on the comparison of sedentary vs. migratory grazing strategies is that migration not only increases resource availability, but also serves to reduce variation in access to resources.
This paper was made possible by the quality of zebra census data from the Department of Resource Surveys and Remote Sensing, Ministry of Environment, Government of Kenya. Partial funding from Mpala Research Centre is gratefully acknowledged. We are grateful to El Karama and Ol Jogi Ltd for permission to conduct research on those properties. We thank Claus Mortensen, Richard Vigne, Guy Grant, Jasper Evans, and Graham Fletcher for rainfall data, and Gilfrid Powys for a history of wildlife management in Laikipia. Nasser Olwero kindly supplied the map. David Augustine and two anonymous referees suggested important improvements to the manuscript.