The influence of rainfall on zebra population ... - Wiley Online Library

Journal of Applied Ecology 2003 40, 125 – 136

The influence of rainfall on zebra population dynamics: implications for management

Blackwell Science, Ltd

NICHOLAS GEORGIADIS*, MACE HACK*†§ and KATHERINE TURPIN*‡ *Mpala Research Center, PO Box 555, Nanyuki, Kenya; †Department of Ecology and Evolutionary Biology, Princeton University, NJ, 08544, USA; and ‡Dartmouth College, Hanover, NH, USA 03755

Summary 1. Where 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. 2. We 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. 3. The 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 RainfallMediated Density-Dependent (RMDD) mechanism. 4. Both 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. 5. Simulations 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. 6. Synthesis 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. Key-words: Africa, herbivore, model, density dependence, carrying capacity, harvest, rainfall dependence, sustainable yield. Journal of Applied Ecology (2003) 40, 125–136

© 2003 British Ecological Society

Correspondence: N. Georgiadis, Mpala Research Center, PO Box 555, Nanyuki, Kenya. Fax: 254 176 32752. E-mail: [email protected] §Present address: Nebraska Game and Parks, PO Box 30370, Lincoln, NE, USA, 68503.

126 N. Georgiadis, M. Hack & K. Turpin

© 2003 British Ecological Society, Journal of Applied Ecology, 40, 125–136

Introduction 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 nonprotected 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 nonprotected 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 estimated 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.

     Laikipia District (9666 km 2) 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

127 Plains zebra dynamics and management

Fig. 1. Map of Laikipia District (with inset of its location in Kenya), georeferenced in UTM coordinates, Zone 37, and featuring the distribution and densities of plains zebras (d) from a sample count in February 2001. Animals were counted within 0·282 km wide transects, spaced 2·5 km apart, arrayed in a 2·5 × 5 km grid (a total of 11·3% of the area was surveyed). The distribution of agriculture (grey shading), and property boundaries (thin black lines) are also featured.


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


Fig. 2. Observed plains zebra time series in Laikipia between 1985 and 2001 (black dots; vertical bars are standard errors), with the best fit model simulations to the period 1985 –99 superimposed for both the Rainfall-Mediated Density-Dependent model (black line), and Rainfall-Dependent model (grey line). Fluctuations in annual rainfall (small open diamonds, dotted line), and the annual harvest (small circles; estimates are filled, known harvests open) are added below.

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%.

    


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 rainfalldependent 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 sexstructures of the model population were set to means from observations in Laikipia during 1999 –2000 (Table 2).

  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


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) Annual mortality rate Population (a) Demogrpahic data Samburu National Reserve, Kenya Athi-Kapiti Plains, Kenya Ngorongoro Crater, Tanzania Ngorongoro Crater, Tanzania Loliondo area, Tanzania Serengeti National Park, Tanzania Nyika National Park, Malawi Kruger National Park, South Africa Means (b) RMDD Model Mean 1985–99 RMDD Model Minimum 1985–99 RMDD Model Maximum 1985–99 RD Model Mean 1985–99 RD Model Minimum 1985–99 RD Model Maximum 1985–99

Foals

Adults

0·33 0·38 0·19 0·19

0·17 0·03–0·09 0·07

0·30

0·11

0·47 0·29 0·28 0·22 0·46 0·30 0·24 0·47

Age at first conception

Birth rate

Refs.*

0·20 0·26 3·4– 4·4

0·19

0·03–0·13 0·10

3·9

0·11 0·10 0·12 0·16

0·08 0·05 0·17 0·09 0·06 0·20

3·63 3·44 4·40 3·73 3·54 4·27

0·16 0·07 0·25 0·16 0·15 0·17

1, 2 3 4, 5 6 7 4, 8 10 12, 13

*References: 1, Rubenstein (1989); 2, Ohsawa (1982); 3, Petersen & Casebeer (1972); 4, Klingel (1969); 5, Klingel (1975); 6, M. Hack; 7, D. Rubenstein, unpublished data; 7, Sinclair & Norton Griffiths (1982); 8, Munthali & Banda (1992); 9, Smuts (1976a); 10, Smuts (1976b).

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

Foals (%)

Sub-adults (%)

Adult females (%)

Adult males (%)

Foals/100 females

Adult sex ratio (M/F)

Ol Jogi Game Sanctuary 6/1999 Mpala 6/1999 El Karama 6/1999 Observed Mean 1999 Model 1999

10 9 5 8 14

10 11 19 13 29

48 49 41 46 34

32 31 36 33 22

20 18 12 17 42

0·68 0·63 0·88 0·73 0·65

Ol Jogi Game Sanctuary 7/2000 Mpala 7/2000 El Karama 7/2000 Observed Mean 2000 Model 2000

13 9 14 12 17

13 18 21 17 26

41 49 45 45 35

33 24 20 26 22

31 20 32 28 50

0·80 0·49 0·44 0·58 0·62

RMDD Model Mean 1985 –99 RMDD Model Mimimum 1985 –99 RMDD Model Maximum 1985 –99

16 7 25

25 17 29

35 33 40

24 22 27

45 17 73

0·70 0·62 0·79

(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. © 2003 British Ecological Society, Journal of Applied Ecology, 40, 125–136

  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


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

  Y    1 Jt =  t−1 ⋅ ( sY + δY ) +  1 −    ⋅ J t−1 ⋅ ( sJ + δJ )  2     α + δa − 1  

Eqn 3. Adult females

Ft =

Eqn 4. Carrying capacity

K = h·Rt

Eqn 5. Rainfall Dependence

 c  δv =  v   K t−1 

Eqn 6. Rainfall-Mediated Density Dependence

c ⋅N  δv =  v t−1   K t−1 

[( F

t−1

]

   1 − q ⋅ p ⋅ ( sF + δ F ) +   ⋅ Jt−1 ⋅ ( sJ + δJ ) + − α δ 1   a  

[ ])

Description

For males: substitute sub-adult male survivorship For males: substitute sub-adult and adult male survivorship

Settings and best fit values

State variables N Total population size Y Number of foals J Number of sub-adults F Number of adult females Parameters, Inputs K Carrying capacity R Rainfall Affected by conditions defining K, e.g. habitat quality, area i Affects how carrying capacity increases with rainfall q Harvest quota p Proportion of females in harvest s Density-independent annual survivorship α Density-independent age at first conception β Density-independent birth rate Density-dependent factor modifying density-independent rates δw c Constant defining sensitivities to rainfall


adults by harem membership, but are not as risky as the first 2 months of life. We therefore assumed densityindependent 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

Calculated annually from eqn 4 Annual means from five gauging stations h Best fit value for RMDD 1·79; set at 1 for RD Best fit values for RMDD 1·6; 1·4 for RD 0 – 0·1 of adult population per year 0·35 or 0·5 Foals 0·95; Females 0·97; Males 0·96 3·4 years 1 Calculated annually from eqns 5 and 6 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

among other populations again provided preliminary guides. For example, mortality can be far higher in foals than adults (Table 1), suggesting greater sensitivity 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 rainfalldependent, and adjusted sensitivities in an iterative manner until variation in each rate most closely matched the range of variation observed in wild populations (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 capacity 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 empirically, 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 residents 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%, respectively, 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 harvested 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: n

M = 1/σln2 N .

∑(ln(N ) − ln(Nˆ )) i

2

i

i =1 max

1/σa2 .

∑ (a

− aˆ j )2 + 1/σβ2 .

j

j =min max

1/σS2 A .

+ max

∑(β

j

− βˆ j )2 +

j =min

∑(S

Aj

− SˆAj )2

j =min

eqn 7 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.

Results   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 mortality, 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 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 available 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 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 capacity 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.

Acknowledgements 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.

References © 2003 British Ecological Society, Journal of Applied Ecology, 40, 125–136

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