Small-scale spatial dynamics in a fluctuating ungulate population


T. Coulson, Institute of Zoology, Zoological Society of London, Regent's Park, London, NW1 4RY, UK


1. The scale at which population dynamics are analysed is important, as results from analyses at different spatial scales can differ and affect interpretation.

2. In this study, detailed census data collected over a 10-year period from a population of Soay sheep (Ovis aries L.) on the Island of Hirta in the St Kilda archipelago, Scotland, is used, together with cluster analysis, to distinguish a temporally stable spatial substructure.

3. Structured demographic accounting of the variance in population change (SDA) is also used to analyse the dynamics of the whole population treated as (a) one unit; (b) one unit subdivided into three subunits; and (c) three independent units.

4. Differences in survival, recruitment and dispersal rates are demonstrated between divisions of the population, which are probably associated with variation in grazing quality.

5. If these groups were not coupled by dispersal and density-independent entrainment, the population dynamics of the three groups would diverge, however, the dynamics of the three subunits are strongly correlated.


Environments are spatially heterogeneous at a hierarchy of scales as a result of a variety of biotic and abiotic factors (Wiens et al. 1993; Gross, Alkon & Demment 1995). At scales ranging from a few square kilometres to continents, spatial heterogeneity can be caused by climatic differences. At finer scales, ranging from metres to a few hectares, causes can be local resource availability (Wedin & Tilman 1993; Hobbs 1996). At these finer scales, spatial variation in nutrient, water and light availability leads to different vegetation communities (Crawley 1986), which, in turn, influences the distribution of herbivores feeding on these communities. Aggregations of individuals encountering different resources may determine social organization and population substructure (Clutton-Brock & Harvey 1978). Consequently, individual life history parameters may vary between segments of the population (Sutherland 1996).

Much empirical work on vertebrate population dynamics assumes that the life history parameters of a study population are homogeneous in space. However, those studies that have incorporated spatial heterogeneity suggest that demographic rates within a population can vary over relatively small areas. The majority of these studies have concentrated on social carnivores (Packer 1986) in which the choice of spatial scale at which to subdivide a population is often apparent from the population social structure. Results from these studies often attribute variation in life-history parameters between groups to the spatial variation in the distribution of prey and other predators; for example Waser et al. (1995) associated mortality and recruitment in social groups of dwarf mongooses (Helogale parvula Peters 1852) to variation in habitat quality within the Serengeti.

There are few studies that have reported small-scale spatial variation in life-history parameters within populations of animals that do not have obvious, discrete population substructures. However, two studies that have addressed the significance of spatial scale within populations have found small-scale variation to be important. Coulson et al. (1997) showed that calf winter survival in a population of red deer (Cervus elaphus L.) on the Isle of Rum, Scotland, varied with local population density—a measure of the number of individuals closely associated within the study site. At high local population density calf survival over the first winter of life was significantly lower than at lower local population density. Ray & Hastings (1996) found that spatial scale at which 79 populations of insects were analysed significantly affected the likelihood of detecting density-dependent dynamics. These examples suggest that spatial heterogeneity can affect demographic rates and may be an important influence on the dynamics of populations, and therefore warrants more attention.

In this paper the prediction that there is spatial variation in life-history parameters between subunits of a continuous population of Soay sheep (Ovies aries L.) is tested. Hierarchical cluster analysis (Gordon 1981) is used to distinguish a spatial substructuring within the population. A population spatial substructure is identified which divides the study population into three stable and distinct subunits. Then, using structured demographic accounting of the variance of relative population change (Brown & Alexander 1991; Brown et al. 1993), a powerful key-factor type of analysis, first different key factors regulating these subunits are identified, and, second, estimates are made of how changes in size of each of these units influence the dynamics of the whole female population.

Materials and methods

Study population

Data were collected from a population of free-living, unmanaged Soay sheep on the island of Hirta in the St Kilda Archipelago (57°49′N, 8°34′W), Scotland. Sheep are distributed throughout the island (638 ha), but for this study they were only studied intensively in ≈230 ha in the catchment of the Village Bay area to the south of the island. The whole island supports a population that fluctuates between ≈600 and 2000 individuals, with the study area population fluctuating between 211 and 513 individuals. The study area is not fenced, so sheep are free to move throughout the island. The population was studied intensively in the 1960s (Jewell, Milner & Morton-Boyd 1974) and since 1985 (Clutton-Brock et al. 1996; Moorcroft et al. 1996; Pemberton et al. 1996).

Lambs were born in April and May (median = 21 April) and ≈80% of marked individuals (n = 1719) were ear-tagged within 1 week of birth and a further 15% (n = 310) were tagged during the summer following birth (Bancroft et al. 1995). Fewer than 2% of sheep seen within the study area were unmarked. Once tagged, individuals were observed until death and breeding attempts were monitored. The majority of mortalities occurred at the end of winter in March and April. Consequently, for this study the sheep year was defined as running from 1 May in year i to 30 April in year i + 1. Mortality rates are influenced by a mixture of density (Clutton-Brock et al. 1991; Grenfell et al. 1992), and severe winter weather (Grenfell et al. 1998).

Between 1986 & 1995 a total of 249 (mean of 24·9 yr−1, sd = 4·33) censuses were conducted, giving a total of 52 239 sheep sightings. On average 78% (SD= 8·4%) of individuals that were known to be alive were seen on each census. Factors known to influence the number of sheep seen in each census are weather conditions and time of day. During wet and windy weather sheep are more likely to seek shelter. The movement patterns of both groups and individuals are such that more sheep tend to be within the study area during the late morning and afternoon than in the early morning and evening.

A census took one day to complete and consisted of three people walking different routes simultaneously and recording the identity and position of all individuals observed. All three routes were fixed and between them they covered all of the study area. The position of each individual was measured to an accuracy of 100 m.

Vegetation composition, biomass and productivity varied within the study area (Jewell et al. 1974; Crawley, personal communication). Crawley (unpublished information) noted that graze quality declines with increasing elevation as Calluna (L.) become more dominant, and that the proportion of nutritious Festuca rubra (L.) and Rumex acetosa (L.) within the grazing sward vary between sites at equivalent elevation over distances as small as 10 m.


Heft identity

A ‘heft’ is a defined as a group of individuals, regardless of sex or age, that utilize the same resources in space. Individuals within a heft may be competing for these resources. A heft can consist of smaller, cohesive groups of individuals that frequently associate; for example, mother–daughter pairs and ram–ram coalitions.

On each census day the distance between all possible pairs of individuals among the animals sighted was calculated. These distances were then averaged over the total number of censuses during which the pair of individuals was seen to give a mean proximity. This was repeated for all pairs of individuals that were seen at least once on the same day. To be included in analyses, an individual had to be seen in at least three censuses. Dissimilarity was defined as the mean distance between two individuals and a dissimilarity matrix was then constructed for each sheep year. The dissimilarity matrix was transformed to a similarity matrix by the formula:

image(eqn 1)

where max. dissimilarity is the largest observed dissimilarity. The transformation was used because variation in small dissimilarities can have a disproportionately larger influence on cluster formation compared to variation in large dissimilarities.

Hierarchical cluster analysis (HCA) with average link clustering was used on the dissimilarity matrix in order to group individuals together (Gordon 1981). This hierarchically clusters a population at scales between n clusters of one individual and one cluster of n individuals. The results from HCA can be displayed in a dendrogram, with a scalar representing the n points at which clusters fuse. The most stable clustering was defined as the grouping or subdivision which remained unchanged over the largest range of scalar values. It was decided to analyse the population at this scale, as temporally stable spatial subdivision of the study population is likely to have relevance to the dynamics of the whole population and not be purely artifactual. The scalar value, in itself, is of no biological importance other than as a pointer to the number of individuals in each cluster, or to the number of clusters into which the population is subdivided. The scalar value can be transformed back into distances, using the maximum observed dissimilarity in any one year and the algorithm used to calculate similarities. The dissimilarity matrices were checked for inversions and non-uniqueness (Morgan & Ray 1995). See Coulson et al. (1997) for a more detailed description of the methodologies used.

As the criteria had been set that an individual had to be seen on at least three different days within a year and that all individuals had to be seen at least once on the same day as every other individual included in the hierarchical cluster analysis, not all animals were assigned to hefts. Once the most stable subdivision of the study area had been identified, some individuals that were not initially assigned to hefts using HCA were classified in another way. To be classified by the use of this alternative method individuals still had to have been seen on at least three different days. Their mean position was used to assign them to groups. If the group they were assigned to was different from either the group they had been in during the previous or following year (if alive), they were not included in analyses. Lambs were only assigned to groups if their mean positions grouped them with their mothers.

Hierarchical cluster analysis led to 89% of identifiable individuals being classified as belonging to a heft. A further 7% were assigned to hefts by the alternative method. Four percent were excluded from analyses. All animals that were assigned to hefts were included in the structured demographic accounting (SDA) analyses described below.

Structured demographic accounting (sda)

Structured demographic accounting of the variance in population change is a form of key factor analysis (Brown & Alexander 1991; Brown et al. 1993). It is only applicable to populations where the fate of every individual is known. Consequently in each year all fecundity and survival terms are exact values and no error term is calculated.

ΔN is defined as the difference in size of a population between year i and year i + 1 measured from the beginning of the reproductive year. Consequently, the relative change in population size between year i and year i + 1, δNi, is equal to ΔNi/Ni. Let Pij be the group-specific performance, which is the portion of the relative change in density of the adult population during the ith year that is attributable to group j. Groups can consist of either age or spatial units of the population. Pij reflects the demographic performance of group j by incorporating two quantities; the recruitment rate of group j in year i, Rij, and the mortality rate of individuals in group j in year i, Mij. Consequently, Pij = Rij − Mij and δNi = Σnj=1Pij, where n is the total number of groups. As survival and recruitment rates need to be exact and do not incorporate error, the fate of each individual had to be known. As paternity was not known for most individuals, male reproductive success could not be accurately calculated for many rams, consequently they were excluded from the SDA analysis and only ewes were considered. It would have been preferable to include males in the analyses, however, it was not possible to do so. There is no evidence that either the adult or lamb sex ratio differs in space within the study area. Only individuals that did not immigrate into or emigrate from the study area were included. The dynamics of the whole study population were described as:

image(eqn 2)

which can be simplified to:

image(eqn 3)

where n is the total number of age/stage groups, aj is the proportion of the population in age group j, bj is the birth rate of individuals in age groups j, sj and wj are, respectively, the summer and winter survival rate of offspring born to individuals in group j, Sj and Wj are, respectively, the summer and winter survival rate of individuals in group j. Summer survival was defined as survival from 1 April to 1 October, and winter survival was defined as survival from 1 October until 1 April in the following year.

Eqn 2 can be considered as two separate components: a recruitment component, Rj (ajbjsjwj) and a mortality component Mj (−aajjSjWj). Initially, for ease of interpretation, results are displayed as the relative change in the population size (from now on referred to the performance of the population), and then decomposed to the two separate components. For the analysis of structured demographic accounting within hefts, a further dispersal component was included (see eqn 4 below).

Structured demographic accounting calculates and decomposes the variance in δN, VN) into the percentage resulting from mortality, recruitment and dispersal. Furthermore, the relative contribution of each age class can be calculated. (Brown & Alexander 1991; Brown et al. 1993). Structured demographic accounting is a powerful form of key factor analysis which solves most of the problems that have recently been highlighted for traditional key factor analysis (Royama 1996). The advantages of SDA over other forms of key factor analysis will not be discussed in any detail here as they have been dealt with by Brown & Alexander (1991 and see Discussion). Similarly, the method by which VN) can be decomposed will not be described, as the method is developed and explained in Brown & Alexander (1991) and Brown et al. (1993).

Initially SDA was used to analyse VN) for the whole population in the study area. First, the population was subdivided into four age groups: lambs, yearlings, 2- to 6-year-olds, and those individuals that were older than 6 years of age. Second, SDA was used on the population subdivided into hefts. Finally, SDA was run for each heft with the population of each heft divided into the four age groups.

When the population was considered as distinct hefts, eqn 2 was modified by substituting age group for heft. Finally when the population of each heft was considered alone, eqn 3 was used with an immigration (Ij) and emigration (Ej) term added:

image(eqn 4)

In the results presented below, only those contributions to VN) that are greater than 10% are discussed.

There are no formal statistics to compare the influence of each demographic component between hefts. However, in the analyses of the dynamics of each heft, the data were bootstrapped with replacement. The resulting bootstrap estimates of the relative contribution of each demographic component to VN) were calculated. To assess whether the difference in the contribution of each demographic component between hefts was significant, the 95% bootstrap estimates were examined . If there was no overlap in the range of these estimates between hefts, the relative contributions were considered significantly different.


Subdivision of the study area

In all years the most stable division of the population was into three distinct hefts. One group occurred predominantly in the east of the study area, one group in the central part of the study area and one group in the south-west (Fig. 1). The most stable subdivision of the population was defined as the one that encompassed the largest range of scalar values used to form clusters. These scalar values were related to the scale at which clusters were fused. The mean width of the range over which the most stable groups remained unchanged was 31% of the total range (difference among years spanned 24–45%Fig. 2).

Figure 1.

Distribution of the three hefts in years 1993–96. Each rectangle is equivalent to the size of the study area. The values along the x and y axes refer to ordinance survey gridlines. One unit is equivalent to 100 m. The position of the hefts was stable over all years of the studies. The points refer to the mean position of animals within each heft. The area to the bottom righthand corner where there are no individuals is the Atlantic ocean—a low quality habitat for sheep. The solid line marks the ocean boundary.

Figure 2.

The distribution of mean and maximum proximities between all dyads. The vertical lines show the range over which the most stable subdivision of the population remains the same. In all cases this division subdivides the population into three hefts.

Mean and maximum dissimilarities were correlated with population size. Mean dissimilarities were larger in years of high density, than in years of low density (Fig. 3a; F1,9 = 5·23, P = 0·048), however, this association did not quite remain significant when bootstrapped at the 95% confidence interval (range of bootstrap slopes −0·0009–0·00529, n = 4999 bootstraps). Maximum dissimilarities also increased with population density (Fig. 3b; F1,9 = 9·81, P = 0·012) and this result remained significant when bootstrapped.

Figure 3.

(a) The association between population size and mean proximity. The numbers by each point refer to the year. (b) The association between maximum proximity between all dyads and population size.

The three hefts showed correlated fluctuations in population size (Fig. 4a; Table 1a). Lamb (Table 1b, Fig. 4b) and adult survival (Table 1c, Fig. 4c) rates were significantly correlated between hefts, with large mortality episodes occurring in sheep years 1988, 1991 and 1994. Lamb survival is lower than adult survival. Birth rates, however, are not significantly correlated between hefts (Table 1d, Fig. 4d).

Figure 4.

Figure 4.

Population size and demographic rates for the Soay sheep population. (a) The time series for the population dynamics of the whole population and the three hefts. (b) Lamb survival rates over time for the three hefts. (c) Adult survival rates over time for the three hefts. (d) Fecundity rates over time for the three hefts.

Figure 4.

Figure 4.

Population size and demographic rates for the Soay sheep population. (a) The time series for the population dynamics of the whole population and the three hefts. (b) Lamb survival rates over time for the three hefts. (c) Adult survival rates over time for the three hefts. (d) Fecundity rates over time for the three hefts.

Figure 4.

Figure 4.

Population size and demographic rates for the Soay sheep population. (a) The time series for the population dynamics of the whole population and the three hefts. (b) Lamb survival rates over time for the three hefts. (c) Adult survival rates over time for the three hefts. (d) Fecundity rates over time for the three hefts.

Figure 4.

Figure 4.

Population size and demographic rates for the Soay sheep population. (a) The time series for the population dynamics of the whole population and the three hefts. (b) Lamb survival rates over time for the three hefts. (c) Adult survival rates over time for the three hefts. (d) Fecundity rates over time for the three hefts.

Table 1.  Correlation coefficients between (a) the population sizes of the three hefts, (b) lamb survival rates between the three hefts, (c) adult survival rates and (d) birth rates (n = 11 years). * significant at P < 0·05, ** significant at P < 0·01, *** significant at P < 0·001, NS = not significant
East 1  
Central 0·934***1 
South-west 0·587*0·574*1
East 1  
Central 0·831***1 
South-west 0·952***0·863***1
East 1  
Central 0·953***1 
South-west 0·937***0·885***1
East 1  
Central 0·729**1 

Structured demographic accounting of the variance

All results from SDA are expressed as the percentage contribution of a term to VN). The results discussed below can be classified in three ways:

1. The influence of each individual term on VN)l; for example, birth rate, b, explains 10% of the variation in VN) (see below).

2. The pairwise covariation between terms; for example, birth rate and winter calf survival negatively covary by 22% when the birth rate is high, winter calf survival tends to be low, and this covariation has an important influence on VN). Simultaneous independent variation is also calculated; however, this proved to be unimportant in this analysis and is not discussed further.

3. Third, the contribution of each demographic component (mortality, recruitment, dispersal) on VN); for example, recruitment involves four factors, a, b, s and w. The contribution of recruitment to VN) is 16%. This contribution includes the influence of the four constituent individual factors and of the two-, three- and four-way covariations between these factors.

Dynamics of the whole population

When the dynamics of the whole population were considered by demographic component (Table 2a–c), mortality explained 40% of the variation in VN), recruitment explained 16% and the interaction between recruitment and mortality explained 44% (Table 2g). The most important factors influencing recruitment were birth rate (10%) and lamb winter survival (30%) but these factors were reduced in importance by their negative covariation (22%). Adult mortality explained more of the overall performance than lamb winter mortality (44%) and these factors reinforced each other through strong positive covariation (61%). However, the contribution of adult survival to VN) was reduced by a negative covariation between it and birth rate (18%).

Table 2.  The breakdown of percentage contributions of age effects, mortality and survival on relative changes in population size over time. Results from structured demographic accounting of the variance of population change are displayed in matrices with rows and columns corresponding to the terms a, b, s, w, S, W, I and E. The diagonals in the matrices (a)–(c) refer to the percentage contribution to VN) of each term, the upper righthand corner to the pairwise covariation between each pair of terms and the lower left hand corner to the simultaneous independent variation between each pair of terms. When the contribution of each age group is displayed [upper matrices in (d)–(f)] the diagonals refer to the percentage contribution of each group, and the off diagonals to the covariation between groups. Also given are the correlation coefficients between each group [lower matrices in (d)–(f)]. Lambs do not directly contribute to performance or mortality as lamb survival is included in the recruitment component of the other age groups (sj and wj). Consequently the values for the contribution of the lamb age class to VN) is always zero. (a) Percentage contribution of lamb and adult survival by season, birth rate and age effects on changes in VN). Values along the diagonals refer to variation in the constituent factors, values in the upper right triangle refer to pairwise covariation between factors, and values in the lower left triangle refer to simultaneous independent variation in each pair of factors. (b) and (c) are subsets of (a) showing the percentage change in VN) resulting from mortality and recruitment, respectively. Matrix (d) shows the percentage variation in performance for the population by age group and the correlation between age group influences, and matrices (e) and (f) break this down into mortality and recruitment components. Matrix (g) shows the percentages of changes in relative population size as a result of covariation between recruitment and mortality
Contribution by componentPerformance Total = 100Mortality Total = 40Recruitment Total = 16
Age group (a) −4   −2     −2 −3    
Birth rate (b) 10−1−22−1−18       10−1−22  
Lamb summer (s)   1           1  
Lamb winter (w)1 1  30−1 61      1 1  30  
Adult summer (S)     −1     −1      
Adult winter (W)1     421    42      
Third orderAge/birth/lamb winter  −2            
Age/birth/adult winter  −2            
Age/lamb win/adult win  −2            
Birth/lamb win/adult win  4            
Contribution by age groupPerformance Total = 100Mortality Total = 40Recruitment Total = 16Recruitment*mortality Total = 44
Lambs (L)0   0   0   0000
Yearlings (Y)06·1  04·5  0 1·4  00·10·60
2–6 years (2–6)016·219·9 06·62·7 0−1·19·1 010·18·115·2
>6 years (>6)015·625·916·3010·47·88·00 2·10·83·403·12·0 5·0
Lambs (L)0   0   0   0000
Yearlings (Y)01  01  01  00·030·15−0
2–6 years (2–6)00·741 00·951 0−0·151 00·790·820·89
>6 years (>6)00·780·72100·870·8410 0·470·07100·390·33 0·48

Population subdivided by age group

When the contributions of recruitment and mortality were broken down according to age group, age groups three (2- to 6-year-olds) and four (>6-year-olds) had the biggest influence on VN), however, all the effects of all age groups were positively correlated (Table 2d–g).

Because the influences of all age groups on the VN) were positively correlated, age groups were combined and the relative contributions of each of the hefts to the dynamics of the whole population considered. By definition, all individuals remained within the population, even if they moved between hefts. It was not possible to consider dispersal between hefts in this analysis as movement between hefts within the population had no influence on the relative change in population size. Table 3 shows the percentage contribution of each heft, the covariation between the hefts and the correlation between each heft in terms of performance, mortality and recruitment.

Table 3.  The contributions of the three hefts to VN) and correlation coefficients between the hefts. The table shows the breakdown in terms of performance, mortality, recruitment and the correlation between mortality and recruitment. The percentage contributions resulting from mortality, recruitment and the covariation between the two differ slightly from Table 2 because this data set was used for the comparison of between- and within-heft differences. Because immigration and emigration were considered in these analyses, years 1987–96 were used. Data from 1986 could not be used because it was not known which individuals had migrated into the population
(a) Performance (Total = 100)(b) Mortality (Total = 44)(c) Recruitment (Total = 19)(d) Recruitment and mortality (Total = 37)
East (E)93 7−121      
Central (C)111043−15 5211   
South-west (SW)11312912912−888 639
East11 1−0·150·230·05      
Central0·5710·681−0·081 0·650·250·67   
South-west0·330·9310·920·811−0·540·631 0·570·330·45

Population subdivided by heft

Of the three hefts, the one in the south-west contributed most to the population performance over time (29%) with the central and east hefts contributing similar percentages (10 and 9%, respectively). The dynamics of the central and south-west heft covaried and reinforced their contributions by a further 31%, whereas covariation between the east and central and east and south-west each contributed 11%. The performances of all three hefts were positively correlated (Table 3a). If performance is decomposed into mortality and recruitment (Table 3b,c) the south-west heft had the dominant influence on mortality (12%). Positive covariation between mortality in the east and south-west heft (12%) and the central and south-west heft (9%) were also important. In contrast, all hefts contributed approximately equally to variation in recruitment, although there was negative covariation between the east and south-west hefts (8%) and positive covariation between the central and south-west heft (8%). The most important covariations between recruitment and mortality was in the south-west heft (9%) and between the central and south-west heft (11%). Both these covariations were reinforcing.

Structured demographic accounting within each heft

When the data were divided into three distinct hefts, different demographic components were found to influence VN) in each heft (Table 4). The influence of recruitment on VN) was the least important component in all three hefts. Mortality was important in all hefts, but was the most important component in the south-west heft. Similarly dispersal was also important in all three hefts and the most important component in the central heft and east heft.

Table 4.  The percentage contribution of mortality, recruitment and dispersal to VN) within each heft. The rows do not add up to 100% as covariation between recruitment factors, mortality factors and dispersal factors can either increase or decrease the influence of mortality, recruitment or dispersal

Birth rate was significantly more important in the east heft than in either of the other two hefts. In the east heft birth rate explained 26% of the variation compared to 5% and 7% for the central and south-west hefts, respectively (95% bootstrap estimates, 20–32% for the east heft, 2–8% for the central heft, 2–12% for the west heft).

Within each heft, the covariation between adult winter mortality and lamb winter mortality was reinforcing, while the covariation between birth rate and adult and lamb mortality was negative (Table 5a–c). These results agree with those from the analysis of the whole population (Table 2a–c). Both lamb and adult winter survival explained significantly more of the variation in the south-west heft than in the east heft (Table 5). Similarly, there were significant differences in the influence of covariations on VN) In the central heft the interaction between birth rate and lamb winter survival was less important than in either of the other two hefts. In the south-west heft the covariation between birth rate and adult winter survival was significantly larger than in either of the other two hefts.

Table 5.  Structured demographic accounting of the variance of population change within each of the three hefts. The underlined values vary significantly between hefts. See legend for Table 2 for further details
Age group (a) −8   −2 123−3−2−4 1−1−23−6 −2 1 3−2 
Birth rate (b)2261−22−1−4 3−131 5−1−9 −10 11−14  7 −18 −232−10
Lamb summer (s)   1  1 1 1  1 1  −4−1−1   2 −1−2 
Lamb winter (w)  2  15 26−2 131 2 116−2 31−26−5  1  20  34−2 11
Adult summer (S)     −1−23     −1−15     1 1
Adult winter (W)1    19 262     26−18−121     33−15 20
Immigration (I)      12−13      642       18−1
Emigration (E)1       222       44       6
  1. win, winter; sum, summer.

Third orderAge/birth/adult winter−2Birth/lamb win/adult win 10Age/birth/adult winter−2
 Age/birth/immigration−3Age/lamb wint/emigration6Age/birth immigration2
 Birth/lamb win/adult win2Age/birth/emigration4Age/birth/adult winter 10
   Lamb sum/lamb win/adult win5Age/birth/emigration4
   Birth/lamb win/emigration−8Lamb sum/lamb win/adult win2
Fourth order  Age/birth/lamb 2in/lamb sum−8  

Although dispersal was very important in influencing the dynamics of all three hefts, both immigration and emigration between hefts showed strong covariation with other components. Generally, these covariations are not consistent between hefts; for example, immigration did not strongly covary with any other factor in the east heft, however, in both the central and south-west heft immigration negatively covaried with adult winter survival (18% and 15%, respectively). Furthermore in the central heft both lamb winter survival and age group negatively covaried with lamb winter survival (26% and 23%, respectively) and positively covaried with birth rate (11%). The effects of all the covariations are to reduce the importance of immigration in the central heft from 64% to 6% and from 18% to 1% in the south-west heft.

Similarly, covariation substantially weakened the effects of emigration in the central heft from 44% to 13%. However, covariation strengthened the importance of emigration in the south-west heft from 6% to 27% and from 22% to 33% in the central heft. In all three hefts emigration negatively covaried with birth rate and in the east and south-west heft it positively covaried with lamb winter mortality. Finally, in the central heft emigration negatively covaried with adult winter survival, while in the south-west the covariation was positive.

Immigration and emigration were not correlated, as immigration was considered as an event that happened at the beginning of each year with immigrants contributing to VN) through contributions to aj, bj, sj, wj, Sj and Wj in the group that they joined. Emigration was considered at the end of each year, with emigrants contributing to the group that they left in that year. Consequently, dispersal was considered as a temporally discrete event occurring at the cusp between 2 years.


We have shown that the dynamics of the population of Soay sheep on St Kilda were strongly influenced by winter mortality of both adults and lambs. These mortality rates both negatively covaried with birth rate and these covariations had a strong influence on the population dynamics. The study area was an arbitrarily chosen part of the island which may not have been the most informative spatial scale at which to analyse the dynamics of the population. The hypothesis that an improved understanding of the dynamics of the population can be obtained through analysis of the spatial substructures was tested. It was found that the most stable spatial scale at which to subdivide the population was into three hefts. The substructure of the population was stable over time, with the dynamics of the three subdivisions being correlated, although contributing differently to the dynamics of the whole population. Theory predicts that small differences in the dynamics of different subpopulations should lead to divergence of the population trajectories, however, this has not occurred. Dispersal between hefts probably entrained the dynamics of the subpopulations. Dispersal did not consistently covary with other components between hefts, suggesting that the reasons why individuals disperse may not entirely be a result of fluctuations in birth or mortality rates.

Populations have often been considered to be spatially homogeneous, and demographic components, such as survival and birth rate, have often been correlated with intrinsic population data such as total density or age structure. However, study populations are not necessarily selected purely on biological criteria; for example, many study populations, including the one analysed here, consist of just part of an island population. Recently, the importance of scale in ecology has received empirical attention (Kadmon 1994; Edmunds & Bruno 1996; Ray & Hastings 1996; Coulson et al. 1997). However, there is a general paucity of work on small-scale spatial dynamics within populations. This is because it is often not apparent how to subdivide a population. Hierarchical cluster analysis is one way of doing this, as the range of scalar values at which clusters are fused can be used as a measure of the stability of subdivisions over a range of spatial scales. Although some studies have reported scale-related effects on fitness and demographic rates (Ray & Hastings 1996; Coulson et al. 1997), more studies are required before it will be possible to generalize about the importance of small-scale spatial heterogeneity within populations.

The importance of different demographic components in influencing change in population size over time has traditionally been explored using key factor analysis (Morris 1959; Varley & Gradwell 1970). However, recently, key factor analysis has been shown to be incomplete. Royama (1996) pointed out three potential problems: first, a key factor cannot indicate quantitative variation in survival or birth rates between different stages; second, the importance of a factor that shows little temporal variation is overlooked; and third, an arbitrary stage division in the life table can lead to spurious results. The development of structured demographic accounting of the variance of population change (Brown & Alexander 1991) overcomes all of these problems, however, it has rarely been used. To our knowledge this is only the third population on which structured demographic accounting of the variance of population change has been applied. The advantages of SDA over traditional key factor analyses are not discussed in any more detail here as they have been dealt with elsewhere (Brown & Alexander 1991; Brown et al. 1993).

By using hierarchical cluster analysis to subdivide a population, and SDA of the variance of population change, two novel approaches are combined to look at small-scale variation in life-history parameters within a population of wild-living mammals.

The dynamics of the Soay sheep population fluctuate over time, with up to 65% of individuals dying in years of high mortality (Clutton-Brock et al. 1991). These persistent fluctuations are caused by overcompensatory, density-dependent winter mortality coupled with density-independent fecundity and neonatal survival (Grenfell et al. 1992). However, overall fecundity is decreased in years of low density following population crashes (Clutton-Brock et al. 1992). The analysis presented here of the dynamics of the whole population reflects this, with winter survival rates being the most important demographic component, and with winter survival rates of lambs and adults being positively and highly correlated. As expected, a weak effect of birth rate, a negative covariation between survival and birth rate and no effect of summer survival were also found. The dynamics of the population were not driven exclusively by one age group.

Analysis of the spatial substructuring of the population demonstrated that the population could be split into three hefts, and that the influence of the dynamics of each of these hefts contributed different amounts to the dynamics of the whole population. Furthermore, when the dynamics of each heft were considered independently, lamb and adult survival was important in all hefts, but increased in importance from the east heft through the central heft to the south-west heft. In comparison, recruitment was an important demographic component in the east, but declined in importance in the other two hefts. Age group remained unimportant in all hefts. It was not possible to make any generalizations about dispersal as neither immigration nor emigration covaried consistently with other demographic components.

These results raise various questions. First, why is the population spatially substructured in the way it is? This paper does not present or analyse data to test any specific hypotheses as to why the population is substructured, however various explanations are suggested. The vegetation on Hirta is spatially heterogeneous (Jewell et al. 1974). Within the low-lying village bay area there are meadows that were used for agriculture until 1930 (Jewell et al. 1974). These meadows now consist of high densities of palatable Festuca rubra and Rumex acetosa. On higher ground, surrounding these meadows, are areas containing lower densities of palatable grazing and higher densities of less palatable Agrostis capillaris and Calluna. Crawley (unpublished information) has shown that palatability declines monotonically with elevation. Consequently, sheep in different parts of the island encounter different quality food resources. Sheep utilize areas that they are familiar with, grazing different parts of a home range on a regular basis (Edwards et al. 1996; Penning et al. 1997). Such behaviour allows graze to recover and may minimize infection by parasitic larvae. Wilson (unpublished data) has shown that faecal egg counts of gut parasites vary spatially within the study area, with lower counts occurring in the east than in either of the areas where the other two hefts occur (see Fig. 1). Gut parasite load has been shown to have a significant association with survival in the Soay sheep population (Gulland 1992; Gulland et al. 1993), consequently, parasite load may contribute to the observed differences in lamb and adult winter survival between hefts. Wild sheep also form dominance hierarchies (Jewell et al. 1974; Hass 1991; Keating 1994; Fournier & FestaBianchet 1995). Consequently, moving from one home range to another could incur costs. First, a new spatial map of vegetation quality will need to be learned and, second, an individual will need to establish its dominance position. Unless an individual has prior knowledge of the area that it is dispersing into, it may disperse into a worse position than it was in prior to dispersal. There is evidence in ungulates that learning from previous foraging events is important in food selection and nutrient gain in both deer (Gillingham & Bunnell 1987) and sheep (Olson et al. 1996). Similarly, there are costs to an individual in finding its position in a dominance hierarchy (Clutton-Brock, Guinness & Albon 1982).

Although we have demonstrated that the most stable division of the study-area population is into three hefts, finer scale subdivisions of the population also exist (Jewell et al. 1974); for example, small ram coalitions and female–lamb groups are common (Jewell et al. 1974). However, although these groupings are cohesive and stable, the scale at which they occur is not stable over a wide range of spatial scales. Many small-scale groupings, although spatially discrete at any instant, could utilize the same resources over time. Consequently, it has proven useful to pool such groups in order to understand the dynamics of the population. When the population size increases, the maximum proximity between two animals increases. This is because animals spread out to utilize a greater area. However, the mean proximity between all dyads also increases with population density. When the population increases, competition between individuals for food is likely to increase. It is possible that this mechanism leads to a more even distribution of animals in space than when the population is smaller, and competition for food is less intense.

Theory predicts that even with very small differences in recruitment and survival rates the dynamics of spatially distinct populations can quickly diverge unless the populations are coupled by dispersal (Lloyd 1995; Ruxton 1996a; Ruxton 1996b). Among the three hefts, the influence of mortality and recruitment vary such that the dynamics would be expected to diverge, however, they have not. There are two reasons for this: first, dispersal between the hefts is frequent enough to couple the hefts; and second, density-independent events are likely to be equivalent between hefts, frequently entraining the dynamics (Grenfell et al. 1998). Although dispersal was a frequent occurrence, there were no consistent patterns between hefts as to what dispersal covaried with. In mammals, dispersal typically shows a sex bias, with males more likely to disperse than females (Greenwood 1980). However, this SDA analysis is only concerned with the dynamics of the female population. Albon et al. (1992) showed that dispersal in female red deer is a gradual process that takes place over several years. They suggested that dispersal was greater at high density than at low density. Dubois et al. (1994) report that space use and dispersal in Corsican mouflon (Ovis musimon L.) ewes is associated with age and dominance, with older individuals having a more static movement regime. If there was age or density effect of dispersal within the Soay sheep of St Kilda, a consistent pattern of covariation of immigration and emigration with survival rates would have been expected (as these are density-dependent) or with age class, however, no such effects were found. It is hoped that ongoing research will highlight why and when individuals disperse.

The present study has shown that within the population spatial substructuring helps to increase our understanding of the dynamics of a study population by identifying small-scale spatial differences in fecundity and mortality rates. We believe that this is because of differences in food quality and quantity. These differences are maintained by the social and energetic risk of dispersal to a different area. Future research requires the identification of the costs and benefits of movement between spatial subdivisions of a population and the actual mechanisms that lead to small-scale spatial variation in demographic rates.


We thank the National Trust for Scotland and the Scottish Natural Heritage for permission to work on St. Kilda and for their help; the Army units that up until April 1998, were stationed on St. Kilda and Benbecula for their assistance. We are indebted to Josephine Pemberton for both her past and ongoing help with the running and maintenance of the Soay sheep project. Important contributions to this work were made by many volunteers who have helped to collect data since the project was begun in 1985. Thanks to David Brown for making available the FORTRAN code to run the SDA analysis, Ken Wilson and Mick Crawley for access to results from unpublished data and to Stephen Jenkins and Josephine Pemberton for comments on an earlier draft of the manuscript. This work was supported by grants from NERC and the Wellcome trust.

Received 23 March 1998;revisionreceived 21 July 1998