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Oecologia

, Volume 145, Issue 2, pp 275–280 | Cite as

The many faces of population density

Population Ecology
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Abstract

Population density, one of the most fundamental demographic attributes, may vary systematically with spatial scale, but this scale-sensitivity is incompletely understood. We used a novel approach—based on fully censused and mapped distributions of eastern grey squirrel (Sciurus carolinensis) dreys, beaver (Castor canadensis) lodges, and moose (Alces alces)—to explore the scale-dependence of population density and its relationship to landscape features. We identified population units at several scales, both objectively, using cluster analysis, and arbitrarily, using artificial bounds centred on high-abundance sites. Densities declined with census area. For dreys, this relationship was stronger in objective versus arbitrary population units. Drey density was inconsistently related to patch area, a relationship that was positive for all patches but negative when non-occupied patches were excluded. Drey density was negatively related to the proportion of green-space and positively related to the density of buildings or roads, relationships that were accentuated at coarser scales. Mean drey densities were more sensitive to scale when calculated as organism-weighted versus area-weighted averages. Greater understanding of these scaling effects is required to facilitate comparisons of population density across studies.

Keywords

Individuals–area relationship Landscape ecology Population ecology Spatial distribution Spatial scale 

1 Introduction

Population density is a fundamental demographic attribute and a cornerstone of ecology. It is integral to the concepts of density dependence (Ray and Hastings 1996), population regulation (Clutton-Brock et al. 1985), and the functional and numerical responses of predators to prey (Morgan et al. 1997). There is mounting evidence, however, that density may vary systematically with the area over which it is computed (Smallwood and Schonewald 1996; Bowers and Matter 1997; Gaston et al. 1999; Schaefer and Mahoney 2003). This scale-dependence, the “individuals–area relationship” (Connor et al. 2000; Gaston and Matter 2002), may hamper our ability to compare densities across studies or populations. Understanding the variability in density has widespread implications for metapopulation dynamics, community-level patterns, management decisions, conservation recommendations, and study design (Matter 2000).

The relationship of population density to area has been explored in a variety of contexts (Connor et al. 2000). The equilibrium theory of island biogeography (MacArthur and Wilson 1967) posits that density should remain constant with area because both species richness and abundance per species increase with area. On the other hand, density may decline with greater spatial extent according to the “generalised individuals–area relationship” (GIAR; Gaston and Matter 2002). This may result from the density compensation phenomenon, i.e. the area independence of the summed density of a group of species (Schoener 1986), or because arbitrary study areas may be centred on high-abundance sites (Smallwood and Schonewald 1996). Finally, in patches of habitat, the resource concentration and enemies hypotheses predict that density should increase with area according to the “patch individuals–area relationship” (PIAR; Gaston and Matter 2002) because of more resources and lower effectiveness of predators in larger patches (Root 1973; Kareiva 1983). Although there is empirical support for each of these hypotheses (e.g. Simberloff and Wilson 1969; MacArthur et al. 1972; Risch 1981), our understanding of the relationship of population density to spatial scale remains unsatisfactory.

Computing mean population densities can also be problematic, and can be expressed using two alternative measures (Lewontin and Levins 1989). When densities are averaged among sampling units, mean density is usually weighted by the proportion of total area in which individuals are living, as in “area-weighted” densities. Alternatively, “organism-weighted” densities give prominence to the proportion of individuals living at each density, and provide a more reasonable measure of density when the condition of individuals is of interest. The discrepancy between area- and organism-weighted densities may be accentuated at finer spatial scales (Lewontin and Levins 1989).

The crux of individuals–area relationships may be the aggregation of conspecifics, a common spatial pattern owing to limited dispersal abilities or positive spatial autocorrelation of conditions and resources in the environment. Populations may be hierarchically structured (Amarasekare 1994) and the relationships between organism abundance and environmental variables may not be predictable across scales (Reed et al. 1993; Schaefer and Messier 1995). The clumping of conspecifics also lends itself to objective delineation of population units based on cluster analysis of the spatial associations among individuals (Bethke et al. 1996; McLoughlin et al. 2002; Mauritzen et al. 2002), a technique which is increasingly used in lieu of arbitrary population bounds. Agglomerative cluster analysis is inherently hierarchical, further implying that populations can be identified at multiple spatial scales.

Here, we used a comprehensive approach, based on fully censused and mapped individuals, to explore the scale-dependence of population density. Individuals–area relationships of moose (Alces alces), beaver (Castor canadensis) lodges, and eastern grey squirrel (Sciurus carolinensis) dreys were quantified with population units identified at multiple scales using hierarchical classification. In the case of dreys, we further investigated the scale-dependence of population density in the following contexts: within arbitrary population bounds centred on high-abundance areas, within patches of green-space, in relation to environmental variables (i.e. density of buildings, roads, and green-space), and finally, as area-weighted versus organism-weighted averages.

2 Materials and methods

2.1 Data collection

Our study comprised three datasets, one generated from field observations and two from published reports. For the first, we conducted a complete census of dreys (i.e. nests) of eastern grey squirrels in 24 km2 along an urban–rural gradient in Peterborough, Ontario, Canada. The study area comprised a downtown core, suburban space, and agricultural fields with hedgerows and scattered tree stands. Dreys are known to relate directly to abundance of individuals (Don 1985). A fully mapped distribution was produced by searching the entire study area systematically and thoroughly for 160 h and recording the location of each drey with a handheld Global Positioning System (GPS). The coordinates were entered into MapInfo Geographic Information System (GIS) with 1:10,000 basemaps that depicted buildings, roads, and green-space within the study area.

We compiled data from two other sources. The first was an aerial census of 383 beaver lodges in 832 km2 near Elliot Lake, Ontario (Coles and Orme 1983). The second comprised four aerial censuses of moose in 36 km2 near the Lower Noel Paul River in central Newfoundland (Bergerud and Manuel 1969). These distributions were geocoded using topographic maps and entered into GIS.

2.2 Data analysis

Following Bethke et al. (1996), we delineated population units by cluster analysis. We used Ward’s (1963) linkage method on the Universal Transverse Mercator grid coordinates of each squirrel drey, beaver lodge, and individual moose. Although classification necessitates some subjectivity in the cutoff level for group identification (Kenkel 1986), we denoted the scales of clustering (i.e. number of recognisable population units within the study area) by scrutinizing each dendrogram and cluster amalgamation schedule for substantial increases in linkage distance. For each of these scales, minimum convex polygons were created around each cluster to delineate population borders (Fig. 1). The density and area of each population unit were displayed as log-log regressions at each scale (Smallwood and Schonewald 1996; Gaston et al. 1999; Schaefer and Mahoney 2003). The 95% confidence interval (CI) of the slope (b) and coefficient of variation (r 2) were determined for these and all following regressions.
Fig. 1

Scales of population delineation of eastern grey squirrel (Sciurus carolinensis) dreys in Peterborough, Ontario, Canada, based upon cluster analysis: a 3 population units, b 8 population units, c 18 population units, and d 32 population units. Minimum convex polygons are displayed around each population unit (solid lines) within the limits of the study area (dashed line)

All further analyses were conducted with respect to squirrel dreys only. First, we modelled the arbitrary delineation of populations centred on high-abundance areas (Smallwood and Schonewald 1996). At the finest scale of delineation (32 population units), circular areas of radii at 200-m intervals were drawn around the centre of each cluster. We included only clusters for which a minimum 1,000-m radius could be fit into the study area. For these, a log-log relationship of drey density to the area was computed for each set of concentric circles, and the mean slope was computed, with each set of concentric circles treated as the experimental unit.

To test the patch individuals–area relationship (Connor et al. 2000), we analysed patches of green-space, as depicted on the 1:10,000 basemaps. Drey density and patch area were quantified, and we conducted separate log-log regressions where non-occupied (zero density) patches were included and excluded.

To explore the relationships between population density and environmental variables, we determined the density of buildings, density of roads, and proportion of green-space within each population unit. To assess the effect of scale on these relationships, log-log regressions were performed at each of the scales of cluster-based population delineations.

Finally, we investigated the relationship of average density to the scale of the population units. We determined the organism-weighted (D O) and area-weighted (D A) densities for the cluster-based population delineations according to the equations from Lewontin and Levins (1989):
$$D_{\rm O} = \sum\limits_{i = 1}^n {di\frac{{N_{i}}}{{N_{\text{T}}}}}, $$
$$D_{\text{A}} = \sum\limits_{i = 1}^n {di\frac{{A_{i}}}{{A_{\text{T}}}}}, $$
where N i and A i are the number of individuals and area of population unit i, respectively; d i =N i /A i , the density of population unit i; N T is the total number of individuals; and A T=A i , the summed area of population units.

DO and DA were log-transformed and each was regressed against the number of population units (n). The discrepancy between these two expressions of mean density was determined as the ratio, DO/DA (Lewontin and Levins 1989). The overall density of the study area (AS) treated as a single population was calculated as NT/AS.

3 Results

The Peterborough study area comprised 1,187 squirrel dreys, from which we recognised four spatial scales of clustering, i.e. from 3 to 32 population units (Fig. 1, Table 1). For the 383 beaver lodges, three scales of clustering were identified, the study area comprising 6–19 population units (Table 1). For moose, we distinguished populations at only one scale in each of the four censuses, with 4–10 population units in each (Table 1).
Table 1

Slopes, intercepts, confidence limits, and coefficients of determination (r 2) of log-log relationships of population density to area for three mammalian species at various scales of population delineation

Species (year of census)

Number of population units

Slope

y-intercept

95% confidence limits for slope

r 2

Lower

Upper

Squirrel dreys

32

−0.801

1.643

−1.035

−0.566

0.618

18

−0.784

1.821

−1.293

−0.274

0.397

8

−0.922

2.126

−1.707

−0.137

0.562

3

−0.703

2.358

−2.77

1.364

0.682

Beaver lodges

19

−0.539

0.696

−0.764

−0.315

0.599

9

−0.518

0.771

−0.744

−0.293

0.8

6

−0.256

0.295

−0.833

0.321

0.246

Moose (1960)

10

−1.083

1.243

−1.433

−0.732

0.859

(1962)

8

−0.714

1.13

−1.001

−0.427

0.852

(1964)

4

−0.702

1.14

−0.977

−0.426

0.971

(1966)

9

−0.607

1.196

−1.072

−0.142

0.564

In these population units derived from cluster analysis, density was dependent on spatial scale, both within and across these scales of population delineation. Density declined with area in all cases (Fig. 2) and the log-log slopes were remarkably similar (Table 1, \( \bar{b} = - 0.694 \)). Except at the coarsest scales for squirrel dreys and beaver lodges, all slopes were significantly different from zero (Table 1). For dreys and lodges, the y-intercepts indicated that, for any given area, populations units at coarser scales generally displayed higher densities than at finer scales (Fig. 2, Table 1).
Fig. 2

Relationships of density of eastern grey squirrel dreys to area at various scales of population delineation

In arbitrary population delineations centred on clusters, drey density was also sensitive to scale. All but one set of these concentric circles exhibited decreasing density with area. The overall regression slope was −0.148 (r 2=0.l44; CI=[−0.269, −0.027]).

Highly contrasting relationships were found between area and drey density within patches of green-space. The log-log regression slope was positive (b=0.545; r 2=0.070; CI=[0.198, 0.892]) when patches of zero abundance were included (Connor et al. 2000). However, when unoccupied patches were excluded (Bowers and Matter 1997), the converse relationship was expressed (b=−0.691; r 2=0.546; CI=[−0.869, −0.513]).

The relationship of drey density to landscape features was scale-sensitive. At each scale of population delineation, as density of buildings or roads increased or proportion of green-space decreased, the density of dreys increased (Fig. 3, Table 2). At coarser scales (fewer population delineations), a steeper slope in the organism–environment relationship was exhibited in all but one case (Fig. 3, Table 2).
Fig. 3

Relationships of eastern grey squirrel drey density (per km2) to building density (per km2) at various scales of population delineation. The regression line is displayed for each scale

Table 2

Slopes, intercepts, confidence intervals, and coefficients of determination (r 2) of log-log relationships of eastern grey squirrel drey density to landscape attributes at four scales of population delineation

Landscape attribute

Number of population units

Slope

y-intercept

95% confidence limits for slope

r 2

Lower

Upper

Building density (number/km2)

32

0.123

1.843

−0.058

0.305

0.060

18

0.495

0.710

0.253

0.738

0.536

8

0.527

0.496

0.157

0.897

0.654

3

0.583

0.295

−0.285

1.451

0.893

Road density (km/km2)

32

0.382

1.788

−0.025

0.789

0.109

18

0.751

1.280

0.155

1.374

0.306

8

0.592

1.325

−0.058

1.242

0.436

3

0.813

1.070

−1.254

2.88

0.741

Proportion of green-space

32

−0.951

2.198

−2.959

1.058

0.030

18

−1.779

2.068

−5.495

1.936

0.060

8

−3.261

1.998

−8.956

2.434

0.234

3

−9.064

2.138

−15.492

−2.636

0.974

Organism- and area-weighted mean drey densities increased at finer scales of population delineation. The log-log regression slope for organism-weighted density was 0.014 (r 2=0.998; CI=[0.012, 0.016]) and for area-weighted density was 0.010 (r 2=0.989; CI=[0.007, 0.013]). At each scale, the organism-weighted density was higher than the area-weighted density, but the discrepancy was accentuated at finer scales (Table 3). Likewise, both measures of average density were higher than the overall density of dreys in the entire study area (49.21 dreys/km2) at all scales, but the difference between them and overall density increased at finer scales.
Table 3

Organism-weighted densities (D O; per km2), area-weighted densities (D A; per km2), and their ratio, for eastern grey squirrel dreys at four scales of population delineation

Number of population units

D O

D A

DO/DA

32

186.86

123.79

1.51

18

124.19

96.06

1.29

8

86.91

72.34

1.20

3

73.48

63.76

1.15

4 Discussion

Understanding variations in organism abundance and the relationships of organism to environment is central to ecology. Because the spatial extent tends to vary among studies, population density has become our common currency of abundance, a shorthand to remove the effects of spatial scale. There is increasing evidence, however, that density is not detached from the extent over which it is computed. Some studies have documented heightened densities in larger habitat patches (Bowers and Matter 1997; Matter 1997; Connor et al. 2000). Many others—for example, of mammalian carnivores (Smallwood and Schonewald 1996) and birds (Gaston et al. 1999)—have reported diminished densities in larger study areas. These two tendencies have been labelled “patch” and “generalised” individual area relationships (PIARs and GIARs), respectively (Gaston and Matter 2002). Patterns may vary even within a species. In caribou and reindeer (Rangifer tarandus), for example, there is a marked divergence in density–area relationships among ecotypes and grazing systems (Schaefer and Mahoney 2003; Van Klink 2003). Such systematic variations may have serious implications. It remains unclear, for instance, to what extent our inferences regarding density-dependence may be confounded by scale-dependence (Ray and Hastings 1996).

These scaling relationships of density may depend critically with how we define and delineate “patches” (in the case of PIARs) and “populations” (in the case of GIARs). Conventionally, such units have been denoted arbitrarily, but more attention is being devoted to “organism-centred” bounds of patches and populations (Kotliar and Wiens 1990), to express density from the “organism’s eye view” (Lloyd 1967; Purves and Law 2002). For example, satellite tracking in conjunction with agglomerative classification has been used to delineate populations of arctic bears (Ursus maritimus, U. horribilis) based on their spatial coherence (Bethke et al. 1996; Taylor et al. 2001; Mauritzen et al. 2002; McLoughlin et al. 2002). Nevertheless, in cluster analysis, groups can often be recognised at multiple levels. Such matters of scale are often relegated to an arbitrary decision. This implies that population units, even objectively defined, might be denoted at several scales, structured as a nested hierarchy. At each of these scales, density and its relationship to environmental variables may differ (e.g. Schaefer and Messier 1995), and the likelihood of scale-dependence remains.

In our study, we united cluster analysis with fully censused and mapped distributions of individuals to delineate population units objectively and hierarchically. Our results underscore the commonness of GIARs suggested by Smallwood and Schonewald (1996) and Gaston et al. (1999). We found GIARs in all three species and at all scales of delineation (Fig. 2, Table 1). Although this negative relationship of density and area was evident when populations were denoted arbitrarily, it was most pronounced in objectively defined population units. This suggests that arbitrarily sized study areas centred on high abundance (Smallwood and Schonewald 1996) may account for only part of the decreasing density pattern in GIARs.

On the other hand, we found inconsistent relationships between drey density and area of patches of green-space, a partial failure to support the PIAR. Bowers and Matter (1997) suggested that the variability in strength and sign of PIARs was due to shifting responses to landscape heterogeneity at different scales. We propose, more fundamentally, that this is more likely a consequence of the vagueness in the definition of habitat “patch”. Eastern grey squirrels, for instance, may exist in suburban and urban environments with little regard for “green-space” and the surrounding “matrix”, as commonly perceived by humans. This underscores the limitations of simplistic, dichotomous, landscape categorisations (McIntyre and Hobbs 1999; Manning et al. 2004), despite their widespread use.

How to express average density, a seemingly simple task, may also present problems in population ecology. Both organism- and area-weighted mean densities were sensitive to scale, but the difference between them appears to grow when the system is viewed at finer scales (Table 3; Lewontin and Levins 1989). Unlike area-weighted densities, organism-weighted densities account for the heterogeneous distribution of individuals (Lewontin and Levins 1989) and consequently are more strongly scale-dependent. Similarly, individuals–area relationships in our study were more pronounced when the clumped distribution was taken into account with cluster analysis, compared to artificial population delineation. Imposing arbitrary population bounds risks sacrificing cross-study comparisons (Matter 2000) because this patchiness—so common in nature, the basis of GIARs and other fundamental ecological patterns—is ignored. By instead revealing scale-dependence from the organism’s perspective, sources of variation in density can be explored as responses to heterogeneous landscapes (Hobbs 2003). Similarly, where the data represent counts of individuals in arbitrary quadrats, mean crowding—the average number of neighbours per individual (Lloyd 1967)—may better convey the local density actually experienced by an individual.

“Heterogeneity rules” (Hobbs 2003). Hence, it should not surprise ecologists that organism density is often linked to the area over which individuals are censused, how populations are demarcated, or where patches are delimited. Such scale-sensitivity is increasingly evident across species and studies. In our view, dealing with the scale-dependence of population density will entail confronting it directly—by accounting for the spatial distribution of organisms at multiple scales—rather than obscuring it with ratios (Schaefer and Mahoney 2003). As an initial step, the area over which density is computed should be reported (Gaston and Matter 2002). Accounting for space, hitherto considered only superficially in population ecology, may represent a crucial step toward deeper understanding of determinants of organism abundance.

Notes

Acknowledgements

This work was supported by the Natural Sciences and Engineering Research Council of Canada.

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Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Biology DepartmentTrent UniversityPeterboroughCanada
  2. 2.Department of BiologyMemorial University of NewfoundlandSt. John‘sCanada

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