Oecologia (2005) 145: 276–281
DOI 10.1007/s00442-005-0114-4
POPULATION ECOLOGY
Stephen J. Mayor Æ James A. Schaefer
The many faces of population density
Received: 28 June 2004 / Accepted: 24 March 2005 / Published online: 7 July 2005
Ó Springer-Verlag 2005
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 nonoccupied 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 Æ Landscape
ecology Æ Population ecology Æ Spatial distribution Æ
Spatial scale
Communicated by John Fryxell
S. J. Mayor Æ J. A. Schaefer (&)
Biology Department, Trent University, 1600 West Bank Drive,
Peterborough, ON K9J 7B8, Canada
E-mail: [email protected]
Tel.: +1-705-7481011
Fax: +1-705-7481205
Present address: S. J. Mayor
Department of Biology, Memorial University of Newfoundland,
St. John‘s, NL A1B 3X9, Canada
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
Stephen J. Mayor James A. Schaefer:
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 organismweighted 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 areaweighted versus organism-weighted averages.
Materials and methods
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
277
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.
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 (r2) were determined for these
and all following regressions.
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
278
Stephen J. Mayor James A. Schaefer:
Finally, we investigated the relationship of average
density to the scale of the population units. We determined the organism-weighted (DO) and area-weighted
(DA) densities for the cluster-based population delineations according to the equations from Lewontin and
Levins (1989):
DO ¼
n
X
di
Ni
;
NT
di
Ai
;
AT
i¼1
DA ¼
n
X
i¼1
where Ni and Ai are the number of individuals and area
of population unit i, respectively; di=Ni/Ai, the density
of population unit i; NT is the total number of individuals; and AT=Ai, 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.
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)
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.
Table 1 Slopes, intercepts,
confidence limits, and
coefficients of determination
(r2) of log-log relationships of
population density to area for
three mammalian species at
various scales of population
delineation
Species (year of census)
Squirrel dreys
Beaver lodges
Moose (1960)
(1962)
(1964)
(1966)
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).
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,
Number
of population
units
32
18
8
3
19
9
6
10
8
4
9
Slope
0.801
0.784
0.922
0.703
0.539
0.518
0.256
1.083
0.714
0.702
0.607
y-intercept
1.643
1.821
2.126
2.358
0.696
0.771
0.295
1.243
1.13
1.14
1.196
95% confidence
limits for slope
Lower
Upper
1.035
1.293
1.707
2.77
0.764
0.744
0.833
1.433
1.001
0.977
1.072
0.566
0.274
0.137
1.364
0.315
0.293
0.321
0.732
0.427
0.426
0.142
r2
0.618
0.397
0.562
0.682
0.599
0.8
0.246
0.859
0.852
0.971
0.564
Stephen J. Mayor James A. Schaefer:
Fig. 2 Relationships of density of eastern grey squirrel dreys to
area at various scales of population delineation
b ¼ 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).
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
(r2=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;
r2=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; r2=0.546; CI=[0.869, 0.513]).
Table 2 Slopes, intercepts,
confidence intervals, and
coefficients of determination
(r2) of log-log relationships of
eastern grey squirrel drey
density to landscape attributes
at four scales of population
delineation
Landscape attribute
Building density (number/km2)
Road density (km/km2)
Proportion of green-space
279
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
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).
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 (r2=0.998; CI=[0.012, 0.016]) and for areaweighted density was 0.010 (r2=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.
Number
of population
units
32
18
8
3
32
18
8
3
32
18
8
3
Slope
0.123
0.495
0.527
0.583
0.382
0.751
0.592
0.813
0.951
1.779
3.261
9.064
y-intercept
1.843
0.710
0.496
0.295
1.788
1.280
1.325
1.070
2.198
2.068
1.998
2.138
95% confidence
limits for slope
Lower
Upper
0.058
0.253
0.157
0.285
0.025
0.155
0.058
1.254
2.959
5.495
8.956
15.492
0.305
0.738
0.897
1.451
0.789
1.374
1.242
2.88
1.058
1.936
2.434
2.636
r2
0.060
0.536
0.654
0.893
0.109
0.306
0.436
0.741
0.030
0.060
0.234
0.974
280
Stephen J. Mayor James A. Schaefer:
2
Table 3 Organism-weighted densities (DO; per km ), area-weighted
densities (DA; per km2), and their ratio, for eastern grey squirrel
dreys at four scales of population delineation
Number of population units
DO
DA
DO/DA
32
18
8
3
186.86
124.19
86.91
73.48
123.79
96.06
72.34
63.76
1.51
1.29
1.20
1.15
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 greenspace, 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 areaweighted 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,
Stephen J. Mayor James A. Schaefer:
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.
Acknowledgements This work was supported by the Natural Sciences and Engineering Research Council of Canada.
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