Global Ecology and Biogeography, (Global Ecol. Biogeogr.) (2008) 17, 203–210
Research
Absences
M.
RESEARCH
Pautasso
Paper
andand
PAPER
density–area
P. J. Weisberg
relationships
XXX
Blackwell
Oxford,
Global
GEB
©
1466-8238
Journal
2007 Ecology
The
compilation
UK
Publishing
Authors
and ©
Biogeography
Ltd
2007 Blackwell Publishing Ltd
RESEARCH
PAPER
Negative density–area relationships:
the importance of the zeros
Marco Pautasso1* and Peter J. Weisberg2
1
Division of Biology, Imperial College London,
Wye Campus, High Street, Wye, Kent TN25
5AH, UK, 2Department of Natural Resources
and Environmental Science, University of
Nevada, Reno, NV 89557, USA
ABSTRACT
Aim Estimates of abundances and densities of birds and mammals have often been
shown to be scale dependent, in that population sizes over large areas are overestimated if extrapolated from surveys of small plots. Previous tests of the mechanisms
suggested to cause this decelerating scaling pattern found evidence of a biased choice
of small plots in patches of homogeneous habitat. Here we show that negative
density–area relationships can also arise as result of not considering plots where
individuals of the species or assemblage of interest are absent in surveys of differing
spatial resolution.
Location We took a complete census of violets (Viola spp.) in 800 m2 of chalk
grassland in Wye, Kent, UK, and used human population censuses for Finnish, Swiss
and Italian municipalities, English districts, states of the USA and European countries.
Methods We used mixed models of logarithmically transformed number of individuals or densities as a function of area.
Results The census of violets shows that by increasing the survey resolution and by
not considering plots without individuals, the effectively occupied area diminishes
and a negative density–area relationship arises. The finding that negative density–area
relationships are also common for people is evidence that the non-random choice of
plots in population surveys of varying areas can be responsible for many observed
negative density–area relationships. The shallower slope of the people–administrative
area relationship for Switzerland and Finland compared with Italy, as well as for
England and the USA compared with Europe, confirms that less than proportionate
individuals–area relationships can be the consequence of larger plot areas containing
a higher proportion of areas without individuals.
Main conclusions Densities should be reported together with the effective areas
for which they were estimated. It should be clearly conveyed whether or not plots
where the surveyed species was absent were included in the density estimation.
*Correspondence: Marco Pautasso, Division of
Biology, Imperial College London, Wye
Campus, High Street, Wye, Kent TN25 5AH,
UK. E-mail: [email protected]
Keywords
Human biogeography, land-use patterns, macroecology, plot area effect, presence–
absence, sampling size, scaling of abundances, spatial autocorrelation, study grain,
urban density functions.
Central to ecology is the accurate quantification or prediction of
population abundance for an area of interest (e.g. Elton, 1933;
McNaughton & Wolf, 1973; Greig-Smith, 1984; McGill, 2006).
Indeed, sound estimations of the size of animal, bacterial, fungal,
plant and other populations are a necessary (although not
sufficient) condition for their effective management and conservation. This requirement has generated a large body of methodological and applied literature on the estimation of population
abundances from sampling data (e.g. Barabesi & Fattorini, 1998;
Karanth & Nichols, 1998; Buckland et al., 2000; Pisani, 2002;
Pollock et al., 2002; du Rau et al., 2003; Witmer, 2005; Sutherland,
2006).
© 2007 The Authors
Journal compilation © 2007 Blackwell Publishing Ltd
DOI: 10.1111/j.1466-8238.2007.00354.x
www.blackwellpublishing.com/geb
INTRODUCTION
203
M. Pautasso and P. J. Weisberg
One often overlooked yet apparently widespread difficulty
with abundance data is that they do not seem to be independent
of spatial scale (e.g. Gaston et al., 1999). This is the case whether
they are expressed as absolute figures (numbers of individuals) or
as densities (numbers of individuals relative to some kind of unit
of area) (Pautasso & Gaston, 2006). In the latter case, densities
tend to decline with an enlargement of plot areas, a pattern that
is described in the literature as a negative density–area relationship. In the former case, numbers of individuals tend to increase
less than proportionately (i.e. with a slope shallower than 1)
with increasing plot area, a pattern known as the generalized
individuals–area relationship (GIAR). One relationship is the
mathematical transformation of the other (see Methods). A less
than proportionate increase of abundances (or a decrease of
densities) with area has biological relevance, because unless it is
taken into account in the course of extrapolations it will inevitably
cause overestimation of the size of populations over large areas when
sampled from small plots (e.g. Schonewald-Cox et al., 1991).
The scale dependence of abundances is well established for
mammals and birds (see e.g. Smallwood & Morrison, 1999;
Smallwood & Smith, 2001, and references cited in Pautasso &
Gaston, 2006). Yet low densities from large plot areas, and high
densities from small plot areas, have been reported also for a
beetle (Matter, 2003) and a butterfly and its associated host plant
(Matter et al., 2003). A negative density–area relationship has
also been documented between the number of large trees, snags
and logs per unit area and the reserve size (Götmark & Thorell,
2003; see also He et al., 2002). Several mechanisms have been
proposed to cause such a pattern including plot choice so as to
sample a constant number of individuals, plot edge effects,
survey efficiency and habitat heterogeneity. The latter, coupled
with the biased choice of small plots in patches of homogeneous
habitat, has been found to be the main factor responsible for less
than proportionate increases of bird abundances from all over
the world (Pautasso & Gaston, 2006). Fundamentally, a negative
density–area relationship can be expected from the interaction of
a spatially aggregated distribution of individuals and non-random
sampling. In this paper, we show that a decrease in densities with
increasing plot area can also result from the omission of plots
where the species of interest is absent from surveys of differing
area. For this purpose, we use a complete census at differing
spatial resolutions of violets (Viola spp.) in 800 m2 of grazed
chalk grassland close to the National Nature Reserve of Wye,
Kent, UK. This is a sampling extent smaller than that of most
macroecological studies, but since the finer grain used in the
present analysis was 0.01 m2, the number of sampling cells at that
grain was on the whole 80,000, which corresponds to more than
half the Earth at a cell resolution of 1° × 1° (a common grain in
macroecological analyses), and which is more than enough to
show the problem that can be caused by not including cells where
individuals are absent in density estimations.
Furthermore, we investigate the presence and form of less than
proportionate individuals–area relationships with human population data. There has been a recent surge in investigations of
biogeographical patterns of human populations (e.g. Terrell,
2006). These have mainly concentrated on the spatial correlation
204
between species richness of various taxa and the settlement of
people (e.g. Balmford et al., 2001; Luck, 2007), but have also
included studies of the human appropriation of net primary productivity (e.g. Imhoff et al., 2004) and of the latitudinal gradient
in worldwide human diseases (Guernier et al., 2004). Although,
ideally, macroecological analyses would be based on regular grid
cells, many studies have to deal with large-scale data obtained for
other purposes and which refer to administrative units (e.g.
McKinney, 2002; Stohlgren et al., 2005; Watts et al., 2007). This
makes the present analyses of human densities as a function of
non-randomly located and irregularly shaped administrative
areas relevant for other analyses in geographical ecology.
Population densities of cities and countries have only rarely
been investigated in terms of density–area relationships (but
see Craig & Haskey, 1978). Many urban and regional studies
followed a seminal work by Clark (1951), who showed that the
population density of several towns tends to decline with distance from the town centre. So-called urban density functions
were thus fitted to data of urban population densities not as a
function of town area, but of a linear distance. This exponential
distance–decay model, with declining densities with distance
from the town centre, translates to a negative density–area relationship when analysed in a two-dimensional way, provided that
small sampling areas are distant from areas of low population.
A different set of studies, comprising the so-called size–density
literature (e.g. Stewart & Warntz, 1958; Stephen, 1972; Best et al.,
1974; Massey & Stephen, 1977), investigated whether human
population density is an explanatory variable for variations in
the size of towns, counties and countries. In these studies, area is
on the y-axis and density on the x-axis, making it difficult to
compare findings with ecological studies where the relationships
are reversed. By contrast, in this paper we analyse human population density as a function of the area of administrative units to
test whether there are differences in the density–area relationships from countries that differ in the presence of large areas
devoid of individuals.
MATERIALS AND METHODS
Violets were censused in eight contiguous replicate plots of
100 m2 in chalk grassland at the boundaries of the National
Nature Reserve on the North Downs of Wye, Kent, UK, at the end
of April 2006, using a botanical square metre with 100 small
squares of 10 × 10 cm. This permits a study of the GIAR at resolutions ranging from 0.01 m2 to 100 m2. At each resolution, the
density–area relationship (and the corresponding individuals–
area relationship) was investigated both including and leaving
out from the density estimation those plots where violets were
absent. In the second case, we refer in the Results to densities on
the basis of the effectively occupied area (Elton’s economic density;
Elton, 1932). The census was carried out on a west-facing slope
of approximately 20° inclination, at an altitude between 40 and
50 m. The vegetation is characteristic of old downland in Kent
(e.g. Wells, 1965), with Brachypodium pinnatum, Festuca ovina
and F. rubra co-dominant, among a rich assemblage of other
grasses and forbs.
© 2007 The Authors
Global Ecology and Biogeography, 17, 203–210, Journal compilation © 2007 Blackwell Publishing Ltd
Absences and density–area relationships
Table 1 Population (in thousand individuals), area (in km2) and overall density (in individuals km–2) of the six countries (or supranational
entity in the case of Europe) analysed. The USA includes Alaska. Europe includes Russia (with its Asian part) and Turkey.
Country
Population
Area
Density
n
r2
a
b
SE
P
c
Finland
Switzerland
Italy
England
USA
Europe
5206
7361
56,996
50,094
300,327
812,400
337,426
41,284
301,526
130,433
9,172,000
23,950,000
15
178
189
384
33
34
444
2762
8101
354
52
54
0.04
0.08
0.15
0.02
0.07
0.86
2.90
2.65
2.20
4.39
4.33
5.16
0.28
0.38
0.80
–0.02
0.45
0.81
0.05
0.02
0.01
0.03
0.08
0.02
< 0.001
< 0.001
< 0.001
n.s.
< 0.001
< 0.001
– 0.72
– 0.62
–0.20
– 1.02
– 0.55
–0.19
n, number of administrative units; r 2, proportion of variance in individuals explained by area of administrative units. Intercept (a), slope (b), slope
standard error (SE), P values (P) of the individuals–administrative area relationship and the slope of the corresponding density–area relationship (c) are
given controlling for spatial autocorrelation.
Human population data and relative areas and geographical
coordinates were obtained from publicly accessible websites.
Data for the 444 Finnish municipalities refer to the National
Census of 2000 (StatFin; http://statfin.stat.fi/). Data for the 2762
Swiss municipalities refer to the 2005 Census (Bundesamt für
Statistik; http://www.bfs.admin.ch/). Data for the 8101 Italian
municipalities refer to the National Census of 1991 (ISTAT;
http://www.istat.it/). Data for the 354 English districts refer
to the 2001 Census (Office for National Statistics; http://
www.statistics.gov.uk/). Data for the 52 states of the USA
(including Puerto Rico and the District of Columbia) refer to the
Census of 2000 (US Census; http://www.census.gov/), whereas
data for 54 European countries (including Russia and Turkey
but not the Svalbard Islands) are estimates for 2002 (Eurostat;
http://epp.eurostat.ec.europa.eu/).
In all cases, number of individuals, area and densities were
log-transformed prior to analysis to conform to the assumptions
of statistical tests. Individuals–area and density–area relationships
were investigated with mixed models ( 9.1) with exponential
covariance structure so as to control for spatial autocorrelation
(as, e.g., in Pautasso & Gaston, 2006). Spatial autocorrelation
needed to be accounted for, as the population, area and density
of administrative areas in a certain region may tend to resemble
those of neighbouring administrative areas, as commonly
reported from other spatial distributional data (e.g. Goslee,
2006). A density–area relationship of slope k translates invariably
into an increase of abundances with increasing plot area of slope
1 + k, so for human data we only present regression estimates
and graphs of the latter relationships.
For the six data sets of human population sizes and areas, all
administrative units contain some individuals, so there were no
zeros in the data sets. But the issue of the importance of the zeros
for negative density–area relationships can be investigated in an
indirect way, because the chosen countries vary in the proportion
of empty territory. On the one hand we compared the individuals–
area relationship of Finnish, Swiss and Italian municipalities. If
less than proportional individuals–area relationships result from
the omission of locations with zero population we expected any
such relationship for Finnish municipalities to have a shallower
slope than for Swiss and Italian ones, given that Finland is a
much less densely populated country (Table 1). For the same
reason, although Switzerland and Italy show similar overall
densities, we predicted the slope of the individuals–area relationship to be shallower for Swiss municipalities compared with
Italian ones due to the large proportion of uninhabitable mountainous territory in Switzerland.
On the other hand, we investigated the pattern at a coarser
scale than with municipalities for English districts, states of the
USA and European countries, although English districts lie at an
intermediate level between municipalities and countries. We
expected the slope of any less than proportionate increase in the
population of English districts with area to be shallower than for
European countries. This expectation followed from the fact that
England, although densely populated, exhibits a marked contrast
between urban areas and countryside, meaning that larger districts
generally include a larger proportion of areas with low population.
We expected these patterns to be stronger in the USA than in the
Old World because of a generally lower population density,
which should translate into a stronger influence of the zeros for
larger areas, leading to a shallower individuals–area relationship.
RESULTS
Over the whole census area (800 m2), 3552 individual violets
were recorded, for an overall density of 4.4 violets m–2 (Fig. 1a).
By leaving out the 86 1-m2 subplots where violets were absent,
the density became 5.0 violets m–2. The density varied in the eight
100-m2 replicates between 3.0 and 5.8 violets m–2 [mean 4.4;
standard deviation (SD) 1.1]. The number of empty square
metres in the eight replicates varied between 0 and 21, and densities
calculated leaving out these zeros in the eight replicates varied
between 3.7 and 5.8 violets m–2 (mean 4.9; SD 0.8).
For each of the eight replicates it was possible to generate
negative density–area relationships by increasing the resolution
of the census and simultaneously leaving out the zeros (Fig. 1b).
At the resolution of 0.1 m2, there are 1000 0.1-m2 cells in 100 m2.
But the number of 0.1-m2 cells with a presence of violets in the
eight replicates varied between 220 and 393. At this resolution,
densities calculated by leaving out the 0.1-m2 cells with no
presence of violets varied between 10.5 and 11.4 violets m–2
(mean 11.0; SD 0.3). At a resolution of 0.01 m2 the effect of
leaving out the zeros was even stronger. Although there are
© 2007 The Authors
Global Ecology and Biogeography, 17, 203–210, Journal compilation © 2007 Blackwell Publishing Ltd
205
M. Pautasso and P. J. Weisberg
Figure 1 (a) The spatial distribution pattern of violets censused in
800 m2 of chalk grassland in Wye (Kent, UK) on the 22–23 April
2006, (b) the negative density–area relationship generated by
leaving out zeros at differing spatial resolutions.
10,000 0.01-m2 cells in 100 m2, violet occurrences ranged from
275 to 532 of these cells in the eight replicates. Densities calculated on the basis of the effectively occupied area varied at this
resolution between 105 and 116 violets m–2 (mean 111; SD 4).
The relationship between density and effectively occupied area
at the resolutions of 0.01, 0.1, 1 and 10 m2 for the eight 100-m2
replicates was a negative one, with slope of –1 (n = 4, r 2 = 0.99,
log(individuals m–2) = 2.61 – log(m2), P < 0.001; Fig. 1b). This
followed from the corresponding flat individuals–area relationship.
For the six human data sets analysed there were less than
proportionate increases of the population with increasing
area (Table 1; Figs 2a–c & 3a–c). These translated into negative
density–area relationships in all cases (Table 1).
DISCUSSION
The complete census of violets analysed shows that a negative
density–area relationship can be caused by increasing the resolution of the survey and at the same time not considering the plots
where there is no occurrence of individuals, since the effectively
occupied area diminishes. This generalization extends the argument
that density–area relationships can follow from small surveys
being biased away from low-density areas (e.g. Scherner, 1981;
Haila, 1988; Gaston et al., 1999). It is important to realize that
206
Figure 2 The less than proportionate increase with increasing area
in the population of (a) Finnish (n = 444, P < 0.001), (b) Swiss
(n = 2762, P < 0.001) and (c) Italian municipalities (n = 8101,
P < 0.001).
leaving aside sampled areas where individuals of the species or
assemblage of interest are not present when calculating densities
(e.g. Gaston et al., 1999; Kolb et al., 2006; Gunton & Kunin,
2007) can make these estimates scale dependent and reduce the
comparability with density estimates obtained from areas varying markedly in size. Information contained in empty (parts of )
plots should not be disregarded (Diggle, 2003). Densities should
thus be reported together with the area over which they were
estimated, and making it clear whether or not sampling plots
where the species of interest was absent were included in the
density estimation.
The reported density–area relationship of violets has a slope
of –1 in log–log space, corresponding to a flat individuals–area
relationship. This assumes that the number of individuals
© 2007 The Authors
Global Ecology and Biogeography, 17, 203–210, Journal compilation © 2007 Blackwell Publishing Ltd
Absences and density–area relationships
Figure 3 The less than proportionate increase with increasing
area in the population of (a) English districts (n = 354, P = 0.47),
(b) states of the USA (n = 52, P < 0.001) and (c) European countries
(n = 54, P < 0.001).
remains constant at differing resolutions (i.e. the census is
complete). In the case of diminished survey efficiency for larger
areas, the negative density–area relationship will tend to be even
steeper, because some individuals may be overlooked at coarser
resolutions. However, this may be compensated for by missed
individuals over larger areas causing whole cells to be recorded
as devoid of individuals, and thus diminishing the effectively
surveyed area where zeros are excluded from density estimation.
For the violets surveyed, a negative density–area is not produced
by coarsening the resolution of the study from 10 m2 to 100 m2
because there are no 10-m2 cells without violets, but leaving out
zeros from density calculations can practically lead to bias at any
grain, provided the plot size is small enough and/or detection
probability low enough for plots devoid of individuals to be
possible. Bellehumeur et al. (1997) provide an example of a study
of tree density at different grains where the smallest grain chosen
is large enough such that there are no plots devoid of individuals.
Further work could investigate the effect of zeros on density–area
relationships of scale-dependent detectability (e.g. Thompson
et al., 1998; MacKenzie et al., 2006). This could be done for
different (i) species, (ii) degrees of aggregation of the sampling
units, and (iii) spatial patterns of the individuals censused.
Density–area relationships are sensitive to the frequency of
species absences omitted from the density calculation with
increasing sample grain size. For the violet census data, omitting
all zeros at a 0.01-m2 grain decreases the effectively sampled area
by 30, whereas omitting only 50% of the zeros decreases the
effectively sampled area only by a factor of 2. These omissions
would lead to overestimation of density by factors of 30 and 2,
respectively, since the number of individuals remains constant.
At a 0.1-m2 grain, the factors are 3 (100% zeros left out) and 0.6
(50% left out). Therefore, although the exact proportion will
vary between different studies, species and scales, it appears that
researchers need to include the majority of the zeros in order
for the effect to be negligible. These values also show that the
importance of accounting for zeros in density estimations
increases for studies with a smaller sampling grain. Conversely,
when the grain of the sample is large enough for no or very few
plots to be lacking the presence of individuals of the species
or assemblage of interest, densities can be assumed not to be scale
dependent due to the effect of leaving out zeros from the estimation (they may still be scale dependent for other reasons,
including non-random placement). Hence, for practical purposes, when planning sampling work for density estimation, if
a non-null proportion of exploratory plots located randomly in
the landscape of interest turn out to be devoid of individuals,
then either the size of the plots should be enlarged or the area
occupied by empty plots should not be overlooked in the
calculation of the density.
For the human population data analysed, in all cases less than
proportionate individuals–area relationships (and therefore
negative density–area relationships) are found. This is strong
evidence for the generality of less than proportionate density–
area relationships. It is important to bear in mind that such a
relationship is not necessarily universal. A priori, there is no reason
why a proportional increase in the number of individuals with
increasing scale should not be observed. For instance, the
number of trees in the forests of the countries of the world
increases with the area of forest in the country in log–log space
with a slope not significantly different from 1, i.e. there is no
negative tree density–forest area relationship (same data as in
Kauppi et al., 2006). Similarly, data used to study the global relationship between forest productivity and biomass show that
aboveground forest biomass increases with plot area in log–log
space, again with a slope not significantly different from 1, i.e.
there is no negative forest biomass–plot area relationship (data
with indication of plot area in Keeling & Phillips, 2007).
However, it is striking how common the less than proportionate increase of human populations with increasing area is. It is
© 2007 The Authors
Global Ecology and Biogeography, 17, 203–210, Journal compilation © 2007 Blackwell Publishing Ltd
207
M. Pautasso and P. J. Weisberg
present in lowly populated regions such as Finland and the USA
as well as in much more densely inhabited countries such as
Switzerland, Italy and England (Table 1). Since all the analysed
data are of high quality, it is unlikely to be a matter of declining
survey efficiency with increasing administrative unit area. Edge
effects are ruled out too, because the population of municipalities
and countries is largely resident. It may be argued that many
people living in the countryside earn their living by commuting
into towns, but this is a reversed edge effect mechanism (Nee &
Cotgreave, 2002), which will tend to make a less than proportionate individuals–area relationship disappear. As for the other
two mechanisms that have been suggested to cause less than proportionate individuals–area relationships, both the choice of
units so as to encompass a constant number of individuals
and the effect of habitat heterogeneity are compatible with the
disproportionate influence of zeros over large areas.
The role of the zeros in potentially generating less than proportionate individuals–area relationships is corroborated by the
matching of the predictions regarding the slopes of the relationships in different regions. The increase of population with
municipality area is shallower in Finland (+0.28) than in Switzerland (+0.38), and in Switzerland than in Italy (+0.80). This is
likely to be caused by the greater occurrence of vast areas devoid
of human presence in Scandinavia and in the Alps compared
with the Italian Peninsula. A similar fit of data and expectations
is found regarding the slope of the individuals–area relationships
for English districts (–0.02), states of the USA (+0.45) and European countries (+0.81). The increase of population size with
district area is actually absent in England, for reasons that might
go back to the rural depopulation following the parliamentary
enclosures starting in the 17th century, one of the most formative
processes of the English landscape and one which has been
estimated to have affected up to 20% of the land surface in
England (Hoskins, 1955; Kain et al., 2004). As for the shallower
slope of the individuals–area relationship for states of the USA
compared with European countries, this result is compatible with
the importance of zeros in generating negative density–area relationships, since Europe (apart from Russia) shows on the whole
a higher population density than the USA and thus a smaller
presence of vast areas with no or little human presence.
While the population living in the USA is increasing, a process
of densification has been reported, with denser areas becoming
more densely populated (Mahmassani et al., 1988; Fonseca &
Wong, 2000). If persisting, this will tend to make the negative
density–area relationship for states of the USA even steeper in the
future. This issue is of relevance for the sake of land planning: do
we wish further sprawl in open countryside and thus a reduction
of the slope of the human density–area relationship, or do we
prefer a steeper slope of the human density–area relationship,
with less land affected by urbanization, but towns with fewer
green areas (Pandit & Laband, 2007)? Further research may
examine whether a dense megalopolis or sprawling urbanized
regions are to be preferred for the global conservation of biodiversity, on the lines of an investigation of the trade-offs between
surface devoted to agriculture and farming intensity (Green
et al., 2005). For urbanization, the issue is complicated by the
208
scale dependence of the human–biodiversity correlation
(Pautasso, 2007).
Human negative density–area relationships are likely to be an
artefact, insofar as administrative units are not randomly chosen
across a nation or a continent, as small municipalities and countries tend to be located in areas of high human density (e.g.
Stephen, 1971). Already in 1951, Clark noted that by enlarging
the administrative size of towns their densities become de facto
lower, although the real densities cannot have changed because
of an administrative decision. Human population densities have
been related to a number of environmental patterns and processes (e.g. species distributions, Allen & O’Connor, 2000; flood
protection, Pottier et al., 2005; air quality assessments in urbanized landscapes, Gupta et al., 2006; declining human fertility,
Lutz et al., 2006; percentage of forest cover, Wright & MullerLandau, 2006; and urban form in developing versus developed
countries, Huang et al., 2007), but whenever these densities refer
to areas varying in size, researchers need to take into account that
higher densities in small sampling areas may not reflect reality
but can be an artefactual consequence of the census. For example,
Huang et al. (2007) report that urban areas in developing countries have higher average densities (c. 15,000 individuals km–2)
than those in developed countries (c. 5000 individuals km–2), but
this result may at least in part reflect the fact that these human
densities were calculated over smaller average areas in developing
countries (c. 400 km2) than in developed ones (c. 1000 km2). Further
work using data from a subdivision of the world’s population
into arbitrary grid cells of differing size (e.g. Tobler et al., 1997,
who find that, on a grid of 5′ of latitude by 5′ of longitude, 78%
of the resulting cells do not contain resident people) may test
whether a negative density–area is absent when sidestepping the
problem of lack of randomness in the choice of the sampling
units (see, e.g., Sutton et al., 2001). If negative density–area
relationships are artefactual (unlike the less than proportionate
scaling of river length with catchment area, which follows from
fractal geometry; e.g. Rosso et al., 1991; see also McKinney, 2005;
Bettencourt et al., 2007; Watts et al., 2007), then randomly
chosen sampling plots across a landscape should enable the
estimation of the density of a population of a whole region without overestimation.
Our analyses show that negative density–area relationships
can arise simply from the omission of plots lacking the species of
interest from the density estimation. This issue might affect the
estimation of biological parameters other than densities (e.g. tree
mortality rates when studies tend to focus on plots where disease
is present and overlook healthy forests: Holdenrieder et al.,
2004). Omission of zeros may not be intentional and, depending
on the sampling strategy, may not be apparent to researchers.
However, in many cases, ecologists consciously choose to calculate
abundances-when-present by leaving out zeros in the estimation
(e.g. Pennington, 1983, Wright, 1991). These average density
values can be misrepresentative of the spatial patterns in a landscape with many absences of the organism(s) of interest. It is
important that researchers explicitly account for artefactual negative density–area relationships when reporting and interpreting
analyses of temporal or spatial patterns of species abundance.
© 2007 The Authors
Global Ecology and Biogeography, 17, 203–210, Journal compilation © 2007 Blackwell Publishing Ltd
Absences and density–area relationships
ACKNOWLEDGEMENTS
Many thanks to T.M. Blackburn, G. Botterill, K. Evans, K.J. Gaston, G. Hirsch, O. Holdenrieder, M. Jeger, D. Storch, P. Warren
and R.J. Whittaker for insights and discussions, to C. Bond,
J.A.F. Diniz-Filho, K.J. Gaston, F. Maroni, and three anonymous
referees for comments on a previous version of the draft, and
to H.-U. Zaugg for kindly providing the areas of the Swiss
municipalities.
REFERENCES
Allen, A.P. & O’Connor, R.J. (2000) Interactive effects of land use
and other factors on regional bird distributions. Journal of
Biogeography, 27, 889 – 900.
Balmford, A., Moore, J.L., Brooks, T., Burgess, N., Hansen, L.A.,
Williams, P. & Rahbek, C. (2001) Conservation conflicts across
Africa. Science, 291, 2616 – 2619.
Barabesi, L. & Fattorini, L. (1998) The use of replicated plot, line and
point sampling for estimating species abundance and ecological
diversity. Environmental and Ecological Statistics, 5, 353–370.
Bellehumeur, C., Legendre, P. & Marcotte, D. (1997) Variance
and spatial scales in a tropical rain forest: changing the size of
sampling units. Plant Ecology, 130, 89 – 98.
Best, R.H., Jones, A.R. & Rogers, A.W. (1974) The density-size
rule. Urban Studies 11, 201– 208.
Bettencourt, L.M.A., Lobo, J., Helbing, D., Kuehnert, C. &
West, G.B. (2007) Growth, innovation, scaling, and the pace of
life in cities. Proceedings of the National Academy of Sciences
USA, 104, 7301–7307
Buckland, S.T., Goudie, I.B.J. & Borchers, D.L. (2000) Wildlife
population assessment: past developments and future
directions. Biometrics, 56, 1–12.
Clark, C. (1951) Urban population densities. Journal of the Royal
Statistical Society A, 114, 490 – 496.
Craig, J. & Haskey, J. (1978) The relationships between population,
area, and density of urban areas. Urban Studies, 15, 101–107.
Diggle, P.J. (2003) Statistical analysis of spatial point patterns, 2nd
edn. Arnold, London.
du Rau, P.D., Barbraud, C. & Mondain-Monval, J.Y. (2003)
Estimating breeding population size of the red-crested pochard
(Netta rufina) in the Camargue (southern France) taking into
account detection probability: implications for conservation.
Animal Conservation, 6, 379–385.
Elton, C.S. (1932) Territory among wood ants (Formica rufa L.)
at Picket Hill. Journal of Animal Ecology, 1, 69 – 76.
Elton, C.S. (1933) The ecology of animals. Methuen, London.
Fonseca, J.W. & Wong, D.W. (2000) Changing patterns of population density in the United States. The Professional Geographer,
52, 504 –517.
Gaston, K.J., Blackburn, T.M. & Gregory, R.D. (1999) Does variation in census area confound density comparisons? Journal of
Applied Ecology, 36, 191–204.
Goslee, S.C. (2006) Behavior of vegetation sampling methods in
the presence of spatial autocorrelation. Plant Ecology, 187,
203 –212.
Götmark, F. & Thorell, M. (2003) Size of nature reserves: densities
of large trees and dead wood indicate high value of small
conservation forests in southern Sweden. Biodiversity and
Conservation, 12, 1271–1285.
Green, R.E., Cornell, S.J., Scharlemann, J.P.W. & Balmford, A.
(2005) Farming and the fate of wild nature. Science, 307, 550–555.
Greig-Smith, P. (1984) Quantitative plant ecology. University of
California Press, Berkeley, CA.
Guernier, V., Hochberg, M.E. & Guégan, J.-F. (2004) Ecology
drives the worldwide distribution of human diseases. PLoS
Biology, 2, e141.
Gunton, R.M. & Kunin, W.E. (2007) Density effects at multiple
scales in an experimental plant population. Journal of Ecology,
95, 435–445.
Gupta, P., Christopher, S.A., Wang, J., Gehrig, R., Lee, Y. &
Kumar, N. (2006) Satellite remote sensing of particulate matter
and air quality assessment over global cities. Atmospheric Environment 40, 5880–5892.
Haila, Y. (1988) Calculating and miscalculating density: the role
of habitat geometry. Ornis Scandinavica, 19, 88 –92.
He, F., LaFrankie, J.V. & Song, B. (2002) Scale dependence of tree
abundance and richness in a tropical rain forest, Malaysia.
Landscape Ecology, 17, 559–568.
Holdenrieder, O., Pautasso, M., Weisberg, P.J. & Lonsdale, D.
(2004) Tree diseases and landscape processes: the challenge of
landscape pathology. Trends in Ecology & Evolution, 19, 446–452.
Hoskins, W.G. (1955) The making of the English landscape.
Hodder & Stoughton, London.
Huang, J., Lu, X.X. & Sellers, J.M. (2007) A global comparative
analysis of urban form: applying spatial metrics and remote
sensing. Landscape & Urban Planning, in press.
Imhoff, M.L., Bounoua, L., Ricketts, T., Loucks, C., Harriss, R. &
Lawrence, W.T. (2004) Global patterns in human consumption
of net primary production. Nature, 24, 870–873.
Kain, R.J.P., Chapman, J. & Oliver, R.R. (2004) The enclosure
maps of England and Wales 1595–1918. Cambridge University
Press, Cambridge.
Karanth, K.U. & Nichols, J.D. (1998) Estimation of tiger densities
in India using photographic captures and recaptures. Ecology,
79, 2852–2862.
Kauppi, P.E., Ausubel, J.H., Fang, J., Mather, A.S., Sedjo, R.A. &
Waggoner, P.E. (2006) Returning forests analyzed with the forest
identity. Proceedings of the National Academy of Sciences USA,
103, 17574–17579.
Keeling, H.C. & Phillips, O.L. (2007) The global relationship
between forest productivity and biomass. Global Ecology and
Biogeography, 16, 618–631.
Kolb, A., Frank, B. & Diekmann, M. (2006) Determinants of
local abundance and range size in forest vascular plants. Global
Ecology and Biogeography, 15, 237–247.
Luck, G.W. (2007) The relationships between net primary productivity, human population density and species conservation.
Journal of Biogeography, 34, 201–212.
Lutz, W., Testa, M.R. & Penn, D.J. (2006) Population density is a
key factor in declining human fertility. Population & Environment, 28, 69 –81.
© 2007 The Authors
Global Ecology and Biogeography, 17, 203–210, Journal compilation © 2007 Blackwell Publishing Ltd
209
M. Pautasso and P. J. Weisberg
MacKenzie, D.I., Nichols, J.D., Royle, J.A., Pollock, K.H., Bailey,
L.L. & Hines, J.E. (2006) Occupancy estimation and modelling.
Inferring patterns and dynamics of species occurrence. Elsevier,
Amsterdam.
Mahmassani, H.S., Baaj, M.H. & Tong, C.C. (1988) Characterization and evolution of spatial density patterns in urban areas.
Transportation, 15, 233 –256.
Massey, D. & Stephen, E. (1977) The size-density hypothesis in
Great Britain: analysis of a deviant case. Demography, 14, 351–361.
Matter, S.F. (2003) Modeling the density-area relationship in a
dynamic landscape: an examination for the beetle Tetraopes
tetraophthalmus and a generalized model. Ecological Modelling,
169, 103 –117.
Matter, S.F., Roland, J., Keyghobadi, N. & Sabourin, K. (2003)
The effects of isolation, habitat area and resources on the
abundance, density and movement of the butterfly Parnassius
smintheus. American Midland Naturalist, 150, 26 – 36.
McGill, B.J. (2006) A renaissance in the study of abundance.
Science, 314, 770 –772.
McKinney, M.L. (2002) Do human activities raise species richness? Contrasting patterns in United States plants and fishes.
Global Ecology and Biogeography, 11, 343 – 348.
McKinney, M.L. (2005) Scaling of park trail length and visitation
with park area: conservation implications. Animal Conservation, 8, 135 –141.
McNaughton, S.J. & Wolf, L.L. (1973) General ecology. Holt,
Rinehart & Winston, New York.
Nee, S. & Cotgreave, P. (2002) Does the species/area relationship
account for the density/area relationship? Oikos, 99, 545–551.
Pandit, R. & Laband, D.N. (2007) Threatened species and the
spatial concentration of humans. Biodiversity and Conversation,
16, 235 –244.
Pautasso, M. & Gaston, K.J. (2006) A test of the mechanisms
behind avian generalized individuals–area relationships.
Global Ecology and Biogeography, 15, 303 – 317.
Pautasso, M. (2007) Scale-dependence of the correlation
between human population presence and vertebrate and plant
species richness. Ecology Letters, 10, 16 – 24.
Pennington, M. (1983) Efficient estimators of abundance, for
fish and plankton surveys. Biometrics, 39, 281–286.
Pisani, C. (2002) The estimation of biological population size at
large scale by incomplete area surveys and replicated counts.
Environmetrics, 13, 155 –166.
Pollock, K.H., Nichols, J.D., Simons, T.R., Farnsworth, G.L.,
Bailey, L.L. & Sauer, J.R. (2002) Large scale wildlife monitoring
studies: statistical methods for design and analysis. Environmetrics, 13, 105 –119.
Pottier, N., Penning-Rowsell, E., Tunstall, S. & Hubert, G. (2005)
Land use and flood protection: contrasting approaches and
outcomes in France and in England and Wales. Applied Geography 25, 1–27.
Rosso, R., Bacchi, B. & LaBarbera, P. (1991) Fractal relation of
mainstream length to catchment-area in river networks. Water
Resources Research, 27, 381–387.
Scherner, E.R. (1981) Die Flächengrösse als Fehlerquelle bei
Brutvogel-Bestandsaufnahmen. Ökologie der Vögel, 3, 145–175.
210
Schonewald-Cox, C., Azari, R. & Blume, S. (1991) Scale, variable
density, and conservation planning for mammalian carnivores.
Conservation Biology, 5, 491–495.
Smallwood, K.S. & Morrison, M.L. (1999) Spatial scaling of pocket
gopher (Geomyidae) density. Southwestern Naturalist, 44, 73–82.
Smallwood, K.S. & Smith, T.R. (2001) Study design and interpretation of shrew (Sorex) density estimates. Annales Zoologici
Fennici, 38, 149–161.
Stephen, E. (1971) Variation in county size: a theory of segmental
growth. American Sociological Review, 36, 451–461.
Stephen, E. (1972) International tests of the size-density hypothesis.
American Sociological Review, 37, 365–368.
Stewart, J.Q. & Warntz, W. (1958) Physics of population distribution. Journal of Regional Science, 1, 90–123.
Stohlgren, T.J., Barnett, D., Flather, C., Kartesz, J. & Peterjohn, B.
(2005) Plant species invasions along the latitudinal gradient in
the United States. Ecology, 86, 2298–2309.
Sutherland, W. (ed.) (2006) Ecological census techniques: a handbook. Cambridge University Press, Cambridge.
Sutton, P., Roberts, D., Elvidge, C. & Baugh, K. (2001) Census
from Heaven: an estimate of the global human population
using night-time satellite imagery. International Journal of
Remote Sensing, 22, 3061–3076.
Terrell, J.E. (2006) Human biogeography: evidence of our place
in nature. Journal of Biogeography, 33, 2088–2098.
Thompson, W.L., White, G.C. & Gowan, C. (1998) Monitoring
vertebrate populations. Academic Press, London.
Tobler, W., Deichmann, U., Gottsegen, J. & Maloy, K. (1997)
World population in a grid of spherical quadrilaters. International Journal of Population Geography, 3, 203–225.
Watts, R.D., Compton, R.W., McCammon, J.H., Rich, C.L.,
Wright, S.M., Owens, T. & Ouren, D.S. (2007) Roadless space
of the conterminous United States. Science, 316: 736–737.
Wells, T.C.E. (1965) Changes in the botanical composition of a
sown pasture on the chalk in Kent 1956–64. Journal of the
British Grassland Society, 22, 277–281.
Witmer, G.W. (2005) Wildlife population monitoring: some
practical considerations. Wildlife Research, 32, 259–263.
Wright, D.H. (1991) Correlation between incidence and abundance
are expected by chance. Journal of Biogeography, 18, 463–466.
Wright, S.J. & Muller-Landau, H.C. (2006) The future of tropical
forest species. Biotropica, 38, 287–301.
BIOSKETCHES
Marco Pautasso is a researcher interested in network
epidemiology, landscape pathology and macroecology.
Peter J. Weisberg is an ecologist interested in broadscale vegetation dynamics, including natural disturbances
such as fire and floods, invasive plant species and plant
community response to resource management practices
and altered disturbance regimes. His methodological
focus includes spatial modelling and scaling issues.
Editor: José Alexandre F. Diniz-Filho
© 2007 The Authors
Global Ecology and Biogeography, 17, 203–210, Journal compilation © 2007 Blackwell Publishing Ltd