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Basic Appl. Ecol. 3, 1–13 (2002)
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Basic and Applied Ecology
Occupancy-abundance relationships and spatial distribution: A review
Alison R. Holt1,*, Kevin J. Gaston1, Fangliang He2
1
2
Biodiversity and Macroecology Group, Department of Animal and Plant Sciences, University of Sheffield, Sheffield, UK
Canadian Forest Service, Pacific Forestry Centre, 506 West Burnside Road, Victoria, B.C., V8Z 1M5, Canada
Received March 16, 2001 · Accepted July 31, 2001
Abstract
One of the most general patterns in community ecology is the positive relationship between the
number of sites or areas in which a species in a taxonomic assemblage occurs regionally and its local
abundance. A number of hypotheses have been proposed to explain this interspecific occupancyabundance relationship, but it has recently been argued that the pattern is most profitably viewed as
a consequence of the spatial distribution of the individuals of each species. In this paper we explore
the link between spatial distribution and the occupancy-abundance relationship, with particular
reference to statistical models that have been suggested to describe the pattern, and discuss its connections with a broad understanding of how organisms are distributed in space. A range of models
describe observed occupancy-abundance relationships reasonably well, but are commonly not well
differentiated over the range of abundances implicit in such relationships. There is little evidence
that species exhibit great commonality in the form of their aggregative behaviour, but this does not
matter in terms of the generation of a positive interspecific occupancy-abundance relationship.
Eines der allgemeinsten Muster in der Ökologie der Lebensgemeinschaften ist die Beziehung zwischen
der Anzahl der Standorte oder Gebiete, an oder in denen Arten einer taxonomischen Gruppe vorkommmen, und ihrer lokalen Abundanz. Es wurde eine Anzahl von Hypothesen vorgeschlagen, um diese
interspezifische Anwesenheits-Abundanz-Beziehung zu erklären. In letzter Zeit wurde jedoch angeführt, dass dieses Muster mit dem besten Ergebnis als eine Folge der räumlichen Verteilung der Individuen einer jeden Art zu sehen ist. In diesem Review erkunden wir die Verbindung zwischen der
räumlichen Verteilung und der Anwesenheits-Abundanz-Beziehung, wobei besonderer Wert auf die
statistischen Modelle gelegt wird. Zudem erörtern wir in welcher Beziehung sie zu einem breiten Verständnis der Verteilung der Organismen im Raum steht. Eine Reihe von Modellen beschreibt die Anwesenheits-Abundanz-Beziehung relativ gut; sie sind aber meistens über den Bereich der Abundanzen,
die in diesen Beziehungen vorkommen, nicht ausreichend differenziert. Es gibt nur wenige Beweise
dafür, dass die Arten eine große Gemeinsamkeit in der Form ihres aggregierenden Verhaltens
aufweisen. Das spielt jedoch keine Rolle in Beziehung auf die Erzeugung einer positiven interspezifischen Anwesenheits-Abundanz-Beziehung.
Key words: Occupancy-abundance relationships – aggregation – macroecology
*Corresponding author: Alison R. Holt, Biodiversity and Macroecology Group, Department of Animal and Plant Sciences,
University of Sheffield, Sheffield, S10 2TN, UK, Phone: ++44-114-222 0034, Fax: ++44-114-222 0002,
E-mail: [email protected]
1439-1791/02/03/01-001 $ 15.00/0
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A. R. Holt et al.
Introduction
The population size and the extent of the occupancy of
a region by a species are correlated, such that there is a
positive interspecific occupancy-abundance relationship. Total population size typically rises faster than
does occupancy, such that more widely distributed
species have higher local densities at the sites at which
they occur than do those more restricted in their distribution (Hanski 1982, Brown 1984, Gaston & Lawton
1990, Hanski et al. 1993, Gaston 1994, 1996). This
pattern has been documented in a large and rapidly
growing number of empirical studies, for groups as diverse as plants (Gotelli & Simberloff 1987, Collins &
Glenn 1990, 1997, Rees 1995, Boecken & Shachak
1998, Thompson et al. 1998, Guo et al. 2000, He &
Gaston 2000a, van Rensburg et al. 2000), spiders (Pettersson 1997), grasshoppers (Kemp 1992, Collins &
Glenn 1997), scale insects (Kozár 1995), hoverflies
(Owen & Gilbert 1989), bumblebees (Obeso 1992,
Durrer & Schmid-Hempel 1995), macro-moths (Gaston 1988, Inkinen 1994, Quinn et al. 1997), butterflies (Hanski et al. 1993, Hughes 2000), beetles (Nilsson et al. 1994, van Rensburg et al. 2000), brackenfeeding insects (Gaston & Lawton 1988a, 1988b),
frogs (B.R. Murray et al. 1998), birds (Fuller 1982,
Hengeveld & Haeck 1982, O’Connor & Shrubb
1986, O’Connor 1987, Gaston & Lawton 1990,
Sutherland & Baillie 1993, Gregory 1995, Gaston &
Blackburn 1996, Blackburn et al. 1997a, 1997b,
1998, Collins & Glenn 1997, Gaston et al. 1997a,
1998a, Elmberg et al. 2000, He & Gaston 2000a, Linder et al. 2000, van Rensburg et al. 2000), and mammals (Brown 1984, Blackburn et al. 1997a, Collins &
Glenn 1997, Johnson 1998). Indeed, few other patterns in community ecology have been found to exhibit such a high level of generality (the species-area relationship being one obvious exception).
Several mechanisms have been proposed to explain
the positive interspecific relationship between local
abundance and regional occupancy (reviewed by Gaston et al. 1997b, 2000, Gaston & Blackburn 2000).
Those that are ecological can be classified very broadly
as range position, resource, and population dynamic
explanations, although they are not necessarily mutually exclusive or even independent (some may simply constitute different levels of explanation from others; Gaston et al. 2000). Range position explanations are based
on the location of a study region relative to the geographic ranges of species, resource explanations concern the effects of resource availability and breadth of
use on the abundances and distributions of species, and
population dynamic explanations reflect the possible
consequences of local population colonisation, growth
and extinction. All such postulated mechanisms have
Basic Appl. Ecol. 3, 1 (2002)
been subject to some empirical scrutiny, and gain some
support from at least a subset of studies (e.g. Burgman
1989, Brown 1995, Gaston et al. 1997b, Warren &
Gaston 1987, Hanski et al. 1993, Gonzalez et al. 1998,
B.R. Murray et al. 1998, Thompson et al. 1998, 1999,
Gregory & Gaston 2000, Guo et al. 2000, Hughes
2000). Indeed, theory and empirical evidence strongly
suggest that positive occupancy-abundance relationships result from the action of several mechanisms, and
that in different systems these vary in their relative importance. Such a conclusion is regarded by many ecologists as being rather unsatisfactory; the explanation of
patterns in terms of single general mechanisms is undoubtedly more elegant. However, a number of
macroecological patterns are increasingly seen as being
best understood as the net outcome of several processes
that pull in essentially the same direction (Gaston 2000,
Gaston & Blackburn 2000, Lawton 2000).
Whilst it is important to seek strictly ecological explanations for the occupancy-abundance relationship,
it is also clear that such a pattern is expected from a
variety of models of the spatial distribution of individuals (Wright 1991, Hanski et al. 1993, Hartley 1998,
He & Gaston 2000a, b). Indeed, it is rather intuitive
that occupancy does not only depend on species abundance but also on spatial distribution as well. Thus, although most studies emphasise the ecological processes that would regulate species abundance, any factors
that would affect species distribution in space would
be equally important in explaining occupancy patterns
(He et al. in press). For a given total population size, a
species experiencing intense aggregation will obviously have a lower level of occupancy than will a less aggregated species. Equally, if two species aggregate in a
broadly similar way not only will the one with the
greater regional population size occupy more sites but
it will also have a higher density per site. The extent to
which spatial distribution strictly provides an explanation of interspecific occupancy-abundance relationships, rather than simply a rephrasing of one macroecological pattern in terms of another, is debatable
(Gaston et al. 1998a, Hartley 1998). Nonetheless, the
link between spatial distribution and interspecific occupancy-abundance relationships is undoubtedly of
potential importance.
If interspecific occupancy-abundance relationships
are usefully regarded in terms of the spatial distribution of individuals, then this argument would seem to
apply yet more forcefully to intraspecific occupancyabundance relationships. Positive intraspecific relationships between occupancy and abundance, or evidence strongly suggestive of their existence, have been
documented by a number of studies. These include investigations of plants (Boecken & Shachak 1998), butterflies (Pollard et al. 1995, van Swaay 1995), fish
Occupancy-abundance relationships and spatial distributions: A review
(Winters & Wheeler 1985, Crecco & Overholtz 1990,
MacCall 1990, Rose & Leggett 1991, Swain & Wade
1993, Swain & Sinclair 1994), and birds (Gibbons et
al. 1993, Smith et al. 1993, Ambrose 1994, Tucker &
Heath 1994, Fuller et al. 1995, Hinsley et al. 1996,
Cade & Woods 1997, Gaston et al. 1997b, 1998b,
Newton 1997, Blackburn et al. 1998, Donald & Fuller
1998, Gaston & Curnutt 1998, Venier & Fahrig 1998,
Tellería & Santos 1999). Unlike virtually all interspecific relationships, in the vast majority of cases intraspecific occupancy-abundance relationships are
based on time series data, with each pair of values of
occupancy and mean local abundance being calculated
for a different season or year. Analyses based on values
calculated in different areas of a species’ geographic
range at the same time (or approximately so) are
scarce (but see Venier & Fahrig 1998), but it seems
reasonable to assume that intraspecific equivalents of
interspecific relationships do exist, albeit with some
differences in their detailed form (see below).
Nonetheless, the link between such patterns and the
spatial aggregation of individuals has been little better
explored than for interspecific occupancy-abundance
relationships.
In this paper, we review and summarise the link between occupancy-abundance relationships and spatial
distribution. We do so by introducing a variety of statistical models that describe the occupancy-abundance
relationship and the possible spatial conditions under
which these models hold.
Occupancy-abundance models
A number of different models have been proposed to
describe occupancy-abundance relationships, albeit
seldom in the context of the macroecological patterns
of primary concern in this paper. Here we outline
some of the more significant of these models.
(i) Poisson
The most basic distributional model of this kind follows from a Poisson distribution, where individuals of
a species are distributed in space at random. Here
p = 1 – e–µ
(1)
where p is the proportion of sites (or areas) occupied,
and µ is the mean abundance of the species across all
sites, occupied or otherwise (Wright 1991). If n individuals of a species are randomly distributed among M
sites, of which m sites are occupied, p = m/M and µ =
n/M. Sometimes, interspecific occupancy-abundance
relationships, and to a lesser extent intraspecific relationships, are reported with abundance averaged over
3
M
occupied sites only (µ′), such that µ′ = –– µ, and
m
p =1 – e–pµ′. The change in the definition of density
makes no qualitative difference to the occupancyabundance relationship, and this will also be true for
the other models that follow (He et al. in press). It
does, however, result in a shallower increase in density
with increasing occupancy, until full occupancy is
achieved.
The Poisson distribution predicts a relationship between µ and the variance in abundance amongst sites,
σ2, of the form
σ2 = µ
(2)
In consequence, departure of the ratio of the variance to the mean (σ 2/m) from a value of 1 has commonly been employed as a test of whether or not a
species is approximately randomly distributed (Collier
et al. 1973, McLean & Ivimey-Cook 1973, Brower &
Zar 1977, Elliot 1977, Southwood 1978, Greig-Smith
1983, Sokal & Rohlf 1995, Zar 1999). If the ratio is
less than one then the distribution is regarded as overdispersed, whilst a ratio of more than one would be
seen as indicating an aggregated (or contagious) distribution. However, whilst the mean and variance are
equal in a Poisson distribution, this is not the only circumstance under which a variance to mean ratio of
one can arise (Pielou 1974, Hurlbert 1990, Dale
1999). In fact there are an infinite number of such distributions (Hurlbert 1990).
In practice, the individuals of a species are seldom
randomly distributed in space, except when they are
very scarce (e.g. Pielou 1977, Taylor et al. 1978,
Greig-Smith 1983, Gaston 1994, Brown et al. 1995,
Hinsley et al. 1996, Venier & Fahrig 1998). This observation can be viewed in two distinct ways. First, because a random distribution is only one of a continuous spectrum of possible patterns of distribution it
may be exceedingly unlikely to occur simply on a
probabilistic basis (Taylor 1961). Second, random distributions may be scarce because there are numerous
abiotic and biotic reasons why species are unlikely to
be distributed in this fashion. The latter seems the
more likely in most cases, but the former should not be
entirely discounted.
(ii) Negative binomial
The statistical distribution most frequently used to
model aggregated patterns of spatial occurrence, and
that against which the fit of very many data sets has
been tested, is the negative binomial distribution (e.g.
Boswell & Patil 1970, Desouhant et al. 1998). Here
µ –k
p=1– 1+–
(3)
k
( )
Basic Appl. Ecol. 3, 1 (2002)
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A. R. Holt et al.
where k is a clumping parameter, with small values
representing strong aggregation moving towards a
random spatial distribution as values of k increase.
Thus, as k increases a species would exhibit increasing
occupancy for a given mean abundance, up to the level
given by eq. (1).
The parameter k is defined to be strictly positive,
but in (3) it could take negative values. When k is negative, model (3) is in fact derived from a positive binomial distribution that describes a regular distribution
of organisms (He & Gaston 2000b). Therefore, the effective domain of k in model (3) is (–∞, –µ) and (0,
+∞), i.e., the left-hand domain describes a regular distribution of organisms whereas the right-hand domain
is an aggregated distribution.
The negative binomial distribution predicts a relationship between µ and σ2, of the form
µ2
σ 2 = µ + –– .
k
(4)
(Routledge & Swartz 1991, Perry & Woiwod 1992,
Gaston & McArdle 1994).
Whilst a large number of possible causal derivations
of the negative binomial have been identified, most
based on the compounding of random processes (e.g.
Boswell & Patil 1970, Taylor 1984), the suitability of
the negative binomial as a general descriptor of the
spatial distributions of species has been much debated.
Even where the negative binomial provides a reasonable fit to observed distributions of abundances, the
appropriate values of k are dependent on the mean
density. This has principally been demonstrated at
small spatial scales but holds more broadly (Finch
et al. 1975, Taylor et al. 1978, 1979, Nachman 1981,
Taylor 1984, Perry & Taylor 1985, 1986, Shorrocks
& Rosewell 1986, Hassell et al. 1987, Rosewell et al.
1990, Feng et al. 1993), and has also been demonstrated using other measures of aggregation (e.g. He et al.
1997, Plotkin et al. 2000).
Indeed, overall, it has been found that no one distribution describes the spatial variation in the abundance
of a single species distribution throughout a wide
range of densities (Taylor et al. 1978, Taylor 1984).
Species can broadly be characterised as typically moving through Poisson, to negative binomial then lognormal distributions with increasing mean abundance, although none of the standard distributions seem to fit
well to some observed patterns of spatial variation in
abundance (McGuire et al. 1957, Brown & Cameron
1982, Perry & Taylor 1985).
(iii) Nachman
With the exception of the Poisson and negative binomial models, other occupancy-abundance models have
Basic Appl. Ecol. 3, 1 (2002)
been proposed largely as empirical descriptors of the
observed patterns. The first of these was originally
suggested by Nachman (1981), and takes the form
p = 1 – e–α µ
β
(5)
or
(6)
where α and β are two positive parameters. More
commonly, (5) is presented as
(7)
This model is suggested as an empirical form of the
Poisson model with α and β ≠ 1 representing a departure from the Poisson distribution (He & Gaston
2000b). Principally the model is used to predict the
densities on crops of pest species from data on their
occupancy of sampling units (plants, tillers, leaves etc.
see Nachman 1981, 1984, Kuno 1986, 1991, Ward et
al. 1986, Ekbom 1987, Perry 1987, Hepworth &
MacFarlane 1992, Feng et al. 1993). On several occasions it has been argued to provide the best available
approach to this problem, and has also found application in modelling host-parasitoid relationships (Perry
1987). Its application at broader scales has also been
discussed (e.g. Gaston 1994, 1999).
The Nachman model predicts a relationship between µ and σ2 of the form
σ2 = α µβ
(8)
This is Taylor’s power function (Taylor 1961). It is
the model that has been used most widely to describe
intraspecific mean-variance relationships in empirical
abundance data, although its suitability has repeatedly been challenged (Taylor 1984, Sawyer 1989, Routledge & Swartz 1991). Crucial to the argument is the
role of sampling error in estimates of variability,
because this may have a substantial influence on the
shape of the relationship that is observed; methods of
removing the effects of sampling error are receiving
growing interest (McArdle & Gaston 1995). A number of models have been proposed that give rise to
power relationships between the mean and the variance of abundances, with variable convergence on
observed parameter values (e.g. Taylor & Taylor
1977, Hanski 1980, 1987, Anderson et al. 1982, Taylor et al. 1983, Binns 1986, Gillis et al. 1986, Perry
1988, Yamamura 1990). Some of these models have
been predominantly rooted in species behaviour and
others in demographics (that of Binns (1986) is based
partially on the negative binomial). Yamamura
(1990) derived a simple model of population growth
and dispersal that gives rise to both equations (7)
and (8).
Occupancy-abundance relationships and spatial distributions: A review
(iv) Hanski-Gyllenberg
Several studies have explored the connection between
the occupancy-abundance and the species-area relationship (Hanski & Gyllenberg 1997, Leitner &
Rosenzweig 1997, Ney-Nifle & Mangel 1999). In so
doing, Hanski & Gyllenberg (1997) use a logistic
model to describe the former, such that
(9)
or
(10)
zweig (1997) suggest a simple power model for occupancy-abundance relationships, such that
p = α µβ
(14)
log p = log α + β log µ
(15)
or
where α is positive and β is a scale parameter.
This model is an empirical form for species to follow a (positive) binomial distribution that describes a
regular distribution of species. It has a variance-mean
relationship of the form
σ2 = α µβ (1 – α µβ)
or
(11)
and
(12)
where α and β are two positive parameters.
The possible spatial distribution behind this model,
as He & Gaston (2000b) point out, is that a species
follows a geometric distribution. The departure from
the geometric model is captured by α and β ≠ 1. For
this model, the relationship between the spatial variance in abundance and mean abundance takes the
form
σ2 = α µβ (1 + α µβ)
(13)
which is larger than the variance in Taylor’s power
model (8), suggesting that the Hanski-Gyllenberg
model is appropriate to describe patterns for species
having stronger aggregation than that under the Nachman model. In other words, the occupancy of model
(5) is larger than that of model (10); this is easily
shown to be true by examining the difference of the
two models ((5)–(10) is always >0).
(v) Power model
Also in the context of exploring the form and determinants of species-area relationships, Leitner & Rosen-
5
(16)
with α µβ < 1, which is a natural condition given that
equation (14) is a proportion. Compared with the
Nachman model (5), the power model is suitable for
species of less aggregated or regular distribution. It can
also be shown that the occupancy of model (14) is
larger than that of model (5), i.e., the difference between (14) and (5) is not less than zero (given
α µβ < 1).
(vi) He-Gaston
Confronted with this profusion of occupancy-abundance models, He et al. (in press) identify a general
model for the relationship, of which the others are special cases, whether their roots are in a recognised statistical distribution or are purely empirical. This model
takes the form
(17)
or
(18)
where α is a positive parameter, β is a scale parameter
(Figure 1), and k is a real parameter defined in the domain of (–∞, –α µβ) or (0, +∞) (He et al. in press). For a
given overall number of individuals and k greater than
zero, as k increases occupancy increases (Figure 2).
Table 1. The He-Gaston occupancy-abundance model and its special forms.
He-Gaston model
αµβ
p = 1 – 1 + ––––
k
(
)
–k
Parameter conditions
Special models
k → ±∞, α = β = 1
(i) Poisson model:
p = 1 – e–µ
α=β=1
(ii) Negative binomial:
µ
p = 1 – 1 + ––
k
k → ±∞
(iii) Nachman model:
k=1
(iv) Hanski-Gyllenberg:
k = –1
(v) Power model:
( )
–k
β
p = 1 – e–αµ
αµβ
p = –––––––
1 +αµβ
p = αµβ
Basic Appl. Ecol. 3, 1 (2002)
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A. R. Holt et al.
For a given overall number of individuals and k less
than zero, occupancy declines as k assumes progressively greater negative values.
When k → ±∞ and α = β = 1, equation (17) is the
Poisson model, when α = β = 1 it is the negative binomial model, when k → ±∞ it is the Nachman model,
when k = 1 it is the Hanski-Gyllenberg model, and
when k = –1 it is the power model (Table 1; He et al. in
press). The corresponding variance-mean relationships
for these various models can accordingly be derived
from the following generalised form
(19)
(vii) Others
Yet more occupancy-abundance models can be proposed although they need not necessarily belong to the
He-Gaston family of models. Here we consider two
such models that have also been used to model occupancy-density relationships in the literature (Ward et
al. 1986, see also Ekbom 1987). The first is the probit
model
p = probit–1 (α + β log µ)
(20)
log µ = α + β probit (p)
(21)
or
where probit (p) is the inverse of the cumulative normal distribution. This model is based on the assumption that µ follows a lognormal distribution.
The second additional model that has been used to
describe occupancy-abundance relationships is the extreme value model of the form
Fig. 1. The He-Gaston model, showing (a) the dependence of the probability
of occurrence (p) on the scale parameter β given that α = 1 and k = 1, and
(b) the dependence of the probability on the spatial parameter k given that
α = 1 and β = 1.
p = 1 – exp[–exp(α + β log µ)]
(22)
(Wilson & Room 1983, see also Ekbom 1987). With a
reparameterisation, it is easy to show that this model
is in fact the Nachman model in a different guise.
Parameter estimation and goodness-of-fit
Because the occurrence of a species in a particular
sample site is a binary (presence/absence) variate, it is
appropriate to assume that the probability of the
species occupying y out of n total number of sites in a
study area follows a binomial distribution
(23)
Density (µ)
Fig. 2. The contour map of the He-Gaston model. The values on the isolines
are the occupancy p.
Basic Appl. Ecol. 3, 1 (2002)
where p is the occupancy defined by the occupancyabundance models such as (1), (3), (5), (9), (14), (17)
or (20), respectively.
Given an assemblage of species, each of them has a
binomial distribution as given by (23). Thus, the esti-
Occupancy-abundance relationships and spatial distributions: A review
mation of parameters for each of these occupancyabundance models is straightforward, simply by maximising the log-likelihood function
(24)
where the subscript denotes the ith of s species in the
assemblage.
The goodness-of-fit of each model is assessed by
comparing the deviance with a χ2 distribution. The deviance is defined as twice the difference between the
maximum log-likelihood achievable in (24) and the
log-likelihood for the model of interest (i.e., one of the
above occupancy models). The maximum log-likelihood achievable (i.e., the full model) is that obtained
by substituting the observed pi into equation (24). If
the model of interest is adequate, the deviance should
be small compared to the χ2s–l, where s is the total number of species and l is the number of parameters in the
model under study. Roughly, the model is considered
adequate if the deviance is smaller than the degrees of
freedom s – l. However, our previous experiences suggest that occupancy data are typically over-dispersed,
thus the χ2 test for goodness-of-fit is usually unreliable
(He et al. in press). Here we are more interested in the
relative performances of the models than a rigorous
statistical test. We therefore use the sum of absolute
differences between the observed proportions of occurrence (pi) and the fitted probabilities ( p̂i) as a goodness-of-fit
.
Intraspecific and interspecific relationships
Several of the six primary models of occupancy-abundance relationships listed above [(i)–(vi)] were originally formulated, and have principally been discussed,
in the context of the occupancy and abundance of single species. Their adequacy for describing interspecific
patterns has not been fully investigated, particularly at
macroecological scales. Plainly some are appropriate
under some circumstances (e.g. Poisson, Negative binomial, Nachman).
Positive empirical intraspecific occupancy-abundance relationships are commonly rather weak, there
are many occasions on which no significant relationships exist, and negative relationships occur at a moderate frequency (e.g. see Ambrose 1994, Marshall &
Frank 1994, Swain & Morin 1996, Blackburn et al.
1998, Boecken & Shachak 1998, Donald & Fuller
1998, Gaston & Curnutt 1998, Gaston et al. 1998b).
This might be expected on two grounds. First, the
range of variation in the mean abundances of individual species is typically rather narrow (particularly for
vertebrates) when contrasted with that between
7
species. Thus, even small to moderate variation around
an underlying occupancy-abundance relationship may
be sufficient to mask its existence. Second, even if occupancy and abundance are linked there may be time lags
in the response of occupancy to increases and decreases
in mean abundance (given that this seems the most
likely directionality of a causal relationship between
the two) which again serve to mask the relationship
(Gaston et al. 1998b). For example, reductions in mean
abundance may not result in the immediate reduction
of occupancy, even if sufficient individuals are no
longer present in sample sites for local populations to
persist there in the longer term.
How such intraspecific occupancy-abundance patterns will translate into, the stronger, interspecific patterns is not clear. Two extreme scenarios can perhaps
be envisaged. If species exhibit sufficient commonality
in the underlying relationships between occupancy
and abundance, even if these relationships are often
masked at the intraspecific level, an interspecific occupancy-abundance relationship may be apparent. Alternatively, the lack of any consistent occupancy-abundance relationship for individual species, and the variety of responses of occupancy to increasing abundance, may mean that no interspecific occupancyabundance relationship is apparent.
Of course, we know that positive interspecific occupancy-abundance relationships are common. So the
important question becomes whether these can be regarded as reflecting sufficient commonality between
species in their aggregative behaviour or whether they
result from some other process.
Whilst the limit to which most published analyses
have been carried is the determination of a correlation
coefficient for logarithmically transformed data, all six
of the primary occupancy-abundance models have, at
some time, been fitted to interspecific data at macroecological scales (in so doing, concerns over the potential
non-independence of species as data points, because of
phylogenetic relatedness, have largely been ignored –
both occupancy and abundance exhibit little phylogenetic constraint, and controlling for such effects has
been found to make little difference to observed patterns; Gaston & Blackburn 2000). Gaston et al. (1998a)
examined the fit of the negative binomial model to data
on the occurrence and the estimated overall population
size of breeding birds in Britain. In the absence of detailed information on spatial patterns of abundance,
they chose values of k at random between values of 0.1
and 4 for each species. For 100 such sets of k-values, the
simulated occupancy-abundance relationships bore
some marked similarities to the real relationship, but at
low population sizes species were more widespread than
predicted and as population size increased they rapidly
became less widespread than predicted. Significantly,
Basic Appl. Ecol. 3, 1 (2002)
8
A. R. Holt et al.
Table 2. Deviance and the absolute difference between the model predictions and the observed proportions of occurrence for comparing the eight occupancy-abundance models for the three bird data sets (see text for details). The models were fitted using the maximum likelihood method.
South-east Scotland
Deviance
df of chi-square
Mean absolute difference
Poisson
7732754
131
0.345829
Power
15940.32
129
0.075894
NBD
16437.44
130
0.082359
Logistic
9663.039
129
0.056499
Nachman
10552.51
129
0.059036
Probit
9600.44
129
0.05608
He-Gaston
9603.058
128
0.056635
Poisson
534767.4
109
0.278118
Power
5936.32
107
0.114866
NBD
3603.662
108
0.083092
Logistic
2614.565
107
0.066555
Nachman
3433.491
107
0.077773
Probit
2585.313
107
0.066507
He-Gaston
2448.985
106
0.063431
Poisson
58708.22
138
0.164297
Power
10835.76
136
0.177248
NBD
4318.067
137
0.095107
Logistic
3333.849
136
0.075219
Nachman
3804.081
136
0.081751
Probit
3328.494
136
0.075949
He-Gaston
3333.508
135
0.07531
Hertfordshire
Deviance
df of chi-square
Mean absolute difference
Lake Constance
Deviance
df of chi-square
Mean absolute difference
over the wide range of population levels represented by
these species (seven orders of magnitude), even substantial variation in the value of k makes only limited difference to the level of occupancy predicted by the negative
binomial model. This effect can also be seen in other
data sets (A.R. Holt, unpubl. analyses).
Implicit in the application of occupancy-abundance
models to interspecific data at broad or macroecological
spatial scales is the notion that all of the sites or areas included can potentially be occupied by individuals of
each species in an assemblage (Gaston et al. 1998a).
This is not always so, particularly if occurrences are
mapped at fine resolutions, when habitat associations
may become apparent, and over very large regions,
when differences in the limits of geographic ranges may
come into play. This effect may have contributed to mismatches between the predicted and observed occupancy-abundance relationships for birds in Britain. In such
cases, if the data are available, proportional occupancy
may best be calculated on the basis of a variable number
of potentially occupiable sites or areas.
The fit of the Nachman, Hanski-Gyllenberg and
He-Gaston models to two data sets was documented
by He & Gaston (2000a). For data on tree species in a
study plot in the Pasoh Forest Reserve of Malaysia,
they found that all three models described the observed occupancy-abundance pattern reasonably well,
for seven different spatial resolutions of mapping occupancy. For data on the breeding birds of Bedfordshire (a county in the UK), at two spatial resolutions, a
similar result was obtained. However, here the Nachman model was apparently superior to the other two,
which both underestimated occupancy at a given
abundance. In another study, He et al. (in press) compared the goodness-of-fit for all six of the primary occupancy-abundance models to the same two data sets.
Basic Appl. Ecol. 3, 1 (2002)
Overall, the three-parameter model gave the best fit to
the data although in some cases the gain in goodnessof-fit may not be significant enough to warrant the superiority of the three-parameter model to some other
two-parameter models such as the logistic or Nachman models. But they did find that the power model
(14) was consistently the least satisfactory.
The fits of all of the models discussed here (including (20) and (22), but (22) is the same as (5)) with regard to three additional data sets are given in Table 2.
These data sets were for the avian assemblages of: (i)
south-east Scotland (R.D. Murray et al. 1998) – 123
species distributed over 1756 tetrads, and mapped between 1988–1994; (ii) Hertfordshire, England (Smith
et al. 1993) – 109 species distributed over 409 tetrads,
and mapped between 1988–1992; and (iii) Lake Constance, Germany (Böhning-Gaese & Bauer 1996) –
141 species distributed over 303 tetrads, and mapped
between 1988–1991. For the first two data sets, the
occurrence records for species were categorised on the
basis of whether breeding was possible, probable or
confirmed. Only probable and confirmed records were
used in the analyses. In all data sets precise distributional data were not provided for some species, for instance for locally protected species such as the barn
owl Tyto alba. These species were not included in the
analysis. For the assemblage of south-east Scotland,
the only one for which it was relevant, seabirds, defined as those species breeding exclusively within
coastal tetrads, were excluded from the analysis.
For the data sets for the avian assemblages from
south-east Scotland, Hertfordshire and Lake Constance
the best fits are provided by the probit model and the
He-Gaston model (Figure 3). The probit model has the
advantage over the He-Gaston model of only having
two parameters, and that many major statistical soft-
Occupancy-abundance relationships and spatial distributions: A review
9
Fig. 3. The shapes of seven occupancy-abundance models fitted to data for the breeding birds of south-east Scotland, Hertfordshire and Lake Constance (see
text for details). The left-hand panel shows the shapes of the different functions, and the right-hand panel shows the observed data and the fit of the probit
and He-Gaston models.
ware packages have algorithms for parameter estimation. The disadvantage is that it cannot readily be related to the other models.
The central message that seems to arise from the
above studies is that a wide range of occupancy-abundance models do fit interspecific data, in some cases
rather well, but that these models are commonly not
well differentiated over the range of abundances implicit in such relationships. Because all of the models
describe certain spatial patterns in either explicit or
empirical manner, and they predict an increase in occupancy with increasing mean abundance, it would be
surprising if real assemblages did not show such a pattern. There is little evidence that species are exhibiting
any great commonality in the form of their aggregative
behaviour. But this does not matter in terms of the
generation of a positive interspecific relationship.
To reflect this observation, He et al. (in press) have
suggested that, the upper bound to interspecific occupancy-abundance relationships is defined by the Poisson model, and that, to a first approximation, the
lower bound is defined by the line p = µ/µmax. They observed this pattern in both the tropical rain forest of
Malaysia and passerine bird community in Bedfordshire, although a highly aggregated or dispersed
species may lie outside these limits. Similarly the avian
assemblages of south-east Scotland, Hertfordshire and
Lake Constance show this pattern (Figure 4).
Basic Appl. Ecol. 3, 1 (2002)
10
A. R. Holt et al.
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Fig. 4. The interspecific occupancy-abundance relationships of the avian assemblages in south-east Scotland, Hertfordshire and Lake Constance (see
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In short, whilst positive interspecific occupancyabundance relationships can be described in terms of
spatial distributions of species, these patterns of distribution can be very diverse and yet predict relationships that are not dissimilar from those actually observed.
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