Occupancy-abundance relationships and spatial distribution: A review

Alison R. Holt1,*, Kevin J. Gaston1, Fangliang He2

1 Biodiversity and Macroecology Group, Department of Animal and Plant Sciences, University of Sheffield, Sheffield, UK 2 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 thenumber of sites or areas in which a species in a taxonomic assemblage occurs regionally and its localabundance. A number of hypotheses have been proposed to explain this interspecific occupancy-abundance relationship, but it has recently been argued that the pattern is most profitably viewed asa consequence of the spatial distribution of the individuals of each species. In this paper we explorethe link between spatial distribution and the occupancy-abundance relationship, with particularreference to statistical models that have been suggested to describe the pattern, and discuss its con-nections with a broad understanding of how organisms are distributed in space. A range of modelsdescribe observed occupancy-abundance relationships reasonably well, but are commonly not welldifferentiated over the range of abundances implicit in such relationships. There is little evidencethat species exhibit great commonality in the form of their aggregative behaviour, but this does notmatter in terms of the generation of a positive interspecific occupancy-abundance relationship.

Eines der allgemeinsten Muster in der Ökologie der Lebensgemeinschaften ist die Beziehung zwischender Anzahl der Standorte oder Gebiete, an oder in denen Arten einer taxonomischen Gruppe vorkom-mmen, und ihrer lokalen Abundanz. Es wurde eine Anzahl von Hypothesen vorgeschlagen, um dieseinterspezifische Anwesenheits-Abundanz-Beziehung zu erklären. In letzter Zeit wurde jedoch ange-führt, dass dieses Muster mit dem besten Ergebnis als eine Folge der räumlichen Verteilung der Indi-viduen einer jeden Art zu sehen ist. In diesem Review erkunden wir die Verbindung zwischen derräumlichen Verteilung und der Anwesenheits-Abundanz-Beziehung, wobei besonderer Wert auf diestatistischen Modelle gelegt wird. Zudem erörtern wir in welcher Beziehung sie zu einem breiten Ver-ständnis der Verteilung der Organismen im Raum steht. Eine Reihe von Modellen beschreibt die An-wesenheits-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 Beweisedafür, dass die Arten eine große Gemeinsamkeit in der Form ihres aggregierenden Verhaltensaufweisen. Das spielt jedoch keine Rolle in Beziehung auf die Erzeugung einer positiven interspezifi-schen 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: a.holt@sheffield.ac.uk

1439-1791/02/03/01-001 $ 15.00/0

Basic Appl. Ecol. 3, 1–13 (2002)© Urban & Fischer Verlaghttp://www.urbanfischer.de/journals/baecol

Basic and Applied Ecology

Introduction

The population size and the extent of the occupancy ofa region by a species are correlated, such that there is apositive interspecific occupancy-abundance relation-ship. Total population size typically rises faster thandoes occupancy, such that more widely distributedspecies have higher local densities at the sites at whichthey occur than do those more restricted in their distri-bution (Hanski 1982, Brown 1984, Gaston & Lawton1990, Hanski et al. 1993, Gaston 1994, 1996). Thispattern has been documented in a large and rapidlygrowing number of empirical studies, for groups as di-verse as plants (Gotelli & Simberloff 1987, Collins &Glenn 1990, 1997, Rees 1995, Boecken & Shachak1998, Thompson et al. 1998, Guo et al. 2000, He &Gaston 2000a, van Rensburg et al. 2000), spiders (Pet-tersson 1997), grasshoppers (Kemp 1992, Collins &Glenn 1997), scale insects (Kozár 1995), hoverflies(Owen & Gilbert 1989), bumblebees (Obeso 1992,Durrer & Schmid-Hempel 1995), macro-moths (Gas-ton 1988, Inkinen 1994, Quinn et al. 1997), butter-flies (Hanski et al. 1993, Hughes 2000), beetles (Nils-son et al. 1994, van Rensburg et al. 2000), bracken-feeding insects (Gaston & Lawton 1988a, 1988b),frogs (B.R. Murray et al. 1998), birds (Fuller 1982,Hengeveld & Haeck 1982, O’Connor & Shrubb1986, 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, Lin-der et al. 2000, van Rensburg et al. 2000), and mam-mals (Brown 1984, Blackburn et al. 1997a, Collins &Glenn 1997, Johnson 1998). Indeed, few other pat-terns in community ecology have been found to exhib-it such a high level of generality (the species-area rela-tionship being one obvious exception).

Several mechanisms have been proposed to explainthe positive interspecific relationship between localabundance and regional occupancy (reviewed by Gas-ton et al. 1997b, 2000, Gaston & Blackburn 2000).Those that are ecological can be classified very broadlyas range position, resource, and population dynamicexplanations, although they are not necessarily mutual-ly exclusive or even independent (some may simply con-stitute different levels of explanation from others; Gas-ton et al. 2000). Range position explanations are basedon the location of a study region relative to the geo-graphic ranges of species, resource explanations con-cern the effects of resource availability and breadth ofuse on the abundances and distributions of species, andpopulation dynamic explanations reflect the possibleconsequences of local population colonisation, growthand extinction. All such postulated mechanisms have

been subject to some empirical scrutiny, and gain somesupport from at least a subset of studies (e.g. Burgman1989, 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, Hughes2000). Indeed, theory and empirical evidence stronglysuggest that positive occupancy-abundance relation-ships result from the action of several mechanisms, andthat in different systems these vary in their relative im-portance. Such a conclusion is regarded by many ecolo-gists as being rather unsatisfactory; the explanation ofpatterns in terms of single general mechanisms is un-doubtedly more elegant. However, a number ofmacroecological patterns are increasingly seen as beingbest understood as the net outcome of several processesthat pull in essentially the same direction (Gaston 2000,Gaston & Blackburn 2000, Lawton 2000).

Whilst it is important to seek strictly ecological ex-planations for the occupancy-abundance relationship,it is also clear that such a pattern is expected from avariety of models of the spatial distribution of individ-uals (Wright 1991, Hanski et al. 1993, Hartley 1998,He & Gaston 2000a, b). Indeed, it is rather intuitivethat occupancy does not only depend on species abun-dance but also on spatial distribution as well. Thus, al-though most studies emphasise the ecological process-es that would regulate species abundance, any factorsthat would affect species distribution in space wouldbe equally important in explaining occupancy patterns(He et al. in press). For a given total population size, aspecies experiencing intense aggregation will obvious-ly have a lower level of occupancy than will a less ag-gregated species. Equally, if two species aggregate in abroadly similar way not only will the one with thegreater regional population size occupy more sites butit will also have a higher density per site. The extent towhich spatial distribution strictly provides an explana-tion of interspecific occupancy-abundance relation-ships, rather than simply a rephrasing of one macroe-cological pattern in terms of another, is debatable(Gaston et al. 1998a, Hartley 1998). Nonetheless, thelink between spatial distribution and interspecific oc-cupancy-abundance relationships is undoubtedly ofpotential importance.

If interspecific occupancy-abundance relationshipsare usefully regarded in terms of the spatial distribu-tion of individuals, then this argument would seem toapply yet more forcefully to intraspecific occupancy-abundance relationships. Positive intraspecific rela-tionships between occupancy and abundance, or evi-dence strongly suggestive of their existence, have beendocumented by a number of studies. These include in-vestigations of plants (Boecken & Shachak 1998), but-terflies (Pollard et al. 1995, van Swaay 1995), fish

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Basic Appl. Ecol. 3, 1 (2002)

(Winters & Wheeler 1985, Crecco & Overholtz 1990,MacCall 1990, Rose & Leggett 1991, Swain & Wade1993, Swain & Sinclair 1994), and birds (Gibbons etal. 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 & Fuller1998, Gaston & Curnutt 1998, Venier & Fahrig 1998,Tellería & Santos 1999). Unlike virtually all interspe-cific relationships, in the vast majority of cases in-traspecific occupancy-abundance relationships arebased on time series data, with each pair of values ofoccupancy and mean local abundance being calculatedfor a different season or year. Analyses based on valuescalculated in different areas of a species’ geographicrange at the same time (or approximately so) arescarce (but see Venier & Fahrig 1998), but it seemsreasonable to assume that intraspecific equivalents ofinterspecific relationships do exist, albeit with somedifferences in their detailed form (see below).Nonetheless, the link between such patterns and thespatial aggregation of individuals has been little betterexplored than for interspecific occupancy-abundancerelationships.

In this paper, we review and summarise the link be-tween occupancy-abundance relationships and spatialdistribution. We do so by introducing a variety of sta-tistical models that describe the occupancy-abundancerelationship and the possible spatial conditions underwhich these models hold.

Occupancy-abundance models

A number of different models have been proposed todescribe occupancy-abundance relationships, albeitseldom in the context of the macroecological patternsof primary concern in this paper. Here we outlinesome of the more significant of these models.

(i) Poisson

The most basic distributional model of this kind fol-lows from a Poisson distribution, where individuals ofa 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 allsites, occupied or otherwise (Wright 1991). If n indi-viduals of a species are randomly distributed among Msites, of which m sites are occupied, p = m/M and µ =n/M. Sometimes, interspecific occupancy-abundancerelationships, and to a lesser extent intraspecific rela-tionships, are reported with abundance averaged over

Moccupied sites only (µ′), such that µ′ = –– µ, and

mp =1 – e–pµ′. The change in the definition of densitymakes no qualitative difference to the occupancy-abundance relationship, and this will also be true forthe other models that follow (He et al. in press). Itdoes, however, result in a shallower increase in densitywith increasing occupancy, until full occupancy isachieved.

The Poisson distribution predicts a relationship be-tween µ and the variance in abundance amongst sites,σ2, of the form

σ2 = µ (2)

In consequence, departure of the ratio of the vari-ance to the mean (σ2/m) from a value of 1 has com-monly been employed as a test of whether or not aspecies is approximately randomly distributed (Collieret al. 1973, McLean & Ivimey-Cook 1973, Brower &Zar 1977, Elliot 1977, Southwood 1978, Greig-Smith1983, Sokal & Rohlf 1995, Zar 1999). If the ratio isless than one then the distribution is regarded as over-dispersed, whilst a ratio of more than one would beseen as indicating an aggregated (or contagious) distri-bution. However, whilst the mean and variance areequal in a Poisson distribution, this is not the only cir-cumstance under which a variance to mean ratio ofone can arise (Pielou 1974, Hurlbert 1990, Dale1999). In fact there are an infinite number of such dis-tributions (Hurlbert 1990).

In practice, the individuals of a species are seldomrandomly distributed in space, except when they arevery 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 ob-servation can be viewed in two distinct ways. First, be-cause a random distribution is only one of a continu-ous spectrum of possible patterns of distribution itmay be exceedingly unlikely to occur simply on aprobabilistic basis (Taylor 1961). Second, random dis-tributions may be scarce because there are numerousabiotic and biotic reasons why species are unlikely tobe distributed in this fashion. The latter seems themore likely in most cases, but the former should not beentirely discounted.

(ii) Negative binomial

The statistical distribution most frequently used tomodel aggregated patterns of spatial occurrence, andthat against which the fit of very many data sets hasbeen tested, is the negative binomial distribution (e.g.Boswell & Patil 1970, Desouhant et al. 1998). Here

µ –kp = 1 – 1 + – (3)

k

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Basic Appl. Ecol. 3, 1 (2002)

( )

been proposed largely as empirical descriptors of theobserved patterns. The first of these was originallysuggested by Nachman (1981), and takes the form

p = 1 – e–α µ β

(5)

or

(6)

where α and β are two positive parameters. Morecommonly, (5) is presented as

(7)

This model is suggested as an empirical form of thePoisson model with α and β ≠ 1 representing a depar-ture from the Poisson distribution (He & Gaston2000b). Principally the model is used to predict thedensities on crops of pest species from data on theiroccupancy of sampling units (plants, tillers, leaves etc.see Nachman 1981, 1984, Kuno 1986, 1991, Ward etal. 1986, Ekbom 1987, Perry 1987, Hepworth &MacFarlane 1992, Feng et al. 1993). On several occa-sions it has been argued to provide the best availableapproach to this problem, and has also found applica-tion in modelling host-parasitoid relationships (Perry1987). Its application at broader scales has also beendiscussed (e.g. Gaston 1994, 1999).

The Nachman model predicts a relationship be-tween µ and σ2 of the form

σ2 = α µβ (8)

This is Taylor’s power function (Taylor 1961). It isthe model that has been used most widely to describeintraspecific mean-variance relationships in empiricalabundance data, although its suitability has repeated-ly been challenged (Taylor 1984, Sawyer 1989, Rout-ledge & Swartz 1991). Crucial to the argument is therole of sampling error in estimates of variability,because this may have a substantial influence on theshape of the relationship that is observed; methods ofremoving the effects of sampling error are receivinggrowing interest (McArdle & Gaston 1995). A num-ber of models have been proposed that give rise topower relationships between the mean and the vari-ance of abundances, with variable convergence onobserved parameter values (e.g. Taylor & Taylor1977, Hanski 1980, 1987, Anderson et al. 1982, Tay-lor et al. 1983, Binns 1986, Gillis et al. 1986, Perry1988, Yamamura 1990). Some of these models havebeen predominantly rooted in species behaviour andothers in demographics (that of Binns (1986) is basedpartially on the negative binomial). Yamamura(1990) derived a simple model of population growthand dispersal that gives rise to both equations (7)and (8).

where k is a clumping parameter, with small valuesrepresenting strong aggregation moving towards arandom spatial distribution as values of k increase.Thus, as k increases a species would exhibit increasingoccupancy for a given mean abundance, up to the levelgiven by eq. (1).

The parameter k is defined to be strictly positive,but in (3) it could take negative values. When k is neg-ative, model (3) is in fact derived from a positive bino-mial distribution that describes a regular distributionof organisms (He & Gaston 2000b). Therefore, the ef-fective domain of k in model (3) is (–∞, –µ) and (0,+∞), i.e., the left-hand domain describes a regular dis-tribution of organisms whereas the right-hand domainis an aggregated distribution.

The negative binomial distribution predicts a rela-tionship between µ and σ2, of the form

µ2

σ2 = µ + –– . (4)k

(Routledge & Swartz 1991, Perry & Woiwod 1992,Gaston & McArdle 1994).

Whilst a large number of possible causal derivationsof the negative binomial have been identified, mostbased on the compounding of random processes (e.g.Boswell & Patil 1970, Taylor 1984), the suitability ofthe negative binomial as a general descriptor of thespatial distributions of species has been much debated.Even where the negative binomial provides a reason-able fit to observed distributions of abundances, theappropriate values of k are dependent on the meandensity. This has principally been demonstrated atsmall spatial scales but holds more broadly (Finchet 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 demonstrat-ed using other measures of aggregation (e.g. He et al.1997, Plotkin et al. 2000).

Indeed, overall, it has been found that no one distri-bution describes the spatial variation in the abundanceof a single species distribution throughout a widerange of densities (Taylor et al. 1978, Taylor 1984).Species can broadly be characterised as typically mov-ing through Poisson, to negative binomial then lognor-mal distributions with increasing mean abundance, al-though none of the standard distributions seem to fitwell to some observed patterns of spatial variation inabundance (McGuire et al. 1957, Brown & Cameron1982, Perry & Taylor 1985).

(iii) Nachman

With the exception of the Poisson and negative bino-mial models, other occupancy-abundance models have

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Basic Appl. Ecol. 3, 1 (2002)

(iv) Hanski-Gyllenberg

Several studies have explored the connection betweenthe occupancy-abundance and the species-area rela-tionship (Hanski & Gyllenberg 1997, Leitner &Rosenzweig 1997, Ney-Nifle & Mangel 1999). In sodoing, Hanski & Gyllenberg (1997) use a logisticmodel to describe the former, such that

(9)

or

(10)

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 speciesfollows a geometric distribution. The departure fromthe geometric model is captured by α and β ≠ 1. Forthis model, the relationship between the spatial vari-ance in abundance and mean abundance takes theform

σ2 = α µβ (1 + α µβ) (13)

which is larger than the variance in Taylor’s powermodel (8), suggesting that the Hanski-Gyllenbergmodel is appropriate to describe patterns for specieshaving stronger aggregation than that under the Nach-man model. In other words, the occupancy of model(5) is larger than that of model (10); this is easilyshown to be true by examining the difference of thetwo models ((5)–(10) is always >0).

(v) Power model

Also in the context of exploring the form and determi-nants of species-area relationships, Leitner & Rosen-

zweig (1997) suggest a simple power model for occu-pancy-abundance relationships, such that

p = α µβ (14)

or

log p = log α + β log µ (15)

where α is positive and β is a scale parameter.This model is an empirical form for species to fol-

low a (positive) binomial distribution that describes aregular distribution of species. It has a variance-meanrelationship of the form

σ2 = α µβ (1 – α µβ) (16)

with α µβ < 1, which is a natural condition given thatequation (14) is a proportion. Compared with theNachman model (5), the power model is suitable forspecies of less aggregated or regular distribution. It canalso be shown that the occupancy of model (14) islarger than that of model (5), i.e., the difference be-tween (14) and (5) is not less than zero (givenα µβ < 1).

(vi) He-Gaston

Confronted with this profusion of occupancy-abun-dance models, He et al. (in press) identify a generalmodel for the relationship, of which the others are spe-cial cases, whether their roots are in a recognised sta-tistical distribution or are purely empirical. This modeltakes 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 do-main of (–∞, –αµβ) or (0, +∞) (He et al. in press). For agiven overall number of individuals and k greater thanzero, as k increases occupancy increases (Figure 2).

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Basic Appl. Ecol. 3, 1 (2002)

Table 1. The He-Gaston occupancy-abundance model and its special forms.

He-Gaston model Parameter conditions Special models

k → ±∞, α = β = 1 (i) Poisson model: p = 1 – e–µ

α = β = 1 (ii) Negative binomial:

k → ±∞ (iii) Nachman model: p = 1 – e–αµβ

k = 1 (iv) Hanski-Gyllenberg:

k = –1 (v) Power model: p = αµβ

αµβ –k

p = 1 – 1 + ––––k αµβ

p = –––––––1 +αµβ

( )µ –k

p = 1 – 1 + ––k )(

For a given overall number of individuals and k lessthan zero, occupancy declines as k assumes progres-sively greater negative values.

When k → ±∞ and α = β = 1, equation (17) is thePoisson model, when α = β = 1 it is the negative bino-mial model, when k → ±∞ it is the Nachman model,when k = 1 it is the Hanski-Gyllenberg model, andwhen k = –1 it is the power model (Table 1; He et al. inpress). The corresponding variance-mean relationshipsfor these various models can accordingly be derivedfrom the following generalised form

(19)

(vii) Others

Yet more occupancy-abundance models can be pro-posed although they need not necessarily belong to theHe-Gaston family of models. Here we consider twosuch models that have also been used to model occu-pancy-density relationships in the literature (Ward etal. 1986, see also Ekbom 1987). The first is the probitmodel

p = probit–1 (α + β log µ) (20)

or

log µ = α + β probit (p) (21)

where probit (p) is the inverse of the cumulative nor-mal distribution. This model is based on the assump-tion that µ follows a lognormal distribution.

The second additional model that has been used todescribe occupancy-abundance relationships is the ex-treme value model of the form

p = 1 – exp[–exp(α + β log µ)] (22)

(Wilson & Room 1983, see also Ekbom 1987). With areparameterisation, it is easy to show that this modelis in fact the Nachman model in a different guise.

Parameter estimation and goodness-of-fit

Because the occurrence of a species in a particularsample site is a binary (presence/absence) variate, it isappropriate to assume that the probability of thespecies occupying y out of n total number of sites in astudy area follows a binomial distribution

(23)

where p is the occupancy defined by the occupancy-abundance models such as (1), (3), (5), (9), (14), (17)or (20), respectively.

Given an assemblage of species, each of them has abinomial distribution as given by (23). Thus, the esti-

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Basic Appl. Ecol. 3, 1 (2002)

Fig. 1. The He-Gaston model, showing (a) the dependence of the probabilityof 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 thatα = 1 and β = 1.

Fig. 2. The contour map of the He-Gaston model. The values on the isolinesare the occupancy p.

Density (µ)

mation of parameters for each of these occupancy-abundance models is straightforward, simply by max-imising the log-likelihood function

(24)

where the subscript denotes the ith of s species in theassemblage.

The goodness-of-fit of each model is assessed bycomparing the deviance with a χ2 distribution. The de-viance is defined as twice the difference between themaximum log-likelihood achievable in (24) and thelog-likelihood for the model of interest (i.e., one of theabove occupancy models). The maximum log-likeli-hood achievable (i.e., the full model) is that obtainedby substituting the observed pi into equation (24). Ifthe model of interest is adequate, the deviance shouldbe small compared to the χ2

s–l, where s is the total num-ber of species and l is the number of parameters in themodel under study. Roughly, the model is consideredadequate if the deviance is smaller than the degrees offreedom s – l. However, our previous experiences sug-gest 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 therelative performances of the models than a rigorousstatistical test. We therefore use the sum of absolutedifferences between the observed proportions of oc-currence (pi) and the fitted probabilities (p̂i) as a good-ness-of-fit

.

Intraspecific and interspecific relationships

Several of the six primary models of occupancy-abun-dance relationships listed above [(i)–(vi)] were origi-nally formulated, and have principally been discussed,in the context of the occupancy and abundance of sin-gle species. Their adequacy for describing interspecificpatterns has not been fully investigated, particularly atmacroecological scales. Plainly some are appropriateunder some circumstances (e.g. Poisson, Negative bi-nomial, Nachman).

Positive empirical intraspecific occupancy-abun-dance relationships are commonly rather weak, thereare many occasions on which no significant relation-ships exist, and negative relationships occur at a mod-erate frequency (e.g. see Ambrose 1994, Marshall &Frank 1994, Swain & Morin 1996, Blackburn et al.1998, Boecken & Shachak 1998, Donald & Fuller1998, Gaston & Curnutt 1998, Gaston et al. 1998b).This might be expected on two grounds. First, therange of variation in the mean abundances of individu-al species is typically rather narrow (particularly forvertebrates) when contrasted with that between

species. Thus, even small to moderate variation aroundan underlying occupancy-abundance relationship maybe sufficient to mask its existence. Second, even if occu-pancy and abundance are linked there may be time lagsin the response of occupancy to increases and decreasesin mean abundance (given that this seems the mostlikely directionality of a causal relationship betweenthe two) which again serve to mask the relationship(Gaston et al. 1998b). For example, reductions in meanabundance may not result in the immediate reductionof occupancy, even if sufficient individuals are nolonger present in sample sites for local populations topersist there in the longer term.

How such intraspecific occupancy-abundance pat-terns will translate into, the stronger, interspecific pat-terns is not clear. Two extreme scenarios can perhapsbe envisaged. If species exhibit sufficient commonalityin the underlying relationships between occupancyand abundance, even if these relationships are oftenmasked at the intraspecific level, an interspecific occu-pancy-abundance relationship may be apparent. Alter-natively, the lack of any consistent occupancy-abun-dance relationship for individual species, and the vari-ety of responses of occupancy to increasing abun-dance, may mean that no interspecific occupancy-abundance relationship is apparent.

Of course, we know that positive interspecific occu-pancy-abundance relationships are common. So theimportant question becomes whether these can be re-garded as reflecting sufficient commonality betweenspecies in their aggregative behaviour or whether theyresult from some other process.

Whilst the limit to which most published analyseshave been carried is the determination of a correlationcoefficient for logarithmically transformed data, all sixof the primary occupancy-abundance models have, atsome time, been fitted to interspecific data at macroeco-logical scales (in so doing, concerns over the potentialnon-independence of species as data points, because ofphylogenetic relatedness, have largely been ignored –both occupancy and abundance exhibit little phyloge-netic constraint, and controlling for such effects hasbeen found to make little difference to observed pat-terns; Gaston & Blackburn 2000). Gaston et al. (1998a)examined the fit of the negative binomial model to dataon the occurrence and the estimated overall populationsize of breeding birds in Britain. In the absence of de-tailed information on spatial patterns of abundance,they chose values of k at random between values of 0.1and 4 for each species. For 100 such sets of k-values, thesimulated occupancy-abundance relationships boresome marked similarities to the real relationship, but atlow population sizes species were more widespread thanpredicted and as population size increased they rapidlybecame less widespread than predicted. Significantly,

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Basic Appl. Ecol. 3, 1 (2002)

over the wide range of population levels represented bythese species (seven orders of magnitude), even substan-tial variation in the value of k makes only limited differ-ence to the level of occupancy predicted by the negativebinomial model. This effect can also be seen in otherdata sets (A.R. Holt, unpubl. analyses).

Implicit in the application of occupancy-abundancemodels to interspecific data at broad or macroecologicalspatial scales is the notion that all of the sites or areas in-cluded can potentially be occupied by individuals ofeach species in an assemblage (Gaston et al. 1998a).This is not always so, particularly if occurrences aremapped at fine resolutions, when habitat associationsmay become apparent, and over very large regions,when differences in the limits of geographic ranges maycome into play. This effect may have contributed to mis-matches between the predicted and observed occupan-cy-abundance relationships for birds in Britain. In suchcases, if the data are available, proportional occupancymay best be calculated on the basis of a variable numberof potentially occupiable sites or areas.

The fit of the Nachman, Hanski-Gyllenberg andHe-Gaston models to two data sets was documentedby He & Gaston (2000a). For data on tree species in astudy plot in the Pasoh Forest Reserve of Malaysia,they found that all three models described the ob-served occupancy-abundance pattern reasonably well,for seven different spatial resolutions of mapping oc-cupancy. For data on the breeding birds of Bedford-shire (a county in the UK), at two spatial resolutions, asimilar result was obtained. However, here the Nach-man model was apparently superior to the other two,which both underestimated occupancy at a givenabundance. In another study, He et al. (in press) com-pared the goodness-of-fit for all six of the primary oc-cupancy-abundance models to the same two data sets.

Overall, the three-parameter model gave the best fit tothe data although in some cases the gain in goodness-of-fit may not be significant enough to warrant the su-periority of the three-parameter model to some othertwo-parameter models such as the logistic or Nach-man models. But they did find that the power model(14) was consistently the least satisfactory.

The fits of all of the models discussed here (includ-ing (20) and (22), but (22) is the same as (5)) with re-gard 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) – 123species distributed over 1756 tetrads, and mapped be-tween 1988–1994; (ii) Hertfordshire, England (Smithet al. 1993) – 109 species distributed over 409 tetrads,and mapped between 1988–1992; and (iii) Lake Con-stance, Germany (Böhning-Gaese & Bauer 1996) –141 species distributed over 303 tetrads, and mappedbetween 1988–1991. For the first two data sets, theoccurrence records for species were categorised on thebasis of whether breeding was possible, probable orconfirmed. Only probable and confirmed records wereused in the analyses. In all data sets precise distribu-tional data were not provided for some species, for in-stance for locally protected species such as the barnowl Tyto alba. These species were not included in theanalysis. For the assemblage of south-east Scotland,the only one for which it was relevant, seabirds, de-fined as those species breeding exclusively withincoastal tetrads, were excluded from the analysis.

For the data sets for the avian assemblages fromsouth-east Scotland, Hertfordshire and Lake Constancethe best fits are provided by the probit model and theHe-Gaston model (Figure 3). The probit model has theadvantage over the He-Gaston model of only havingtwo parameters, and that many major statistical soft-

8 A. R. Holt et al.

Basic Appl. Ecol. 3, 1 (2002)

Table 2. Deviance and the absolute difference between the model predictions and the observed proportions of occurrence for comparing the eight occupan-cy-abundance models for the three bird data sets (see text for details). The models were fitted using the maximum likelihood method.

South-east ScotlandPoisson Power NBD Logistic Nachman Probit He-Gaston

Deviance 7732754 15940.32 16437.44 9663.039 10552.51 9600.44 9603.058df of chi-square 131 129 130 129 129 129 128Mean absolute difference 0.345829 0.075894 0.082359 0.056499 0.059036 0.05608 0.056635

HertfordshirePoisson Power NBD Logistic Nachman Probit He-Gaston

Deviance 534767.4 5936.32 3603.662 2614.565 3433.491 2585.313 2448.985df of chi-square 109 107 108 107 107 107 106Mean absolute difference 0.278118 0.114866 0.083092 0.066555 0.077773 0.066507 0.063431

Lake ConstancePoisson Power NBD Logistic Nachman Probit He-Gaston

Deviance 58708.22 10835.76 4318.067 3333.849 3804.081 3328.494 3333.508df of chi-square 138 136 137 136 136 136 135Mean absolute difference 0.164297 0.177248 0.095107 0.075219 0.081751 0.075949 0.07531

ware packages have algorithms for parameter estima-tion. The disadvantage is that it cannot readily be re-lated to the other models.

The central message that seems to arise from theabove studies is that a wide range of occupancy-abun-dance models do fit interspecific data, in some casesrather well, but that these models are commonly notwell differentiated over the range of abundances im-plicit in such relationships. Because all of the modelsdescribe certain spatial patterns in either explicit orempirical manner, and they predict an increase in oc-cupancy with increasing mean abundance, it would besurprising if real assemblages did not show such a pat-tern. There is little evidence that species are exhibiting

any great commonality in the form of their aggregativebehaviour. But this does not matter in terms of thegeneration of a positive interspecific relationship.

To reflect this observation, He et al. (in press) havesuggested that, the upper bound to interspecific occu-pancy-abundance relationships is defined by the Pois-son model, and that, to a first approximation, thelower bound is defined by the line p = µ/µmax. They ob-served this pattern in both the tropical rain forest ofMalaysia and passerine bird community in Bedford-shire, although a highly aggregated or dispersedspecies may lie outside these limits. Similarly the avianassemblages of south-east Scotland, Hertfordshire andLake Constance show this pattern (Figure 4).

Occupancy-abundance relationships and spatial distributions: A review 9

Basic Appl. Ecol. 3, 1 (2002)

Fig. 3. The shapes of seven occupancy-abundance models fitted to data for the breeding birds of south-east Scotland, Hertfordshire and Lake Constance (seetext 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 probitand He-Gaston models.

In short, whilst positive interspecific occupancy-abundance relationships can be described in terms ofspatial distributions of species, these patterns of distri-bution can be very diverse and yet predict relation-ships that are not dissimilar from those actually ob-served.

Acknowledgements. A.R.H. is supported by a NERC stu-dentship, and K.J.G. is a Royal Society University ResearchFellow. F.H. is supported by the Biodiversity and EcosystemProcesses Networks of the Canadian Forest Service. A.S.L.Rodrigues and two anonymous referees kindly commentedon the manuscript.

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