Revisão ocupação e abundancia

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

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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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

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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

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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

( )

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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

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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-

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(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

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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.

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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 ( pˆ i) 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

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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,

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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.

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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

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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).

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Fig. 4. The interspecific occupancy-abundance relationships of the avian assemblages in south-east Scotland, Hertfordshire and Lake Constance (see text for details), and the triangle defined by the upper bound of the Poisson model (1) and the lower bound of the model p=µ/µmax.

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. Acknowledgements. A.R.H. is supported by a NERC studentship, and K.J.G. is a Royal Society University Research Fellow. F.H. is supported by the Biodiversity and Ecosystem Processes Networks of the Canadian Forest Service. A.S.L. Rodrigues and two anonymous referees kindly commented on the manuscript.

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