Present address and correspondence: T.M. Blackburn, School of Biosciences, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK. Tel: + 44 121414 5893 Fax: + 44 121414 5925. E-mail:firstname.lastname@example.org
1 Associated with the development of the field of macroecology has been the recognition and analysis of a number of different patterns in the large-scale abundance and distribution of species. The mechanistic bases of these patterns have usually been considered in isolation, yet the patterns are necessarily linked, as the same individual animals contribute to all of them.
2 Here, a model linking macroecological patterns in abundance, distributional extent and body mass is developed, based on how the finite amount of energy available to the species in a region is divided between them. The energy available to a species is assumed to support some quantity of biomass, which must then be allocated to either many small-bodied or fewer larger-bodied individuals. This identifies a necessary link between population size and body mass, which predicts when the variety of relationships between these variables in the published literature are expected to occur.
3 Although framed in terms of energy use by species, the model does not assume that energy per se is necessarily limiting populations. How individuals use space determines the form of relationships between population size and distributional extent, distributional extent and body mass, and population density and body mass. The model additionally allows a number of falsifiable predictions about the anatomy of macroecological patterns.
4 Support for the assumptions of the model is discussed.
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One possible reason for the limited attention that has been paid to the links between different macroecological patterns is that discussions of their causes have commonly been rooted in different theoretical frameworks (Gaston & Blackburn 1999). Thus, for interspecific abundance–body size relationships attention has been directed predominantly to energetic explanations (e.g. Damuth 1981), for interspecific abundance–occupancy relationships to explanations rooted in niches and metapopulation dynamics (e.g. Brown 1984, 1995; Hanski, Kouki & Halkka 1993; Maurer 1999), and for interspecific range size–body size relationships to explanations based on minimum viable population sizes (e.g. Brown & Maurer 1987). In fact, as for most macroecological patterns, in all three of these cases the full breadth of explanations which have been considered is much wider than is suggested by most of the literature (Gaston & Blackburn 1996b, 1999, 2000; Gaston et al. 1997; Blackburn & Gaston 1999), and the necessity that there are causal linkages between the patterns is readily apparent. Indeed, in many cases differences in the predominant mechanisms discussed for different macroecological patterns reflect differences in the extent to which proximate or ultimate causality is being sought. The breadth of theoretical frameworks in which explanations of at least some patterns are rooted spans speciation and extinction dynamics, population dynamics, energetics and niche structures. What is needed at the present time are explorations of the interactions between multiple macroecological generalities within particular of these frameworks.
In this paper, we examine how a number of macroecological patterns might be linked into a coherent whole by simple constraints on the subdivision of available energy. We begin by considering the case in which a single species occurs in a region with a finite resource base. We then address the consequences of the occurrence of an additional one or more species in that area. Finally, we discuss the assumptions that were needed to construct the model – for simplicity we ignore these during its exposition – and some of the implications of the model for understanding of macroecological patterns. Throughout, we will be concerned primarily with what have come to be regarded as some of the principal macroecological variables; namely, the local and regional abundances, the regional distributions, the body sizes, the biomasses and the energy use of species.
The model presented here owes much to earlier studies that have considered how absolute limits to the abundance and distribution of species might delineate regions of this parameter space within which species are constrained to lie (e.g. Brown & Maurer 1987, 1989; Maurer & Brown 1988; Brown 1995; Maurer 1999), and that have considered how energy might be allocated among species within that constraint space (e.g. Lawton 1990). However, it extends this work in that many of the boundaries of constraint space derive from the model, rather than being imposed upon it, and that explicit predictions about the form of interactions between variables are derived.
The single species case
Consider a hypothetical land area sufficiently large and isolated that the predominant determinants of the number of species found there are the processes of speciation and extinction; the effects of immigration and emigration are negligible. This is analogous to Rosenzweig’s (1995, p. 264) definition of a ‘province’, and we will use this term to refer to the hypothetical area.
Imagine a single species within the province, exploiting all of the resources available to it. The term ‘resources’ here potentially encompasses a wide variety of factors necessary for the maintenance and propagation of life, but for simplicity we refer here to available energy. The species may be a primary producer obtaining energy directly from solar radiation, or it may be a consumer, deriving it ‘second-hand’ from producers. We assume that energy availability imposes a limitation on the species at some point. The absolute amount of life in the province must ultimately be limited to that which can be supported by the harnessing of all energy arriving from the sun. This limit will inevitably be lower than that expected from levels of solar radiation alone. Some energy is conducted and convected away from the province, so that not all can be captured. Initial plant, and subsequent animal, conversion efficiencies are less than perfect. Constraints on energy use are imposed by the availability of other essential resources, such as water, minerals and space. Energy exploitation is restricted further by the physical limits of plant and animal physiologies (e.g. Phillipson 1981; Sibly & Calow 1986; West, Brown & Enquist 1997; Enquist, Brown & West 1998). Nevertheless, ignoring the small amount of energy contributed by geothermal sources, energy arriving from the sun sets an upper limit to the total amount of life that the province can support, and hence also on our imaginary species.
The species in the province uses available energy to fuel its productivity. In effect, and allowing for the constraints described above, it converts the energy into biomass. The total biomass of the species is dictated by energy availability and the energy requirements of individuals. The latter depend on metabolic rates which in turn depend on, among other things, the body mass of the individual. Across a wide range of organisms, metabolic rate is a power function of body mass, with the exponent being positive but less than 1. This means that large-bodied organisms use less energy per gram of body mass than do small-bodied organisms (although the form of the relationship may be different when comparison is between small- and large-bodied individuals within the same species – see below). Total biomass is the product of number of individuals and their body mass. Another way of viewing this is that energy availability sets the carrying capacity, and this is being fully exploited.
From these interactions, it follows that our imaginary species uses energy to support a certain amount of biomass, that this biomass may be divided among many small-bodied or fewer larger-bodied individuals, and that the total biomass supported will be greater if individuals are large-bodied. The expected population size would scale with body mass to the power −x, where x is the allometric power exponent for the scaling of metabolic rate (we use the term allometry solely to refer to the scaling of variables with body mass). The energy used by the species will then be constant, and independent of its mass (it has to be, because the species is assumed to use all available energy, however large it is). However, population size and body mass are constrained to be negatively related, so that they must fall somewhere along a straight line with slope −x in log abundance–log body mass space (Fig. 1).
The population of the imaginary species will be distributed across the province. The extent of that distribution is the product of the number of individuals and the inverse of their density (number per unit area), which we call here the average individual area requirement (if the home ranges of individuals overlap, as they almost invariably do, this will be smaller than home range size). As we are assuming that there is but a single species in the province, using all available energy, it seems logical to conclude that the distribution is likely to cover the entire province. For this to be true whatever the body mass of the individuals, the relationship between average individual area requirements and body mass would have to be a power function with a positive exponent equal to x, the allometric exponent of metabolic rate (the relationship between population density and body mass would then be a power function with exponent −x, i.e. substitute population density for population size and area use for energy use, both individual and total, in Fig. 1). Then, increases in population size would be balanced exactly by decreases in average individual area requirements, leaving constant the total size of the distribution of the species, the product of these two variables.
So far we have assumed implicitly that energy input to the province is constant in both space and time. This is not necessarily the case. All else being equal, if there is spatial variation in energy input, at equilibrium the local biomass, individual area requirement, and density of individuals will reflect local variation in energy availability, but with no change in the average values of these quantities across the entire province. Any marked spatial autocorrelation in levels of energy input, with areas closer together being more similar (e.g. a hotspot from which levels decline), will generate similar spatial patterns in these quantities. Assuming that changes in individual body size occur on a much longer time scale than changes in individual energy availability, then temporal variation in energy input will serve to generate synchronous variation in total provincial biomass and population size of the species, while its body mass stays roughly constant.
In sum, we have a province occupied by a single species which utilizes all available energy. This energy supports the biomass of the species, which is divided into a certain number of individuals of a certain size. Size and number of individuals necessarily trade off within the constraint imposed by the amount of energy. The individuals are distributed across the province giving the extent of the distribution of the species.
A multispecies assemblage
In the real world, systems with only a single species are seldom encountered – the energy available in a province is normally divided amongst a large number. Nevertheless, consideration of the single species example is instructive when turning to the interactions that would be expected among macroecological variables in a multispecies assemblage. The logical development of macroecological patterns is clearest if first we consider the addition of a second species to the hypothetical province, at the same trophic level as the first, and then consider the case where the assemblage consists of many related species.
Population size and body mass
The second species to enter our imaginary assemblage appropriates a proportion of the energy available in the system, reducing the amount that was available to the first species when it was alone. We assume for simplicity that the total amount of energy used by the two species is the same as that used by the first alone, but this is not necessary for the model (or biologically). The two species may use more or less energy than one alone; for example, because of differences in their ability to exploit available energy, or because of the costs of interspecific interactions.
Whatever the total amount of energy available to each species, that energy will support a certain amount of biomass that, as has already been shown, depends on the metabolic rate (and thus the body mass) of individuals of the species. As for the single species case, the biomass of each species may be divided into many small-bodied or fewer large-bodied individuals.
For each species, the necessary trade-off between population size and body mass in the allocation of biomass means that the species is constrained to fall somewhere on a line with slope −x in log population size–log body mass space, where x is the allometric exponent of metabolic rate. In practice, where the species falls on the line will depend on its body mass. The elevation of the slope for each species is proportional to the total amount of energy it can utilize (Fig. 2). In the special case where the total amount of energy available is divided equally between the two species in the system, and they have identical allometric scaling of metabolic rate, they will both lie on the same log population size–log body mass line. Therefore, for a given amount of energy, a linear negative interspecific relationship is expected between log population size and log body mass. Otherwise, the interspecific population size–body mass relationship between the two species in this system can take any form, and may be positive. By definition, positive relationships occur whenever both the body mass and population size of the species using less energy are lower than those of the species using more energy. Therefore, they occur whenever a species using the same amount of energy as species 2 falls on that part of its population size–body mass trade-off function bounded by the horizontal and vertical lines connecting that function to the position occupied by species 1 on its trade-off function (Fig. 2). The length of this bounded region increases as the difference between the elevations of two trade-off functions increases. If both trade-off functions have similar elevation, then the species on the lower function will only fall in the bounded region if it is of slightly smaller body mass. Thus, positive population size–body mass relationships between two species are more likely when available energy is less equably divided, or when the two species are similar in body size.
The constraint applied to the population size–body mass relationship by the amount of energy available to any given species sets a theoretical negative upper boundary to this relationship (Fig. 2). This is the slope on which a single species of any body mass utilizing all available energy in a province would lie. In practice, the upper boundary in a real multispecies assemblage will be defined by the population sizes and body masses of those species that appropriate most available energy. Nevertheless, the trade-off between population size and body mass for any given amount of energy means that this limit is also likely to be negative (although it may in principle be positive, as in some of those situations described in the previous paragraph, for example).
For a given taxon, the range of body mass values possible will ultimately be limited by design constraints. There are physical limits on any given way of life (e.g. Calder 1984; Schmidt-Nielsen 1984; Reiss 1989). This forces population size–body mass relationships to lie within certain body mass limits (see Brown & Maurer 1987; Maurer 1999), restricting the ways in which a given amount of biomass can be subdivided into individuals.
Whether, as well as constraints on the upper boundary and the lower and upper body mass, there is a necessary constraint on the lower boundary of population size-body mass relationships is less clear (Lawton 1990; Silva & Downing 1994; Blackburn & Gaston 1997a). In theory, some species in an assemblage may appropriate very little energy, so that the elevation of the population size–body mass trade-off line on which they must lie is very low. However, since no extant species may have a population size of less than one individual there may be few attainable population size–body mass combinations for such species. They would inevitably have to have both low population sizes and small body masses. Since species with small populations are vulnerable to extinction, there may be a minimum population size below which the long-term persistence of species is unusual. Whether this varies across species is unclear. If it did, it seems most likely that small-bodied species would require larger population sizes to persist. However, some studies suggest that medium-sized species are most extinction-resistant for a given population size (e.g. Johst & Brandl 1997). Thus, it remains to be resolved whether there is any necessary constraint on the lower boundary of population size–body mass relationships, beyond that imposed by the requirement that at least one individual must exist of any extant species.
Within the constraints described above, the members of a multispecies assemblage can lie almost anywhere with respect to each other in a population size–body mass plot (Lawton 1990). Whether the overall interspecific population size–body mass relationship is negative depends on the relative positions of all species along the slopes on which they lie by virtue of their energy availability. However, unless energy is particularly inequably divided among the species in an assemblage, negative interspecific relationships are likely. With equable energy division and a reasonable spread of body masses relative to population size variation for a given body mass, the interspecific slope should be close to −x.
The populations of species in the multispecies assemblage must be distributed across space. Consider first the two-species case, in which both species appropriate equal amounts of energy. We assume that there is a single relationship across all species for the allometric scaling of metabolic rate. Then, if the body masses of the species are equal, so too must be their population sizes, and hence their distributional extents. If species 2 is larger than species 1, then the shape of the population size–distribution relationship depends on the allometric scaling of average individual area requirements. This is because distributional extent is the product of average individual area requirements and population size, which both vary with body mass. If the absolute values of the allometric exponents of average individual area requirements and population size are equal (i.e. the actual values of the exponents sum to zero), then the distributional extents of both species will be constant, and the population size–distribution relationship will have a slope of 0. If the absolute value of the allometric exponent of average individual area requirements is less than that for population size, then the population size–distribution relationship will be positive: the larger-bodied species will have both smaller population size and distributional extent. However, if the absolute value of the allometric exponent of average individual area requirements is greater than that for population size, then the population size–distribution relationship will be negative: the larger-bodied species will have smaller population size but larger distributional extent.
When species differ in energy appropriated, the situation is more complicated. Population size is inversely related to mass across species using a constant amount of energy, but the product of population size and mass is greater for species using more energy. Therefore, the allometric scaling of distributional extent depends on population size, as well as on the allometric scaling of average individual area requirements (Fig. 3). Now, whether the population size–distribution relationship for our two-species assemblage is positive or negative depends on exactly how much energy a species appropriates, the allometric exponent of average individual area requirements and the body masses of the species. It is clear from Fig. 3, however, that positive relationships will occur for many parameter values, and especially when species do not differ greatly in body mass.
If the scaling constraints applied to the average individual area requirements of species are relaxed, then absolute limits can be defined on where species can lie in population size–distribution space (Fig. 4). First, there is an upper limit to distributional extent set by the size of the province. Second, there is a lower boundary to distributional extent set by the minimum area into which populations of different sizes can be squeezed. Unless individuals can be stacked on top of each other, this area will inevitably increase with population size. Third, there is a lower limit to the population size–distribution relationship set by the maximum amount of area that can be occupied by populations of a given size. This probably increases with population size in the real world. Finally, there is an upper limit to the population size that the province can maintain, which equals the total number of individuals that would occupy the province if the assemblage consisted of only a single species of body mass that minimized individual energy use.
These constraints imply that a set of species occupying random points in population size–distribution space would probably show a positive relationship. Moreover, constraints on energy allocation within this space suggest that positive relationships are more likely between closely related species. These will differ to some degree in the amount of energy appropriated, but will be physically and physiologically quite similar: the relative with the larger population size should then have the greater distributional extent (Fig. 3).
As Fig. 3 shows, the relationship between distributional extent and body mass can take a variety of forms, both for a given amount of appropriated energy, and across species appropriating different amounts. Ultimate constraints are placed on the relationship by the size of the inhabited province and again by the minimum and maximum body mass attainable by species in a taxon (Fig. 5). The extent of the distribution required to house a minimum viable population of a species probably also increases with body mass, suggesting that the lower bound of the distribution–body mass constraint space will be positive (see Brown & Maurer 1989). This observation does carry some caveats, however. First, the sign of this boundary slope will depend on precisely how minimum viable population size and average individual area requirements vary with body mass. Second, some species in an assemblage may not have viable abundances, and so will fall below the constraint line. This is particularly likely at present given the extinction crisis (May et al. 1995). Third, the concept of viability requires reference to a time frame. Extinction is inevitable, so even huge populations have a finite probability of becoming extinct over any time period. The concept of a minimum viable population is therefore somewhat nebulous, even when extinctions are at the ‘normal’ background level. Nevertheless, we can envisage a population size below which time to extinction will be short relative to the average lifespan of a species in the taxon in question, and that the distribution required to house this population increases with body mass. Then, the boundary constraints described imply that a set of species occupying random points in distribution–body mass space might show a positive relationship, but that this form is only marginally more likely than any other.
Although we have framed much of the preceding discussion on the determinants of distributional extent in terms of average individual area requirements, as pointed out earlier the average individual area requirement is the inverse of population density. Thus, we can consider how density may vary in our two-species assemblage. In fact, it is relatively unconstrained. While it necessarily varies allometrically as the inverse of average individual area requirements, the exponent is not restricted to any particular value. This was not true in the single species case under the assumption that that species occupied all available area, where distributional extent had to be mass invariant. Once there are two species in the assemblage, this constraint on density only applies in situations where we assume all available energy is used by the two species, and that energy is homogeneously available across the province. Then density must not be so high that some areas in the province are uninhabited by either species. However, this constraint is removed if either of these assumptions is relaxed.
Population density is constrained in one important way. If, for a given amount of appropriated energy, average individual area requirements scale allometrically with exponent x, where −x is the allometric exponent for population size, then distributional extent is mass invariant. Since average individual area requirement is the inverse of population density, density must scale as mass to the power −x in this case. Thus, population size and density both scale with mass in the same way. However, if distributional extent varies with body mass, then population size and density must scale with different allometric exponents, for a given amount of appropriated energy. In a multispecies assemblage, with energy reasonably equably divided among the species then, as we have already noted, the interspecific population size–body mass exponent will be close to −x. However, the interspecific population density–body mass exponent will only take the same value if distributional extent is mass invariant. If distributional extent increases with mass, it must do so because individual area requirements increase (and hence density declines) with body mass at a faster rate than population size declines. A similar argument applies if distributional extent decreases with mass.
We have assumed that in a hypothetical multispecies assemblage the energy available in the province as a whole is divided among the species, and that relationships between macroecological variables result from constraints on how individual species use the energy they appropriate. A logical consequence is that there is a limit to the number of species that the province will support. The maximum richness attainable equals the amount of available energy divided by the amount of energy required to support the minimum biomass that constitutes a viable population of any species. The precise value of this latter quantity depends on how minimum viable population size and individual energy requirements trade-off against body mass. Thus, the province would have lower species richness if the elevation of the allometric relationships of either minimum viable population size or individual energy requirements were higher, or if the total amount of energy available in the province was lower. This latter situation would arise either if the amount of energy available per unit area was lower, if the province covered less area, or both.
Support for the model
The key element underlying the model of the links between macroecological patterns outlined thus far is that these associations are constrained ultimately by the energy available to an individual species. The amount of energy constrains the amount of biomass that can be supported, which in turn limits the population size that a species of given body mass can attain. From this, it would be easy to conclude that we believe energy to be the key factor limiting species’ populations. That is not the case. The model makes no assumptions about what limits any given species, which may be any of a variety of factors (e.g. availability of space or resources, competition, predation, disease). All the model requires is that once the population of a species is limited, then that species utilizes a certain amount of the energy available in the environment. This is trivially true. What is important are the constraints placed on a population using this amount of energy.
We do assume, however, that the total amount of energy available in a province must set an upper boundary to the total amount of biomass that the province can support, which in turn must limit the total number of species and individuals that inhabit it. This also is trivially true. Whether this upper boundary is ever actually reached is a rather more significant question. Provincial species richness may be depressed below the theoretical maximum dictated by energy availability because of a variety of effects acting in ecological and evolutionary time, including seasonality in energy availability, and periodicity in the global climate. Nevertheless, there is evidence that species richness is correlated with both energy availability per unit area and geographical area at large spatial scales (e.g. Wright 1983; Turner, Gatehouse & Corey 1987; Turner, Lennon & Lawrenson 1988; Rosenzweig 1992, 1995; Wright, Currie & Maurer 1993; Turner, Lennon & Greenwood 1996; Blackburn & Gaston 1997b; Rosenzweig & Sandlin 1997; O’Brien 1998; O’Brien, Whittaker & Field 1998), the product of the last two being the provincial energy availability. If some factors do indeed act to depress species richness below the theoretical maximum, it seems that they do not act sufficiently differentially with respect to provincial energy availability to disrupt its relationship to species richness. That said, it would be desirable to have some direct tests of the relationships between provincial energy availability, biomass and number of species.
We also assume that the amount of energy available to a species constrains its biomass. This must be true to the extent that energy is required to produce and maintain biomass. Thus, available energy certainly sets an upper limit to biomass. Beyond this necessary constraint, the amount of biomass that a given amount of energy can support depends on energy requirements per gram. This varies systematically with body mass (see above), and will also vary across species of a given body mass to the extent that the efficiency of energy conversion varies. We know of no published evidence to support the assertion of a positive correlation between energy availability and biomass across species, although such a relationship does pertain across assemblages of primary consumers (McNaughton et al. 1989; Cyr & Pace 1993). Note, however, that energy availability may not be reflected in the standing crop of biomass for any given species, as specific biomass may continually be removed by consumers. This effect may also serve to disrupt relationships between energy availability and biomass across provinces, but will not affect the necessary trade-off between population size and body mass for a given standing crop of biomass.
Although the model is framed in terms of values potentially attainable for various macroecological parameters (population size, distributional extent, body mass, density) by a species utilizing a certain amount of energy in the environment, the energy it appropriates is not constant. Rather, this will fluctuate, as do the effects of whatever factors happen to limit its population. In addition, some macroecological traits of species are more plastic than are others. The population size and distributional extent of a species will in general change much more quickly than will its body mass (e.g. Elgar & Harvey 1987; Promislow & Harvey 1989; Read & Harvey 1989; Gaston & Blackburn 1997; Blackburn et al. 1998; Gaston 1998; Webb, Kershaw & Gaston 2000). Thus, in reality, in ecological time species will not move along constraint lines of the type illustrated in Figs 2 and 3. Instead, they will tend to move between them as the energy appropriated by their populations changes. Moreover, where one of the axes refers to body mass, movement in constraint space will tend to be perpendicular to it.
In our model the most fundamental constraint on the way in which a given amount of energy can be used is that there must be a trade-off between population size and body mass: both cannot increase for a species without also an increase in the amount of energy it appropriates. The slope of the trade-off will be the inverse of that for the scaling of per gram energy use (or metabolic rate) with body mass. In the macroecological literature, metabolic rate is usually argued to be a power function of body mass with exponent approximately 0·75. In fact, there is considerable debate about the exponent of any such general relationship (e.g. Kleiber 1962; Peters 1983; Hayssen & Lacy 1985; Bennett & Harvey 1987; Elgar & Harvey 1987; McNab 1988; Heusner 1991). Moreover, although basal metabolic rate is normally considered, it is not clear whether this or field metabolic rate is the more appropriate indicator of actual energy use, or indeed what the relationship between the two measures may be (e.g. Bennett & Harvey 1987; Daan, Masman & Groenewold 1990; Bryant & Tatner 1991; Degen & Kam 1995; Ricklefs, Konarzewski & Daan 1996). If the appropriate rate does indeed scale with an exponent of 0·75, the allometry of population size ought to be a power function with exponent −0·75. However, there are at least two caveats to this conclusion.
First, the exponent 0·75 for the allometry of metabolic rate is calculated across species. The slope may be somewhat different within any given species (e.g. Heusner 1982; Schultz 1988). As we frame our model in terms of the population size–body mass trade-off for a species using a fixed amount of energy, might the inverse of an intraspecific metabolic rate exponent not better approximate the exponent of the population size-body mass trade-off? In fact, we consider the interspecific exponent more likely to be appropriate in this case. In effect, intraspecific relationships can be viewed as showing how energy use varies for individuals of different body mass (or for individuals at different stages in development) in the static case where the species is occupying a point in population size–body mass space. However, the constraint lines we imagine in this space connect points for which the average body mass of the species differs. A species moving along such a constraint line would necessarily change its average body mass, and hence its metabolic rate. We modelled these lines in terms of the changes that they would imply for a species moving along them, but this was purely for illustrative purposes. In real ecological systems, the constraint lines will connect different species appropriating the same amount of available energy at a given point in time, rather than the same species at different points in time. Thus, they are probably better modelled by interspecific allometries.
Second, the theoretical trade-off between population size and body mass applies only to situations where available energy is constant. Therefore, strictly its exponent should be the inverse of the allometric scaling of metabolic rate across species using equal amounts of energy. As interspecific allometries are not plotted with respect to species’ total energy usage, we do not know what value this exponent will take. Nevertheless, there is reason to believe that it will not differ greatly from that observed across all species. The elevation of constraint lines will be similar for species using similar amounts of energy. Therefore, as we discussed earlier in a different context, where energy is reasonably equably divided amongst species, and there is a wide spread of body masses for all levels of energy use, the exponents of relationships across species using different amounts of energy will not differ greatly from those across species for which energy use is constant (Fig. 6). The general tendency for the allometric exponent of metabolic rate to be around 0·75 in a variety of studies suggests that the use of this figure may be not unreasonable.
An interspecific allometric exponent of 0·75 for metabolic rate implies that that for population size ought to be −0·75. Exponents approximating this value have indeed been found for some assemblages (e.g. Nee et al. 1991; Greenwood et al. 1996), although in others they have not (e.g. Cotgreave, Middleton & Hill 1993; Gaston & Blackburn 1996a). There are at least three reasons why observed exponents might differ from the predicted value. First, energy may be particularly inequably divided in an assemblage. Second, where species actually lie along constraint lines may not be independent of energy appropriated. For example, observed exponents would be greater (i.e. less negative) than −0·75 if species appropriating small amounts of energy tended to be small-bodied, while species appropriating large amounts of energy tended to be large-bodied. Third, the species may display a limited range of body masses or abundances. In the first two cases, the differences arise because the slope prediction strictly applies only across species using equal amounts of energy, and the distribution of species across the interspecific relationship violates the conditions described in the previous paragraph whereby the slope of the interspecific relationship will approximate the theoretical prediction. In the third case, the restricted set of species results in a poor estimate of the true form of the relationship.
Note, however, that an interspecific allometric exponent of −0·75 for population size does not mean that populations are energy limited. Rather, all it shows is that the allometric exponent of per gram energy use (metabolic rate) is 0·75. Whatever limits the resources available to species, be it for instance competition, prey abundance, water availability or nest sites, once they have appropriated the energy that this limiting factor allows, then they can only allocate it to more small- or fewer large-bodied individuals, and the form of this trade-off must be mediated by metabolic rate. Thus, the exponent of the interspecific population size–body mass relationship says nothing about the factors that limit population sizes.
Our model allows predictions to be made not only about the slope of the interspecific population size–body mass relationship, but also about its elevation. Maximum elevation is observed in the case where a province is inhabited by a single species utilizing all available energy (although of course the slope of this relationship would only be described by plotting evolutionary changes in the species’ body mass). It follows that, for a given amount of total available energy, provinces with more species will have population size–body mass relationships of lower elevation. The converse is that the elevation will be higher for a given number of species inhabiting a more energy-rich region. The difficulty in testing these predictions will arise in obtaining estimates of the energy availability to any given taxon that are independent of the biomasses of the species.
The sign of the population size–distribution relationship for a two-species assemblage depends on exactly how much energy a species appropriates, the allometric exponent of average individual area requirements, and the body masses of the species. Clearly, there is enough latitude for the relationship to take a variety of forms. None the less, our model does suggest some constraints. First, absolute limits on where species can lie in population size–distribution space suggest that points falling randomly within that space should tend to show a positive relationship (Fig. 4), albeit that since the area encompassed by these limits is not especially restricted, the positive relationship may not be particularly strong. Second, Fig. 3 shows that positive relationships between species are more likely the more similar are the species to each other in body mass and scaling of average individual area requirements (Brown 1984), and the more they differ in terms of total energy appropriation. All these things serve to restrict the amount of constraint space in which the species with the lower population size can have the larger distributional extent. Third, if individual area requirements scale with the same absolute allometric exponent as population size, then population size–distribution relationships are always positive (the more abundant species is always more widespread), except in the special case that the species use exactly the same amount of energy, when the slope of the relationship is zero (Fig. 3).
Current evidence supports the first two of these assertions. Positive population size–distribution relationships are the norm across species in broad taxonomic assemblages. These relationships are often stronger than the rather large area within the absolute constraints would lead one to expect (e.g. Gaston & Blackburn 1996a; Gaston 1996; Blackburn et al. 1997), but this may be in part because species even in such broad assemblages may use energy in only a fraction of the ways possible. In particular, if species total energy use is relatively even, and they scale similarly in average individual area requirements, then species would be expected to occupy a much more restricted part of the total population size–distribution space theoretically inhabitable. Population size–distribution relationships also seem to show less variance when species are more similar to each other, at least to the extent that taxonomic relatedness indicates similarity (Gaston 1994; Maurer 1999).
As was the case for population size–distribution relationships, absolute limits on where species can lie in distribution–body mass space suggest that points falling randomly within that space should tend to show a positive relationship (Fig. 5). If anything, though, constraints in the case of this latter relationship are even looser. Of the four absolute constraints bounding the relationship, only the lower limit of distributional extent does not parallel an axis. However, the slope of this boundary is not necessarily positive (see above), and is not an absolute constraint: species may fall below it, albeit that their populations should be below the minimum level for long-term viability. Moreover, as is clear from Fig. 3, there is no constraint on the likely form taken by closely related species: widely distributed species are as likely as not to be smaller-bodied than more narrowly distributed relatives. In fact, positive relationships among close relatives may be slightly more frequent because of the constraint imposed by the positive slope of the lower boundary to range size (Fig. 5). However, that is only true to the extent that species actually approach this lower bound.
Empirical evidence bears out these assertions. Interspecific distribution–body mass relationships tend to be positive (reviewed by Gaston & Blackburn 1996b). When relationships have been plotted for entire faunas of a given taxonomic group, polygonal relationships are obtained that resemble the constraint space in Fig. 5 (Brown & Maurer 1987; Blackburn & Gaston 1996a). Species apparently below the lower bound of range size in such plots tend to be those classified as at risk of extinction, as expected if this lower boundary described the space required by minimum viable populations. In the one complete fauna where within-taxon relationships were plotted, positive and negative relationships were about equally common (Blackburn & Gaston 1996a).
A tendency for positive relationships to pertain between the extent of a species’ distribution and its body mass implies the form of the allometric scaling of density. As was shown above, distributional extent is mass invariant if average individual area requirements scale allometrically with exponent x, where −x is the allometric exponent for population size. Positive distribution–body mass relationships thus imply an allometric exponent of greater than x for average individual area requirements, and thus of less than −x for its inverse, density. Since we saw earlier that x is typically taken to be approximately 0·75, the allometric exponent for population density should be less than −0·75.
The precise form of the relationship between population density and body mass has been the subject of intense debate. Slopes close to −0·75 on log-log plots have been claimed for a variety of taxa (e.g. Damuth 1981, 1987; Peters & Wassenberg 1983; Marquet, Navarrete & Castilla 1990). However, it has been suggested that some such slopes are biased because small-bodied rare species are differentially excluded (Brown & Maurer 1987; Morse, Stork & Lawton 1988; Lawton 1989). If so, the true slope would be greater (i.e. less negative) than observed. In contrast, others have claimed that population density–body mass relationships are better characterized by slopes less than −0·75. The basis of these arguments is that the method of ordinary least squares by which slopes are normally quantified is a poor estimator when there is error variance in both dependent and independent variables. Using the reduced major axis, which assumes error variance in both variables, slope estimates are typically closer to −1·0 (Griffiths 1992, 1998). At present, the issue remains unresolved (see Blackburn & Gaston 1997a, 1999; for recent reviews). Nevertheless, the generally weak and variable nature of the relationship between distributional extent and body mass suggests that population density–body mass slopes of around −0·75 may not be unreasonable.
Additional predictions of the model
The model allows a number of falsifiable predictions about the anatomy of macroecological patterns in addition to those already mentioned:
1.All else being equal, between-province variation in the maximum population sizes and range sizes attained by species should be positively related with the areas of those provinces. There is evidence that this is indeed the case for the geographical range sizes of mammals between continents (Letcher & Harvey 1994; Smith, May & Harvey 1994).
2.Population size and population density will scale with identical allometric exponents only when there is no relationship between distributional extent and body mass. This follows from the observation that population size is a product of density and distributional extent. This prediction is dependent on accurate estimation of the various allometric exponents.
3.Negative population size–distribution relationships are more likely within taxa the more dissimilar the body masses of the species. This follows from the relationships illustrated in Fig. 3, where it is clear that for a given body mass, a species using more energy will be both more widely distributed and have larger population size than a species using less energy. Only where species differ greatly in body mass will negative relationships be likely to pertain.
4.If the lower boundary of distribution–body mass space needs to be positive to allow minimum viable populations of large-bodied species to be housed, then positive distribution–body mass relationships are more likely within taxa the closer a taxon lies to this boundary. That is because only at this lower boundary is the form of the distribution–body mass relationship constrained to take a specific form. Note, however, that tests of this prediction will be complicated by species currently persisting below the minimum viable level.
5.The relationship between provincial energy availability and species richness assumed by the model suggests that provincial species richness will be modified by spatial variation in the minimum viable population size and individual energy requirements of species. These latter two variables both affect the amount of energy that is required to maintain a viable population of a species in a region, and hence the number of species between which all available energy can be divided. If there are provinces where one or other, or both, of these variables need to be higher for species to persist, then the species richness of those provinces will be lower for a given total energy availability. Since the best current evidence suggests that species living at high latitudes tend to have larger population sizes and body masses (at least for birds and mammals:, e.g. Cousins 1980, 1989; Zeveloff & Boyce 1988; Currie & Fritz 1993; Blackburn & Gaston 1996b; Gaston & Blackburn 1996a; Johnson 1998), implying that they may require larger population sizes and greater amounts of energy per individual to persist, species richness at high latitudes may be lower than expected on the basis of simple energy availability. In other words, each species should use a greater proportion of available energy at higher latitudes.
The model as outlined provides a framework within which the principal macroecological patterns can be accommodated. It shows how some simple but necessary relationships among features of ecological assemblages apply constraints to the form that macroecological patterns can take. Some of these constraints are weak, and hence so are the relationships they predict. However, such is the nature of many macroecological patterns. The model agrees with current empirical evidence about the general form of such relationships, and there is support for many of the assumptions made about the constraints within which such patterns must operate. Indeed, the most important of these must necessarily be true, suggesting that the broad structure of the model will be robust.
Although framed in terms of energy use by species in a large-scale assemblage, it is worth re-emphasizing that we do not imply that it is necessarily energy per se that is limiting populations. Rather, whatever it is that limits a population results in that population appropriating a certain amount of the energy available in the environment. Certainly, ultimate constraints on where species can lie in parameter space are provided by energy availability, and by the energy required to maintain viable populations (or the area required from which to harvest that energy). However, most species do not lie close to these ultimate boundaries, strongly implying that some additional factor is determining the position of species in parameter space. For example, while we have shown that population size and distributional extent are likely to be positively related, real relationships are not a simple consequence of the random distribution of individuals across the environment (e.g. Wright 1991; Gaston et al. 1997; Gaston, Blackburn & Lawton 1998b; Venier & Fahrig 1998). Similarly, Harte, Blackburn & Ostling (in press) show that the bivariate relationship between distributional extent (R) and size of the grid cells over which distribution is mapped (A, where R = WA, and W = number of grid cells occupied) can potentially take many forms depending on the distribution of individuals across the environment, but that if species are distributed according to the principle of self-similarity, then the form of that relationship is tightly constrained. It is the search for these additional, proximate constraining factors that has dominated much of the literature pertaining to some (but not all) macroecological patterns.
In a similar vein, one should be wary of interpreting the model as implying a significant role for interspecific competition in structuring macroecological patterns. The exploitation of available energy by one species necessarily precludes its exploitation by another (at least at the same trophic level). In this very general sense there is competition for energy. However, it need not follow that the latter species would have been able to use this energy had it not been appropriated by the former. Indeed, it seems unlikely to us that direct interspecific competition of this kind is particularly important in placing constraints on the shapes of macroecological patterns. It may, none the less, be a determinant of precisely where individual species lie within these constraints.
The principal assumption required for the model described here is that individual energy use, or metabolic rate, scales across species with an allometric exponent of x, which is typically close to 0·75. Population size–body mass relationships follow from this allometry, as do predictions for exceptional situations, because of a necessary trade-off between the sizes and numbers of individuals of species using equal amounts of energy. Predictions for interactions between body mass and distributional extent, between population size and distributional extent and between these variables and density follow from the relationship of x to h, the power exponent of the relationship between average individual area requirements and body mass. Additional assumptions, such as those about upper and lower limits to body masses of species in monophyletic taxa and about patterns in minimum viable population sizes, are not essential to the model, albeit that the model will be affected by their precise form. Some such constraints are likely to exist in nature, although their causes are currently unresolved.
Although we have not explicitly considered multitrophic assemblages, we think that the model we have outlined could be readily adapted to situations where species are distributed across more than one trophic level. Most of the relationships and trade-offs described will hold within trophic levels, but the absolute position of species in parameter space is likely to vary in relation to the amount of energy available to different trophic groups. In that regard, there is already some evidence that the elevations of abundance–body mass relationships are lower across species of predatory than herbivorous mammals, as expected from the reduction in available energy as it is transferred between trophic levels (e.g. Peters & Wassenberg 1983; Peters 1983). Further development of the multitrophic case would be interesting, but premature here given the preliminary nature of this treatment.
Perhaps the most important conclusions arising from this consideration of the links between macroecological patterns are twofold. First, if such links, which must inevitably exist, are to be understood, then far more attention must be paid to developing theories for these patterns that make more precise predictions about the position of constraints on the patterns. Most theory to date has been largely qualitative (for a very notable exception, see West et al. 1997; Enquist et al. 1998; West, Brown & Enquist 1999a, 1999b). However, as we have seen, most relationships are more likely to take some forms than others (e.g. positive or negative) due to the limits imposed by their ultimate constraints. Hypotheses that predict this form will not be great advances unless they can also predict exactly where in constraint space species lie.
Second, and related to the previous point, far more attention needs to be paid to what we have elsewhere referred to as the detailed ‘anatomy’ of macroecological patterns (Gaston & Blackburn 1999). Again, studies of patterns have tended to report broad correlation coefficients, and their associated significance levels, and perhaps the overall slope of any relationships. It is becoming increasingly apparent that considerably more detail is required, particularly as pertains to the shape of the envelope within which data points lie. This may necessitate the further development of appropriate statistical techniques (e.g. Blackburn, Lawton & Perry 1992; Thomson et al. 1996; Blackburn & Gaston 1998; Scharf, Juanes & Sutherland 1998). Quantitative theory needs to be combined with detailed knowledge of the relationships to be explained if we are to distinguish between alternative hypotheses, and to forge firm links between macroecological patterns.
The rationale behind the macroecological approach has been the belief that a broad geographical perspective on the structure of ecological assemblages will yield insights which have not been forthcoming from a reductionist approach concerned with the behaviour of individual organisms at local sites (Brown 1995; Lawton 1999; Maurer 1999; Gaston & Blackburn 1999, 2000). The links between macroecological patterns are becoming apparent, but some retention of a reductionist philosophy will be required in their examination.
We thank S.L. Chown and P.H. Warren for the criticism that prompted this work, and J. Alroy, J.H. Lawton and P.H. Warren for commenting on it. T.M.B. is especially grateful to S. Gaston for her unfailing patience, understanding and generous hospitality over many years. T.M.B. was supported partly by a Leverhulme Special Research Fellowship, and partly by core funding from the NERC Centre for Population Biology. K.J.G. is a Royal Society University Research Fellow.
Received 6 June 2000; revision revision received 29 September 2000