The scale-dependence of population density–body mass allometry: Statistical artefact or biological mechanism?

Abstract

The relation between population density and body mass has vexed ecologists for nearly 30 years as a consequence of high variability in the observed slope of the relation: No single generalisation of the relation has been accepted as universally representative. Here, we use a simple computational approach to examine how observational scale (the body mass range considered) determines variation in the density–mass pattern. Our model relies on two assumptions: (1) resources are partitioned in an unbiased manner among species with different masses; (2) the number of individuals that can be supported by a given quantity of resources is related to their metabolic rate (which is a function of their mass raised to the power of a scaling coefficient, b). We show that density (1) scales as a function of body mass raised to the power of −b on average, but (2) the slope of the relation varies considerably at smaller scales of observation (over narrow ranges of body mass) as a consequence of details of species’ ecology associated with resource procurement. Historically, the effect of body mass range on the slope of the density–mass relation has been unfailingly attributed to a statistical effect. Here we show that the effect of body mass range on the slope of the density–mass relation may equally result from a biological mechanism, though we find it impossible to distinguish between the two. We observe that many of the explanations that have been offered to account for the variability in the slope of the relation invoke mechanisms associated with differences in body mass and we therefore suggest that body mass range itself might be the most important explanatory factor. Notably, our results imply that the energetic equivalence rule should not be expected to hold at smaller scales of observation, which suggests that it may not be possible to scale the mass- and temperature-dependence of organism metabolism to predict patterns at higher levels of biological organisation at smaller scales of observation.

Introduction

The relation between organism density (D) and body mass (M) has been a consistent source of discussion in the ecological literature for more than 25 years (Damuth, 1981, Lawton, 1989, Currie, 1993, Blackburn and Gaston, 1997, Arneberg et al., 1998, Griffiths, 1998, Cyr, 2000, Makarieva et al., 2004, White et al., 2007). While the origin and consequences of density–body mass allometry have stimulated theorists (Damuth, 1981, Peters and Raelson, 1984, Brown and Maurer, 1987, Enquist et al., 1998, Lawton, 1990, Enquist et al., 2003, Brown et al., 2004, Makarieva et al., 2004, Long et al., 2006), the true form of the relation has pre-occupied empiricists and remains a subject of considerable analysis and debate (White et al., 2007). The empirically driven debate concerning the form of the relation has cast serious doubt on its general utility and had dire consequences for developing theories (Peters and Wassenberg, 1983, Lawton, 1989, Lawton, 1990, Cotgreave, 1993, Currie and Fritz, 1993, Blackburn and Lawton, 1994, Marquet et al., 1995, Cyr, 2000, Loeuille and Loreau, 2006). There is, however, some general agreement that the revelation of the true nature of D–M allometry has been confused by the use of a variety of system definitions (White et al., 2007), data collection methodologies (Lawton, 1989, Damuth, 1991, Cotgreave, 1993, Blackburn and Gaston, 1997, Cyr et al., 1997), analytical techniques (LaBarbera, 1989, Griffiths, 1998, deBuryn et al., 2002), and differences in observational scale (Lawton, 1989, Nee et al., 1991, Currie, 1993, Blackburn and Gaston, 1997, Arneberg et al., 1998, Griffiths, 1998, Silva et al., 2001, Ackerman and Bellwood, 2003, White et al., 2007).

Despite the ongoing debate regarding the general form of the relation, body mass remains the best single predictor of population density, explaining 75–85% of the variation on average, and up to 90% in some cases (Damuth, 1981, Damuth, 1987, Arneberg et al., 1998). This strong link between population density and body mass is thought to follow from the mass-dependence of metabolic rate: The number of individuals of a species that can be supported in a given region (i.e. D), must be a function of the quantity of resources available to individuals of a given size (R) divided by the quantity of resources individuals of that size use, their total metabolic rate (B), such that D = (R/B) (Enquist et al., 1998; but see Li et al., 2004). Since metabolic rate is mass-dependent, we can substitute M for B. The mass-dependence of B is best described as a power function, B ∝ M^b, where b is the slope of the regression on log–log axes. Thus, D is expected to be proportional to R, assuming that R is mass-independent, and M raised to the negative power of b: D = (R/B) ∝ (R/M^b) ∝ (R·M^−b). Although the value of b is debated (McNab, 1988, Dodds et al., 2001, White and Seymour, 2003, Bokma, 2004, Kozlowski and Konarzewski, 2004), it is generally accepted that B ∝ M^3/4 (Dobson et al., 2003, Agutter and Wheatley, 2004, Savage et al., 2004). Thus, assuming that R is mass-independent, D is generally expected to scale as a function of M^−3/4, which is exactly the relation that Damuth, 1981, Damuth, 1987 originally found for terrestrial mammals.

The apparent simplicity of the pattern revealed by Damuth, 1981, Damuth, 1987 initiated a widespread search for similar patterns in other systems. The results of this search are somewhat inconclusive, with a variety of patterns reported (White et al., 2007). Three general forms of the D–M relation have been identified: (1) negative linear; (2) positive linear; (3) polygonal (“constraint envelope”; Marquet et al., 1995). The three general patterns exhibit a degree of scale-dependence (Lawton, 1990, Cotgreave, 1993, Currie, 1993, Blackburn and Gaston, 1997, Griffiths, 1998, White et al., 2007). At large scales (e.g., across broad taxa, wide ranges of body mass, and large geographic areas), the relation tends to be negative and linear with a fairly consistent slope of approximately −3/4 (Damuth, 1981, Damuth, 1987, Blackburn and Gaston, 1997, Enquist et al., 1998, Schmid et al., 2000, Belgrano et al., 2002, Meehan et al., 2004; but see Silva et al., 2001). At small scales (e.g., within lower taxa, narrow ranges of body mass, and small geographic areas) there is much less consistency in the pattern: some studies report strong negative linear relations (Marquet et al., 1990, Schmid et al., 2000, Ackerman and Bellwood, 2003); others report positive linear slopes (Patterson, 1992, Cotgreave and Harvey, 1994, Arneberg et al., 1998); and still others report a polygonal distribution of data points, with no or only a weakly detectable negative linear trend (Griffiths, 1986, Brown and Maurer, 1987, Morse et al., 1988, Nee et al., 1991, Cyr and Pace, 1993, Marquet et al., 1995, Navarrete and Menge, 1997, deBuryn et al., 2002).

Much of the debate about the form of the D–M relation has been attributed, in various ways, to the effect of scale (Damuth, 1991, Cotgreave, 1993, Currie, 1993, Blackburn and Gaston, 1997, Arneberg et al., 1998, Griffiths, 1998, Cyr, 2000). Many authors have recognised that the body-size range considered and the observed slope (and shape) of the D–M relation are intertwined (Peters and Wassenberg, 1983, Lawton, 1989, Currie, 1993, Cyr and Pace, 1993, Blackburn and Lawton, 1994, Blackburn and Gaston, 1997, Arneberg et al., 1998, Griffiths, 1998, Cyr, 2000, Ackerman and Bellwood, 2003). Arneberg et al. (1998), for example, undertook a meta-analysis of empirical values obtained for the scaling exponent and variance explained for D–M relations from the literature and found that much of the variation in the D–M pattern (b and r²) was associated with the range of body mass included in any particular study. Other overviews of the literature have reached similar conclusions (Lawton, 1989, Currie, 1993, Blackburn and Gaston, 1997, Griffiths, 1998), with the notable exception of Damuth (1993), who found no correlation between body mass range and observed slope. Here we use a simple computational approach to examine how observational scale (defined as the range of body mass considered) determines variability in the D–M pattern observed and compare the simulated results against observations collected from the literature. Specifically, we examine the effect of observational scale on variability in the D–M pattern using four kinds of simulation models. The first two models were designed to address whether the way in which resources are partitioned among members of a community of co-existing species affects variability in the D–M pattern observed. Thus, we created two kinds of “real-type” communities in which the species present must split the total amount of resources available using (1) the broken stick and (2) the niche preemption model of resource partitioning. Our third model, “compendia-type” communities, was created to address the fact that many of the studies that have examined the D–M relationship in the past involve the use of data collected from many literature sources. As such, our third model was designed to mimic literature-based examinations of D–M scaling by drawing species randomly from the many different communities created in the broken stick “real-type” communities simulation, in order to examine the possibility that some of the variability in the D–M relation might result from differences in sampling methodology or system definition. Our fourth model, “statistical artefact” communities, was designed to investigate the idea that the variability in the D–M relation observed at smaller scales of observation is a statistical artefact resulting from the examination of small sections of global regressions: We investigate the statistical artefact hypothesis by randomly selecting species from small sections of the global density–body size distribution generated in the broken stick “real-type” communities simulations. Finally, we compare the results of each of the four models to observations from the literature.