## Introduction

Many properties of species scale with body mass according to a power law relationship *Y* = *am*^{Δ}, where *Y* is the trait value and *m* is species body mass. Such properties include ingestion rate, metabolic rate, growth rate, birth rate, death rate and generation time (Peters 1983). One of the best documented ecological relationships is that between body mass and population density (or numerical abundance). The power parameter Δ is generally acknowledged to be negative (but see Marquet, Navarrete & Castilla 1995), but its magnitude is not so clear. Within a single trophic level an exponent Δ ≈ –3/4 has been suggested (Damuth 1981; Schmid, Tokeshi & Schmid-Araya 2000; Cermeño *et al*. 2006). Across trophic levels an exponent Δ ≈ –1 is widely reported (Peters 1983; Boudreau & Dickie 1992; Schmid *et al*. 2000). These values are often treated as a benchmark in empirical studies (e.g. Long *et al*. 2006), although it is notable that the empirical literature has recorded a wide range of values of the exponent (e.g. Peters & Raelson 1984; Juanes 1986; Robinson & Redford 1986; Long *et al*. 2006; Blanchard unpublished).

Metabolic theory provides an argument to explain the scaling of abundance (implicitly an equilibrium abundance of a single population *i*) with body mass *m*_{i} at trophic level* i* (Brown & Gillooly 2003; Brown *et al*. 2004). The relationship is:

Here α is the ratio of total metabolic rate at adjacent trophic levels, β is ratio of body sizes at adjacent trophic levels (the predator : prey size ratio), and *K* is a constant. The values of α and β are assumed to remain unchanged at all adjacent pairs of trophic levels. The reasoning behind the relationship starts from an observation that an individual's metabolic rate scales approximately as *m*^{3/4} (Kleiber 1932; Peters 1983; West, Brown & Enquist 1997), and hence the total metabolic rate *B*_{i} at trophic level *i* scales as The argument assumes that, irrespective of trophic level, the ratio of *B*_{i}s at adjacent trophic levels all have the same value α. So *B*_{i} at trophic level *i* is related to the total rate of metabolism of basal species, *B*_{1}, by *B*_{i}/*B*_{1} = α^{i–1}. In a similar way, the body size of an organism at trophic level at *i* is related to the body size *m*_{1} of one at trophic level 1 as *m*_{i}/*m*_{1} = β^{i–1}. Taking logarithms, this means that *i* – 1 = log(*m*_{i}/*m*_{1})/log(β), and therefore that using the laws of logarithms, this gives *B** _{i}*/

*B*

_{1}= (

*m*

*/*

_{i}*m*

_{1})

^{logα/logβ}. Algebraic manipulation of these relationships results in eqn 1. Taking values α = 0·1 and β = 10 000, the power parameter is –1, corresponding to the approximate empirical relationship (Brown

*et al*. 2004).

This paper considers the effect of population dynamics on the relationship between abundance and body mass at dynamic equilibrium. The motivation for this is that, in addition to basic matters of metabolism, the transfer of energy through ecological communities depends on the behaviour of predators and prey, and how this behaviour impacts on their population dynamics.

There are two different ways in which dynamics can be incorporated, depending on what is meant by body size. The first takes body size as being fixed for a species, and is used in the metabolic theory and much research on food webs (e.g. Cohen, Jonsson & Carpenter 2003; Brown *et al*. 2004; Woodward *et al*. 2005; Brose, Williams & Martinez 2006). The second allows body size of individuals to change as feeding takes place so a species (or some aggregation of species) is represented by a spectrum of sizes at different densities (Silvert & Platt 1978, 1980; Camacho & Solé 2001; Benoît & Rochet 2004; Maury *et al*. 2007a,b). We use the former approach here, as this is in keeping with the approach used in metabolic theory.

Although metabolic theory does recognize that dynamics could leave their mark on equilibrium population densities (Brown, Allen & Gillooly 2007), the quantitative effects of food-chain dynamics on eqn 1 have not previously been studied. There are also empirical matters left unresolved by the metabolic argument such as the large amount of unexplained variation around the predicted relationship (Peters & Raelson 1984; Juanes 1986; Robinson & Redford 1986; Horn 2004; Long *et al*. 2006), and differences in the relationships at different trophic levels (Long *et al*. 2006; Blanchard unpublished). Loeuille & Loreau (2006) and Rossberg *et al*. (2008) give related analyses on the emergence of the energetic equivalence rule; these use more complex communities and are not amenable to a formal analysis of scaling.

The simple population-dynamic approach developed here is based on a chain of three species with Lotka–Volterra dynamics, but it extends to a larger class of dynamical systems at equilibrium. This makes the comparison with metabolic theory conservative: discrepancies between metabolic theory and population dynamics are likely to be greater when more details of predator–prey interactions are introduced. The analysis shows that population dynamics generate a relationship between abundance and body size close to that observed in nature, and that the metabolic model (eqn 1) (Brown & Gillooly 2003; Brown *et al*. 2004) emerges as a good approximation to this relationship. A feature of the results is that different forms of intraspecific competition in the basal species generate different scalings within trophic levels elsewhere in the food chain, and can offer an explanation for nonlinear relationships between log abundance and log mass sometimes observed in nature. Small differences in scaling of abundance with body size within trophic levels could, however, be hard to detect in practice. It is important to be aware that the reduction of interactions in food webs to a single ratio β of predator to prey mass used in metabolic theory and the corresponding analysis here greatly simplifies a complex ecological process.