Journal of Animal Ecology 2008, 77, 1056–1062 doi: 10.1111/j.1365-2656.2008.01405.x © 2008 The Authors. Journal compilation © 2008 British Ecological Society Blackwell Publishing Ltd Abundance–body size relationships: the roles of metabolism and population dynamics Hannah M. Lewis 1 *, Richard Law 1 and Alan J. McKane 2 1 Department of Biology, PO Box 373, University of York, York YO10 5YW, UK; and 2 Theory Group, School of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, UK Summary 1. Species’ abundance scales approximately as an inverse power of body mass. This property has been explained on the basis of metabolic rates of organisms of different sizes. 2. This paper considers the additional effect of population dynamics on the abundance–body size relationship, on the grounds that mass flow through food webs also depends on interactions between predators and their prey. To do this, an analysis of simple dynamical food-chain models was carried out, using rate parameters which scaled with body mass according to empirically based rules. 3. The analysis shows that a function for the abundance–body size relationship derived from metabolic theory is a good first approximation to a function derived for food chains at dynamic equilibrium, although the mechanistic interpretation of terms in the functions is not the same. 4. The results are sensitive to assumptions about the scaling of the self-limitation of basal species with respect to body size. Depending on the assumption made, the abundance–body size relationship may have a power parameter –1 at all trophic levels, or be described by different functions at different trophic levels. Key-words: abundance-mass scaling, energetic equivalence rule, linear biomass hypothesis, metabolic theory. 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 par- ameter 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: eqn 1 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 relation- ships results in eqn 1. Taking values α = 0·1 and β = 10 000, *Correspondence author. E-mail: hml103@york.ac.uk x i * x Km i i * log log . = = - Δ Δ where α β 3 4 xm i i * . / 34 BB i / 1 mm i ; log( / )/log( ) 1 =α β