## Introduction

Natural ecosystems comprise a large number of species engaging in a vast number of predator–prey interactions of variable strength (Berlow *et al*. 2004; Wootton & Emmerson 2005). Two categories of interaction strength most commonly studied include (1) the magnitude of energy flowing from prey to predator, or predator consumption rates; and (2) the change in abundance of one species given a change in abundance of another, or the dynamic coupling of two species. Both measures provide insight into the stability of complex food webs (Brose, Williams & Martinez 2006b; Montoya, Pimm & Sole 2006; Navarrete & Berlow 2006; Neutel *et al*. 2007; Otto, Rall & Brose 2007; Rall, Guill & Brose 2008). Thus a critical step for moving beyond descriptive, community-specific approaches in ecology is to uncover general principles that determine interaction strengths (Berlow *et al*. 2004; Brose, Berlow & Martinez 2005; Wootton & Emmerson 2005). While not always directly related, high consumption rates (high energy fluxes) can establish the potential for strong dynamic coupling between two species (Bascompte, Melian & Sala 2005; Brose *et al*. 2005). Here we focus on general principles that determine energy fluxes, because recent allometric models based on metabolic scaling theory (Brown *et al*. 2004) have proposed that predator consumption rates follow a power-law increase with predator body mass (Yodzis & Innes 1992; Emmerson & Raffaelli 2004; Reuman & Cohen 2005; Wootton *et al*. 2005). The elegant simplicity and empirical tractability of this theory has led to its recent widespread application in theoretical studies (McCann, Hastings & Huxel 1998; Emmerson & Raffaelli 2004; Williams & Martinez 2004; Brose *et al*. 2006b).

By defining physiological constraints, metabolic theory predicts the per capita (per individual) and total consumption rates necessary to sustain a population. However, being based only on predator body masses, this approach cannot distinguish the energy fluxes among the individual feeding links of generalist predators (hereafter, per link fluxes). It thus predicts per link fluxes by distributing per capita consumption rates equally across the feeding links. In contrast, foraging theory uses traits of the prey species, such as average body masses (where the average is an evolutionarily stable mean over the individuals of a population), to predict how behavioural aspects of predator–prey interactions determine the relative strength of these feeding links while ignoring the overall energy flux at the level of predator individuals. Here we propose a framework that integrates metabolic theory at the levels of predator individuals (per capita) with foraging theory at the level of feeding interactions (per link). Our focus on predator–prey energy fluxes complements recent analyses of how allometric foraging models predict food-web topology (Beckerman, Petchey & Warren 2006). We first introduce the models, then test their predictions using laboratory data on metabolism and consumption of arthropods. Macroecological abundance–mass relationships scale-up these predictions to natural communities with variance in predator and prey abundance.

### models

#### Per capita metabolism and consumption

At the level of individuals, metabolic theory predicts that the metabolic rates of species *i*, *I _{i}*[J s

^{−1}] scale with body mass,

*M*(g) as:

_{i}where *I*_{0} and *a* are constants (Brown *et al*. 2004). Moreover, it predicts that per capita consumption rates of consumer individuals of species *i*, *C _{i}*[J s

^{−1}], follow a similar mass-dependence to the metabolic rates:

where *C*_{0} and *b* are constants (Carbone, Teacher & Rowcliffe 2007).

#### Per link consumption

Direct application of this per capita relationship (equation 2) to individual feeding interactions between predator *i* and prey *j* yield per link consumption rates, *K _{ij}*[J s

^{−1}]:

where *K*_{0} is a constant and

(Yodzis & Innes 1992; Brown *et al*. 2004). This ‘metabolic model’ predicts energy fluxes through individual links by distributing the predator's per capita consumption rate equally across its feeding links.

In contrast, foraging theory suggests that per link predation rates *P _{ij}*[ind s

^{−1}] (the number of individuals of prey

*j*consumed per individual of predator

*i*) follow a hump-shaped relationship with predator–prey body-mass ratios

*R*=

_{ij}*M*/

_{i}*M*:

_{j}where *P*_{max}[ind s^{−1}] is the maximum predation rate, *R*_{max} is the body-mass ratio at which this maximum is achieved, and γ is a scaling constant (Wilson 1975; Persson *et al*. 1998; Wahlstrom *et al*. 2000; Bystrom *et al*. 2003; Aljetlawi, Sparrevik & Leonardsson 2004; Finstad, Ugedal & Berg 2006). This phenomenological foraging model describes (1) predation rates increasing with body mass ratios when predators are small relative to their prey (*R _{ij}* <

*R*

_{max}), which is explained by an increasing ability of the predator to subdue and handle prey; and (2) predation rates decreasing with body mass ratios when predators are large relative to their prey (

*R*>

_{ij}*R*

_{max}), which results from a decreasing detectability and catchability of smaller prey (Persson

*et al*. 1998; Aljetlawi

*et al*. 2004).

#### Macroecological energy fluxes

The metabolic model (equation 3) and the foraging model (equation 4) predict laboratory energy fluxes at a fixed abundance, [ind]. This implies similar laboratory encounter rates for each prey: where are constants that are independent of body mass. In natural communities, however, field encounter rates, *E _{ij}*[ind s

^{−1}] =

*E*

_{0}

*N*depend on field prey abundance,

_{j}*N*[ind] (Emmerson, Montoya & Woodward 2005) that scales with body mass as:

_{j}where *N*_{0} and *c* are macroecological constants and often *c* ≈ 0·75 (Brown *et al*. 2004; Meehan 2006). Synthesizing either the metabolic model or the foraging model with this macroecological abundance–mass relationship scales up predictions from laboratory to field conditions. The per capita energy flux, *F _{ij}*[J s

^{−1}], through the link from prey population

*j*to an individual of predator population

*i*, can be derived under the metabolic model by normalizing the per link consumption rates (equation 3) by the laboratory encounter rates, , and multiplying them with field encounter rates,

*E*:

_{ij}This approach distributes per capita consumption rates across feeding links using encounter rates that apply to averages over individuals within populations, while ignoring differences among individuals. Under the foraging model (equation 4), *F _{ij}*[J s

^{−1}] is equal to predation rates,

*P*[ind s

_{ij}^{−1}] divided by the laboratory encounter rate, , and multiplied by the field encounter rate,

*E*, and prey energy content [the product of prey mass,

_{ij}*M*[g ind

_{j}^{−1}] and the energy content per wet mass ɛ[J g

^{−1}]]:

The total energy flux from prey population *j* to predator population *i*, *T _{ij}*[J s

^{−1}], is defined as the product of the per capita energy flux (equation 6a or b) and predator abundance, which yields:

under the metabolic (equation 7a) and foraging (equation 7b) model, respectively.