## Introduction

Understanding constraints on species’ interaction strengths is critically important for predicting population dynamics, food-web stability and ecosystem functions such as biological control (Berlow *et al.* 2004; Wootton & Emmerson 2005; Montoya, Pimm & Solé 2006). Empirical and theoretical evidence suggests that predator and prey body masses are among the most important of these constraints (Emmerson & Raffaelli 2004; Woodward *et al.* 2005; Wootton & Emmerson 2005; Berlow, Brose & Martinez 2008; Berlow *et al.* 2009). Conceptually, these nonlinear interaction strengths are described by the magnitude and shape of functional responses that quantify *per capita* consumption rates of predators depending on prey abundance. One generalized functional response model is based on Holling’s disk equation (Holling 1959):

where *F* is the *per capita* consumption rate, *N* is prey abundance, *T*_{h} is the handling time needed to kill, ingest and digest a resource individual, *b* is a search coefficient that describes the increase in the instantaneous search rate, *a*, with resource abundance, *N*:

where *q* is a scaling exponent that converts type II into type III functional responses (Williams & Martinez 2004; Rall, Guill & Brose 2008). The hill exponent, *h*, used in some prior studies (Real 1977) is equivalent to *q* (*h* = *q* + 1).

Functional responses can be linear (type I, *T*_{h} = 0, increase up to a threshold abundance), hyperbolic (type II, *T*_{h} > 0, *q* = 0) or sigmoid (type III, *T*_{h} > 0, *q* > 0). While many early studies focused on type I and type II functional responses and ignored the scaling exponent, type III functional responses with scaling exponents larger than zero could occur more frequently than previously anticipated (Sarnelle & Wilson 2008). Under hyperbolic type II functional responses predation risks for prey individuals decrease with prey abundance causing inverse density-dependent prey mortality, which can lead to unstable boom-burst population dynamics (Oaten & Murdoch 1975a, b; Hassell 1978). In contrast, increasing predation risks under sigmoid functional responses can yield an effective *per capita* top-down control that often prevents such unstable dynamics (Gentleman & Neuheimer 2008; Rall *et al.* 2008). Slight differences in functional response parameters can thus have drastic consequences for population and food-web stability in natural ecosystems (Oaten & Murdoch 1975a; Williams & Martinez 2004; Fussmann & Blasius 2005; Brose, Williams & Martinez 2006b; Rall *et al.* 2008).

Allometric scaling theories provide a conceptual framework how body masses could determine foraging interactions (Peters 1983; Brown *et al.* 2004). The maximum consumption rates realized at infinite prey densities are proportional to the inverse of handling time and independent of the success of the attacks (Yodzis & Innes 1992; Koen-Alonso 2007). Consequently, the ¾ power-law scaling of maximum consumption with predator body mass (Peters 1983; Carbone *et al.* 1999) suggests that handling time should follow a negative ¾ power-law with predator body mass. This trend is qualitatively supported, though studies reported linear (Hassell, Lawton & Beddington 1976; Spitze 1985), power-law or exponential relationships (Thompson 1975; Hassell *et al.* 1976; Aljetlawi, Sparrevik & Leonardsson 2004; Jeschke & Tollrian 2005).

The characteristic components of search rates include the reactive distance between predator and prey (i.e. the distance between predator and prey individuals at which a predator individual responds to the presence of the prey) and the capture success. While the reactive distance increases with the body masses of the predators (i.e. large predators have a larger visual range than small predators), the capture success decreases with predator mass above an optimum body mass ratio (Aljetlawi *et al.* 2004; Brose *et al.* 2008). A further explanation for the low capture success is that the predator’s motivation to capture small prey of limited energy content is low (Petchey *et al.* 2008). Together, these patterns in reactive distances and capture success may explain the hump-shaped relationships between search rates and predator–prey body-mass ratios with a maximum search rate at intermediate (optimum) body-mass ratios documented in prior studies (Hassell *et al.* 1976; Wahlström *et al.* 2000; Aljetlawi *et al.* 2004; Vonesh & Bolker 2005; Brose *et al.* 2008). However, these studies were either restricted to search rates of single predator–prey interactions (with variance in individual size) or studied multiple predator–prey search rates at a single, constant prey density. Thus, none of these prior studies has addressed body-size constraints on functional responses across species.

In this study, we quantified systematic effects of predator and prey masses on functional response parameters (handling time, search coefficient and scaling exponent) across different predator–prey interactions. While more complex functional response models accounting for digesting time and interference behaviour exist (Skalski & Gilliam 2001; Jeschke, Kopp & Tollrian 2002, Schenk, Bersier & Bacher 2005; Kratina *et al.* 2009), testing for their body-size dependence was left for subsequent studies. Instead, the allometric functional response model addressed here provide an empirical basis for an understanding of body-size constraints on interaction strengths, food-web topology (Petchey *et al.* 2008) and dynamics (Brose *et al.* 2006b; Otto, Rall & Brose 2007; Brose 2008; Rall *et al.* 2008).