## Introduction

The functional response, which gives the predator's per capita feeding rate on prey, plays a key role in ecology, and during the last two decades it has become clear that theoretical predictions on, for example, predator–prey dynamics (Arditi & Ginzburg 1989; Abrams & Ginzburg 2000), spatial patterns of predators (Van der Meer & Ens 1997; Alonso, Bartumeus & Catalan 2002), and the structure and dynamics of communities and ecosystems (DeAngelis, Goldstein & O’Neill 1975; Ginzburg & Akçakaya 1992; Fussmann & Blasius 2005), are highly sensitive to the precise form of the employed functional response. Particularly, the question of whether or not the functional response should be predator-dependent, in order to incorporate the effect of mutual interference among predators, has received a lot of attention (Arditi & Akçakaya 1990; Abrams & Ginzburg 2000; Skalski & Gilliam 2001; Fussmann, Weithoff & Yoshida 2005; Jensen, Jeschke & Ginzburg 2007).

The prey-dependent Holling's type II functional response, also known as the disc equation (Holling 1959), is still the most widely used functional response model (Skalski & Gilliam 2001; Jeschke, Kopp & Tollrian 2002). The model is based upon the idea that the predator can be in either one of two mutually exclusive states: it is either searching for prey or handling prey. The model furthermore assumes that the capture rate of the predator (and hence the transition from the searching to the handling state) is proportional to the number of prey in the system. Beddington (1975) extended Holling's approach by incorporating the possibility of multiple predators that can mutually interfere. He assumed that predators when meeting a competitor start to interfere, that is they then enter an interference state and he arrived at a functional response in which the predator's per capita intake rate is predator dependent, that is it is a function of both prey density and predator density. More or less at the same time, DeAngelis *et al*. (1975) independently proposed the same function, but considered it as an empirical relationship and did not attempt to derive the term. The model is often referred to as the Beddington–DeAngelis model (Skalski & Gilliam 2001). Later Ruxton, Guerney & De Roos (1992) embroidered upon Beddington's approach and corrected some major inconsistencies in the derivation of the model. Yet, they arrived at the same equation.

Skalski & Gilliam (2001) took a practical approach to the question of how relevant predator-dependent functional responses actually are, and collected 19 data sets from the literature where predator intake rates were measured for at least two prey densities and two predator densities. These data sets were used to examine which model, Holling or Beddington–DeAngelis, gives a better description of predator intake rates. In their comparison, they also included few other functional response models, for example, the Hassell–Varley model, that were not derived from mechanistic principles. They remarked that mechanistic models do, however, lead to clearer science, reiterating a plea we earlier made (Van der Meer & Ens 1997). From their analysis, they concluded that in most cases the Beddington–DeAngelis model provided a better description of predator feeding than Holling's type II functional response.

Skalski & Gilliam (2001) used only experimental data where both predator and prey numbers were under experimental control. Experiments are indeed to be preferred over observational field data where many factors may be confounded with prey and predator densities. However, such experimental data are usually (and, certainly for larger animals, almost necessarily) from experiments with very few predators, quite often as few as two (Swennen, Leopold & De Bruijn 1989; Mansour & Lipcius 1991; Van Gils & Piersma 2004; Vahl *et al*. 2005; Smallegange, Van der Meer & Kurvers 2006; Smallegange & Van der Meer 2007). This practice contradicts the assumption of infinitely large prey and predator populations, made in the derivation by Ruxton *et al*. (1992) and, as we show below, also made by Beddington (1975). So here, we do have a problem: the theory is based on infinitely large populations, but the experimental data required to link the theory to reality are usually from experiments with very few predators. Proper analysis of such experiments requires a functional response model valid for a finite number of predators. Here we present such a model using the theory of stochastic processes. The underlying behavioural rules within the model are entirely analogous to those used by Ruxton *et al*. (1992). In passing, we correct a mistake in Ruxton *et al.*'s derivation for infinitely large populations, with the consequence that the resulting model is not entirely equivalent (though rather similar) to the Beddington–DeAngelis model. Finally, using experimental data on shore crabs feeding on blue mussels, we show how the stochastic model can be fitted to data.