## Introduction

Prey selection has profound effects on stability of predator-prey dynamics (Fryxell & Lundberg 1994) and coexistence between competing prey (Paine 1966; Holt & Kotler 1987). The mechanisms underlying food preferences are therefore essential components of community ecology. Ever since the publication of a seminal paper by MacArthur & Pianka (1966), the mainstream prey-selection studies have approached the problem from an optimality perspective, of which the so-called contingency model (CM) is best known (Pulliam 1974; Charnov 1976; also called the prey model by Stephens & Krebs 1986). For a predator that aims to maximize its long-term average energy intake rate, the CM predicts which prey types should be accepted. A prey type *i* is characterized by its metabolizable energy content *e _{i}* and required handling time

*h*. The optimal decision rule, the so-called ‘zero-one’ rule, is simple: prey types whose profitability (

_{i}*e*/

_{i}*h*) exceeds or equals long-term average energy intake rate should be included in the diet, while those prey types with lower profitabilities should be ignored.

_{i}The CM has been developed for so-called ‘handling-limited’ foragers (Farnsworth & Illius 1998), i.e. animals that spend all of their time foraging (handling and searching) and whose maximum rates of energy intake are ultimately constrained by the rate at which prey can be externally handled (Jeschke, Kopp & Tollrian 2002). However, as recently concluded by Jeschke *et al*. (2002), rates of energy intake are mostly constrained by digestion rates rather than by handling rates. Whenever rate of digestion constrains rate of energy intake (i.e. a digestive bottleneck; Kenward & Sibly 1977), a forager may need to take digestive pauses before any new prey item can be ingested (e.g. Van Gils *et al*. 2003b). A forager obeying ‘contingency-rules’ would maximize its energy intake per unit foraging time (i.e. short-term intake rate, cf. Fortin, Fryxell & Pilote 2002) but not necessarily per unit total time (i.e. foraging plus digestive pauses; long-term intake rate, cf. Fortin *et al*. 2002). In fact, whenever prey types differ in the rate at which they can be assimilated (W), a digestively constrained forager obeying the CM does not maximize its energy intake over total time (Verlinden & Wiley 1989). If natural selection acts primarily upon energy intake over total time (Stephens & Krebs 1986), a digestively constrained forager should be selective towards prey types that can be digested rapidly. Time which otherwise would be lost to digestive pauses can then be used to search for easy-to-digest prey types (Verlinden & Wiley 1989; Hirakawa 1997a).

The so-called digestive rate model (DRM) predicts optimal diets that maximize long-term energy intake rate in such digestion-constrained situations (Verlinden & Wiley 1989 as amplified by Hirakawa 1997a; Hirakawa 1997b; Farnsworth & Illius 1998; as adjusted by Fortin 2001). DRM is structurally similar to the CM in the sense that prey types can be ranked in terms of rate of energy uptake. In fact, the CM's ‘zero-one’ rule emerges from the DRM as an optimal solution for the (restricted) case that rate of energy intake is ‘handling-limited’. The best-known alternative diet model that considers a digestive constraint, the linear programming model (LPM; Belovsky 1978, 1984, 1986), has been criticised for its inability to consider many prey types and for being circular (Owen-Smith 2002). DRM lacks these limitations.

It is interesting to note that although digestive constraints are believed to be widespread (Masman *et al*. 1986; Prop & Vulink 1992; Kersten & Visser 1996; Guillemette 1998; De Leeuw 1999; Jeschke *et al*. 2002; Karasov & McWilliams 2004), and food preferences are often explained in the light of digestive bottlenecks (Kenward & Sibly 1977; Bustnes & Erikstad 1990; Zwarts & Blomert 1990; Kaiser *et al*. 1992), actual explicit tests of optimal diet models that consider a digestive bottleneck have mainly been restricted to herbivores (Westoby 1974; Belovsky 1978; Owen-Smith & Novellie 1982; Vivås, Sæther & Andersen 1991; Shipley *et al*. 1999; see Hoogerhoud 1987 for an application to molluscivorous cichlids). In most of these cases, LPM or modified versions of the CM were tested; tests of DRM have been rare up to now (but see Fortin *et al*. 2002; Illius *et al*. 2002).

In this study we provide, to the best of our knowledge, the first explicit test of a DRM in a nonherbivore. Our study species is the red knot (*Calidris canutus*), a medium-sized shorebird that lives in intertidal habitats where it feeds mostly on molluscs. Due to their habit of swallowing prey whole, a lot of bulky, indigestible ballast (shell) material enters the digestive tract (80–90% of total prey dry mass; Zwarts & Blomert 1992). As their buried prey are detected relatively efficiently by a pressure-sensory system in the bill tip (Piersma *et al*. 1998) and as handling times are relatively short (Zwarts & Wanink 1993; Piersma *et al*. 1995), digestive processing rates often cannot keep up with rates of prey encounter and ingestion. Thus, energy intake of red knots tends to be digestion- rather than handling-limited (Van Gils *et al*. 2003a, 2003b, 2005; Van Gils & Piersma 2004). It turns out that the constraining link in the chain of digestive processes is the rate at which shell mass is crushed and processed, and that knots can alleviate this constraint to a certain extent by flexibly increasing the size of the crushing organ, the muscular gizzard (Dekinga *et al*. 2001; Van Gils *et al*. 2003a). The relative ease with which we can (1) measure available prey densities (Zwarts, Blomert & Wanink 1992; Piersma *et al*. 1993b; Piersma, de Goeij & Tulp 1993a; Piersma, Verkuil & Tulp 1994) (2) experimentally manipulate prey densities (Piersma *et al*. 1995; Van Gils *et al*. 2003b) (3) reconstruct diet composition (Dekinga & Piersma 1993) (4) experimentally quantify handling and searching times (Piersma *et al*. 1995), and (5) noninvasively estimate gizzard mass as a predictor of digestive processing capacity (see below; Dietz *et al*. 1999), makes the knot an ideal species to study factors determining prey choice. Our objective here is to test, both under controlled laboratory conditions and in the wild, whether prey choice by red knots follows the predictions of the CM or those of the DRM.

### the digestive rate model

The DRM applied here (Hirakawa 1995) assumes that knots aim to maximize long-term average metabolizable energy intake rate *Y*, under the constraint that ballast mass intake rate *X* does not exceed a specific threshold. Metabolizable energy intake rate *Y* (W) while foraging is given by (notation follows Stephens & Krebs 1986 and Hirakawa 1995):

where λ_{i} is encounter rate (s^{−1}) with items of prey type *i* (= 1, … , *n*), *p _{i}* is the probability that type

*i*is accepted upon encounter,

*e*is the metabolizable energy content (J) of type

_{i}*i*, and

*h*is the time (s) required to handle type

_{i}*i*. Recall that this is the familiar Holling's disc equation in a multiple prey context (Holling 1959; Charnov 1976). Similarly, ballast material intake rate

*X*(g s

^{−1}) while foraging is given by:

where *k _{i}* is the ballast mass of prey type

*i*. The digestive constraint is expressed as

where *c* is maximum digestive capacity (g s^{−1}), expressed as the upper limit on long-term ballast intake rate *X*.

Hirakawa (1995) presented a (graphical) solution procedure of the problem to find **P** = (*p*_{1}, *p*_{2}, ... , *p** _{n}*) that maximizes

*Y*under constraint

*X*

*c*. First, for each prey type

*i*, profitability (i.e. the ratio

*e*/

_{i}*h*) is plotted vs. the ratio ballast mass/handling time (

_{i}*k*/

_{i}*h*). Second, in the same graph, the so-called ‘feasible region’ is plotted, that is the region that contains the

_{i}*X*-

*Y*data for all possible prey choice strategies, that is for all possible vectors

**P**. The term ‘feasible’ refers to what intake rates are feasible while foraging (short-term) within the constraints of food environment

**F**= (

**f**) = ([

_{i}*e*,

_{i}*k*,

_{i}*h*, λ

_{i}*]). Hirakawa (1995) provides a very elegant graphical procedure to find the boundaries of the feasible region in a relatively simple and straightforward manner. Basically, the procedure starts with including the prey type with the highest ratio*

_{i}*e*/

_{i}*k*(which will be called ‘quality’ or ‘digestive quality’ cf. Verlinden & Wiley 1989) as the first prey type in the diet. This gives:

_{i}and

Hence, graphically, the bivariate point *X*,* Y* | (**P** = 1, 0, ... , 0) is located on the line that connects the origin (0, 0) with the point (*k*_{1}/*h*_{1}, *e*_{1}/*h*_{1}). See Fig. 1. Including a second prey type in the diet gives:

and thus

and similarly for *Y* | (**P** = 1, 1, 0, ... , 0). Hence, point *X*,* Y* | (**P** = 1, 1, 0, ... , 0) is located on the line that connects point* X*,* Y* | (**P** = 1, 0, ... , 0) with point (*k*_{2}/*h*_{2}, *e*_{2}/*h*_{2}), and so forth (Fig. 1). By a clever choice of new points to be included in the diet (or excluded again), one can graph the boundary lines of the feasible region quite easily. For details we refer to the original paper (Hirakawa 1995). It can further be shown that the boundary line at the optimal *X*, *Y* point (recall that the optimal point has maximum *Y*, defined as *Y*_{max}, while still obeying *X* ≤ *c* in the long run) separates those prey types that are included in the diet from those that are excluded (the so-called ‘Optimal Diet Line’ ODL; Fig. 1).

### testing the drm against the cm

We provide three tests of the DRM against the CM. The first is an experimental test of preference for 11 different prey types (different size classes of 5 different prey species) that were offered pairwise, unburied, and *ad libitum* to single (captive) knots (Fig. 2A–B). Given the birds’ gizzard masses and high short-term intake rates (since no search time was required), these experimental conditions ensured long-term energy intake rate to be digestion- rather than handling-limited. Under these conditions, the CM, which ignores digestive constraints, predicts that only the most profitable (*e*/*h*) prey should be preferred (Fig. 2A). In contrast, the DRM predicts that only prey of the highest digestive quality (*e*/*k*) should be taken (Fig. 2B).

The second test is an experiment where two buried prey types (two size classes of a single species) were offered pairwise in low densities to single (captive) red knots. The most profitable prey type was lowest in digestive quality and occurred in higher densities than the less profitable but higher quality prey (Fig. 2C–D). Given experimental conditions (gizzard masses and short-term intake rates), the CM predicts the higher quality prey type should be completely ignored (*P =* 0), while the lower quality prey type should be completely accepted (*P =* 1; Fig. 2C). In contrast, the DRM predicts almost the opposite: the higher-quality prey type should be completely accepted (*P =* 1), while the lower quality prey type should only be partially accepted (0 < *P <* 1; Fig. 2D).

The third test is performed on free-ranging red knots in the wild that fed in a natural patch containing multiple prey types of two species. The CM predicts both species to be eaten in equal amounts, while the DRM predicts the diet to be composed mainly of the higher quality prey species.