Individual variation in intake rate: the relative importance of foraging efficiency and dominance

Authors


R.A. Stillman, Centre for Ecology and Hydrology, CEH Dorset, Winfrith Technology Centre, Dorchester, Dorset DT2 8ZD, UK. E-mail: RAST@CEH.AC.UK

Summary

1. Individual variation in the intake rate of foraging animals arises largely from variation in two characteristics: their intrinsic ability to forage in the absence of competitors (foraging efficiency) and the detrimental effect of competitors on their intake rate (susceptibility to interference). We use a model to explore the relative importance of foraging efficiency and dominance, which influences susceptibility to interference, in determining the intake rate of foraging animals at a range of competitor densities.

2. The model is parameterized and tested for a shorebird–prey system, oystercatchers Haematopus ostralegus L. feeding on mussels Mytilus edulis L., in which interference occurs because dominant individuals steal prey from subdominant ones. In both oystercatchers which open mussels by stabbing their bill between the shell valves (stabbers) and those which hammer a hole through the shell (hammerers), foraging efficiency is predicted to be the major determinant of intake rate at competitor densities below 100–150 birds ha-1 and dominance the major determinant at higher densities. In reality, a very similar relationship is found in stabbers, but in hammerers foraging efficiency remains the most important determinant of intake rate across the full range of observed competitor densities, in contrast to the model's prediction. Oystercatchers typically forage at densities in the range 100–250 birds ha-1, in the region in which foraging efficiency and dominance are of approximately equal importance to stabbers, while foraging efficiency is most important to hammerers. We suggest reasons for the difference in the model's predictive power for stabbers and hammerers.

3. We use the model to make the general predictions that the relative importance of foraging efficiency is higher when (i) it varies more between individuals, (ii) prey encounter rate is high, (iii) handling time is short, (iv) prey stealing (kleptoparasitic) attacks occur over short distances, (v) the probability of stealing prey is low, and (vi) movement speed while foraging is low. In the oystercatcher–mussel system, prey encounter rate is lower and handling time longer than in most other shorebird–prey systems. This suggests that the importance of foraging efficiency in determining intake rate is likely to be greater in many other shorebirds.

4. Most studies of between individual variation in intake rate have focused on the importance of dominance rather than variation in foraging efficiency. Both the model predictions and field data suggest that variation in foraging efficiency may be a more important source of variation in intake rate than its previous depth of study would suggest.

Introduction

Individual variation in the intake rate of foraging animals arises largely from variation in two individual characteristics: foraging efficiency, their intake rate in the absence of competitors, and susceptibility to interference, the detrimental effect of competitors on their intake rate (Goss-Custard & Durell 1987; Sutherland 1996; Goss-Custard & Sutherland 1997). Although individuals may differ in both ways, studies have tended to concentrate on variation in susceptibility to interference (e.g. van der Meer & Ens 1997) and, in particular, on how this variation is influenced by dominance (e.g. Ens & Goss-Custard 1984; Sutherland & Parker 1992; Sutherland 1992; Sutherland & Dolman 1994; Stillman et al. 1996; Stillman, Goss-Custard & Caldow 1997). Fewer studies have investigated variation in foraging efficiency between individuals (although see Partridge (1976), Goss-Custard and Durell (1988), Sutherland & Parker (1992), Illius et al. (1995)), Cresswell (1998) and Caldow et al. (1999)) for examples of such studies).

The relative importance of foraging efficiency and dominance in determining intake rate is likely to depend on competitor density (Caldow et al. 1999). Foraging efficiency will, by definition, be the sole determinant of intake rate at low competitor densities at which interference due to competitive interactions is absent. The relative importance of dominance will increase with increased competitor density if the strength of interference also increases and if susceptibility to interference is related to dominance. However, the precise importance of each characteristic will depend on specific details of a system such as (i) the extent of individual variation in foraging efficiency, (ii) the rate at which the strength of interference increases with increased competitor density, (iii) the extent to which susceptibility to interference is related to dominance and (iv) the competitor densities at which animals usually forage.

In this paper we use a model to predict the shape of the relationship between competitor density and the relative importance of foraging efficiency and dominance in determining the intake rate of foraging animals. An earlier version of the model accurately predicted the strength of interference in oystercatchers Haematopus ostralegus L. feeding on mussels Mytilus edulis L. and the presence or absence of interference in a range of shorebird–prey systems (Stillman et al. 1997). Importantly, the new model extends the previous one by allowing individuals to differ in both their foraging efficiency and dominance, instead of only in dominance. The model is once again parameterized for oystercatchers feeding on mussels, because suitable data were not available for other species. The extent to which dominance influences susceptibility to interference in oystercatchers depends on the feeding method used to open mussels (Stillman et al. 1996), and so the model was parameterized and tested separately for individuals which open mussels by stabbing their bill between gaping shell valves (stabbers) and those which hammer a hole through the shell (hammerers) (see Hulscher (1996) for a full description of oystercatcher feeding methods). Previously the model was only tested for stabbers (Stillman et al. 1997). Although the model is tested for oystercatchers, we also use it to make more general predictions of the relative importance of foraging efficiency and dominance in determining individual variation in intake rate in other predator–prey systems.

The model

An earlier version of the model has been described by Stillman et al. (1997) and only a brief description is given here. Predators forage in a square patch and capture prey either directly from the patch itself or by stealing from competitors. Prey are uniformly distributed within the patch and are not depleted during simulations. Simulations progress in discrete, 1-s time intervals during each of which each predator occupies one of four behavioural states: searching for or handling prey, avoiding or fighting with competitors. At the start of simulations, all predators are set to the searching state and given a random direction of movement. During simulations, the model follows their movement and changes in behavioural state as they encounter prey and competitors.

Searching predators move in a straight line at constant speed, v, and during each time interval have a constant probability, PC, of capturing prey directly from the patch. After capturing prey, the handling predator remains stationary and spends a fixed amount of time, TH, overcoming the prey's defences before consuming it instantaneously (it is assumed that prey are consumed in this way to avoid disputes over partially consumed prey). If a searching predator comes within the avoidance distance, DA, of any other it may avoid it. Avoiding predators change direction to move directly away from a competitor at a constant speed, v, for a fixed period, TA, before resuming searching. While avoiding, predators cannot capture or steal prey. If a searching predator comes within the fight distance, DF, of another which is handling prey it may start a fight in an attempt to steal the prey. Fights last for a fixed amount of time, TF, and their outcome depends on the relative dominance of the two competitors. After an individual attacks another and wins a fight, it has a probability, PS, of also successfully stealing the prey, otherwise the victim keeps the prey even though it lost the fight.

The responses of predators to encounters with competitors (avoidance and fighting) are calculated using optimal decision rules. Full details of these rules are given in Stillman et al. (1997) but briefly, two alternative responses are identified to the two possible types of encounter.

1. Fight encounters: a predator may respond either by initiating a fight or by continuing to search.

2. Avoidance encounters: a predator may either avoid a competitor or continue along its current search path. All possible outcomes of a given response are identified (e.g. when responding by initiating a fight, one outcome is that the fight is won and prey stolen, another that the fight is lost). The probability of each outcome occurring and its associated prey gain and time cost are then calculated. The mean gain (inline image) and time cost (inline image) associated with a response (R) are found by weighting the gains and time costs of all outcomes by their probability of occurrence:

image(eqn 1)
image(eqn 2)

where n= total number of possible outcomes to the response, Pi= probability of ith outcome, Gi= energy gain associated with ith outcome, and Ci= time cost of ith outcome. The expected intake rate associated with a response (IR) is then calculated from the ratio of inline image and inline image:

image(eqn 3)

This procedure is repeated for each possible response, and the optimal response (which the animal will adopt) defined as that which maximizes expected intake rate. If expected intake rates are equal, the animal has a 50% probability of adopting each response. See Stillman et al. (1997) for the actual equations used to calculate the costs and gains associated with each response.

Different individuals in the model vary in their foraging efficiency and dominance. Foraging efficiency (E) influences the chance that a searching individual will capture prey from the patch.

image(eqn 4)

where inline image= the capture probability of an individual of average foraging efficiency. Individuals of higher foraging efficiency capture prey more frequently. Dominance (D) influences an individual's chance of winning fights with competitors, and hence its susceptibility to interference. A strict dominance hierarchy is assumed, so that the dominant individual in a fight always wins regardless of the absolute difference in dominance between the participants. Foraging efficiency is assumed to be normally distributed with unit mean and fixed coefficient of variation (Ecv), and dominance uniformly distributed between 0 and 1. The foraging efficiency and dominance of an individual are assumed to be independent of each other. The assumptions of normally distributed foraging efficiency, a strict dominance hierarchy and the independence of foraging efficiency and dominance hold for oystercatchers (Ens & Goss-Custard 1984; Goss-Custard & Durell 1987; Caldow et al. 1999), the species against which the model's predictions are tested.

All simulations were run with 100 birds competing in the patch. Variation in competitor density was achieved by changing the patch width. Simulations were run for 11 000 time intervals with results calculated from the last 10 000 intervals. Results varied slightly between replicate simulations, and so 10 simulations were run for each combination of parameter values and the results averaged. The relative importance of foraging efficiency was calculated using the following importance index (IE).

image(eqn 5)

where

image

= proportion of variation in intake rate explained solely by foraging efficiency and

image

= proportion of variation in intake rate explained solely by dominance. The values of

image

and

image

were calculated from two linear regressions of intake rate against either foraging efficiency or dominance and a multiple linear regression of intake rate against both foraging efficiency and dominance:

image

= proportion of variation explained by both variables in the multiple regression minus proportion of variation explained by dominance;

image

= proportion of variation explained by both variables minus proportion of variation explained by foraging efficiency. The index ranged from 1, when foraging efficiency was the sole determinant of intake rate, to 0, when dominance was the sole determinant.

The following parameter values were used to model mussel-feeding oystercatchers using the stabbing or hammering feeding method. Hammerers had a higher average prey encounter rate (PC= 0·010 s-1 for stabbers and 0·014 s-1 for hammerers), faster searching speed (v = 0·29 m s−1 for stabbers and 0·32 m s-1 for hammerers) and longer handling time than stabbers (TH = 69 s for stabbers and 109 s for hammerers) (J. D. Goss-Custard, unpublished). The observed between-individual coefficient of variation in interference-free intake rate is higher in stabbers (0·15) than hammerers (0·10) (Goss-Custard et al. 1995). The coefficients of variation in foraging efficiency used in the model (Ecv = 0·24 for stabbers and 0·22 for hammerers) were calculated so that the variation in interference-free intake rate between model birds was the same as that between real birds. All other parameters were the same for both feeding methods. Fight duration (TF = 7 s) was the average length of time taken to initiate and complete a dispute over a mussel (Goss-Custard, Cayford & Lea 1998; R. A. Stillman, unpublished) and fighting distance (DF = 3 m) that over which oystercatchers launch attacks (R. A. Stillman, unpublished). The probability of an aggressor stealing a mussel after winning a fight (PS = 0·21) was that observed for oystercatchers on the Exe estuary (R. W. G. Caldow, unpublished). Avoidance distance (DA = 2 m) was an average of the minimum distance to which foraging oystercatchers approach each other (≈2·5 m; Moody et al. 1997), the nearest-neighbour distance at which they most frequently change direction (≈2·5 m; Vines 1980) and the distance at which the victim of a kleptoparasitic attack starts to avoid an aggressor (≈1·3 m; R. A. Stillman, unpublished). Avoidance time has not been measured, but had relatively little influence on the previous model's predictions (Stillman et al. 1997) and so was set to the value used previously (TA = 2 s).

The model was tested using data collected between October 1982 and March 1987 on beds 4 and 26 of the Exe Estuary from unmarked and marked mussel-feeding oystercatchers specializing in either the stabbing or hammering feeding method (the same dataset was also used by Goss-Custard & Durell 1987, 1988 and Stillman et al. 1996). Full details of the methods are given in Goss-Custard & Durell (1987, 1988) but briefly, direct observation was used to record the intake rate of mussel flesh of a focal bird during 5 min of active foraging. The study area was marked into a grid of 25 × 25 m cells, allowing the numbers of oystercatchers feeding in the same cell as the focal bird to be counted at the start and end of the 5-min observation. The numbers present at the start and end of the observation were averaged and multiplied by 16 to calculate competitor density ha-1. The numbers of encounters with other competitors which each bird won and lost were also recorded. The data set for each bird comprised its feeding method, a series of paired observations of intake rate and competitor density, and a list of encounters won and lost. The foraging efficiency of each individual was calculated as its average intake rate in observations collected at low competitor densities (≤64 competitors ha-1; see Stillman et al. (1996) for the reasoning for using this cut-off). The dominance of each individual was calculated as the proportion of all encounters which were won. The importance index was calculated using a similar approach to that used for the model. The entire dataset was divided into a number of competitor density ranges, within each of which the average intake rate of each individual was calculated. Linear regressions relating average intake rate within a competitor density range to the foraging efficiency and dominance of each individual were then used to estimate the parameters of eqn 5 from which the importance index was calculated.

The major difference between the current model and that described by Stillman et al. (1997) was that, in the current version, individuals varied not only in their dominance but crucially in their foraging efficiency as well. Moreover, recent empirical work allowed refinement of a number of parameter values: (i) avoidance distance was reduced from 2·5 to 2 m; (ii) fighting distance was increased from 2·5 m to 3 m; (iii) fighting time was reduced from 10 to 7 s; and (iv) the probability of successfully stealing prey after winning a fight was reduced from 1 to 0·21. The model was also parameterized for both stabbers and hammerers, rather than just stabbers. Hammerers had a higher prey encounter rate, searching speed and handling time, but less between-individual variation in foraging efficiency than stabbers.

Results

The model accurately predicted the strength of interference in stabbers across the full range of observed competitor densities (Fig. 1a; predicted intake rate within 95% confidence interval of observed in 13 of 14 competitor density ranges). Its predictions were less accurate for hammerers, underestimating the observed strength of interference between 150–600 birds ha-1 (Fig. 1b; predicted intake rate above 95% confidence interval of observed in five competitor density ranges within 150–600 birds ha-1, but within 95% confidence interval in remaining nine competitor density ranges). Both the model's predictions and empirical observations for both feeding methods showed that interference was negligible at low competitor densities, and only reduced average intake rate to any extent at densities above 100-250 birds ha-1.

Figure 1.

Effect of competitor density on the average intake rate of mussel-feeding oystercatchers using either the (a) stabbing or (b) hammering feeding method. Two sets of simulations were run with oystercatcher populations comprised entirely of either stabbers or hammerers. Relative intake rate is intake rate expressed as a percentage of that achieved in the absence of competitors. The lines show model predictions and the symbols observed means and 95% confidence limits within competitor density ranges. The observed data are for marked and unmarked oystercatchers within the following competitor density ranges: 0–5 (n = 208 for stabbers, n = 77 for hammerers); 5–15 (n = 193, 89); 15–25 (n = 368, 173); 25–50 (n = 641, 409); 50–75 (n = 432, 331); 75–100 (n = 213, 225); 100–200 (n = 311, 375); 200–300 (n = 115, 167); 300–400 (n = 61, 99); 400–500 (n = 33, 57); 500–750 (n = 75, 100); 750–1000 (n = 40, 36); 1000–1500 (n = 31, 23) and > 1500 birds ha-1 (n = 28, 15). ●, Predicted; ○, observed.

The sources of between-individual variation in intake rate varied with competitor density. Below about 200 birds ha-1 interference was negligible and so the only possible source of variation in intake rate was the variation in foraging efficiency. Within this density range, the model predicted that intake rate differed more between the most and least efficient individuals, efficient ones having a higher intake rate, than it did between the most and least dominant individuals (Figs 2 and 3a). Although less clear, a similar relationship was found in the observed data. Within the four competitor density ranges up to 200 birds ha-1, efficient individuals had a significantly higher intake rate than inefficient individuals in two ranges for stabbers (Fig. 2a) and two for hammerers (Fig. 2c) (non-overlapping 95% confidence intervals), while no significant differences were found between individuals differing in dominance (Fig. 2b,d). Above 200 birds ha-1 the strength of interference increased and the model predicted that intake rate differed more between the most and least dominant individuals, dominant ones having a higher intake rate, than between the most and least efficient ones (Figs 2 and 3b). Again the observed relationship was less clear. Within the three competitor density ranges above 200 birds ha-1, no significant differences in intake rate were found between stabbers differing in foraging efficiency (Fig. 2a), but in one range efficient hammerers had a significantly higher intake rate than inefficient ones (Fig. 2c). No significant differences were found between hammerers differing in dominance (Fig. 2d), but in one range dominant stabbers had a higher intake rate than subdominant ones (Fig. 2b). Although the number of significant results was low, they suggest that predicted and observed were similar in stabbers, the influence of efficiency decreasing and the influence of dominance increasing with increased competitor density. In contrast, predicted and observed appeared to differ in hammerers, the influence of efficiency in reality remaining high, rather than decreasing with density, and the influence of dominance remaining low rather than increasing with density.

Figure 2.

Effect of competitor density on the intake rate of mussel-feeding oystercatchers using either the (a, b) stabbing or (c, d) hammering feeding method. The left-hand figures (a, c) show differences in intake rate between birds of above- or below-average foraging efficiency and the right-hand figures (b, d) differences between birds of above- or below-average dominance. Relative intake rate is intake rate expressed as a percentage of that achieved in the absence of competitors. The lines show predicted intake rate and the symbols observed means and 95% confidence limits for marked birds within the following competitor density ranges: 0–5 (n = 16/35 for inefficient/efficient stabbers, n = 34/17 subdominant/dominant stabbers; n = 14/35 for inefficient/efficient hammerers, n = 14/35 subdominant/dominant hammerers); 5–25 (n = 101/106, 137/70, 118/92, 62/148); 25–100 (n = 337/299, 384/252, 411/358, 239/530); 100–250 (n = 99/96, 94/101, 196/227, 148/275); 250–500 (n = 45/30, 36/39, 117/113, 88/142); 500–1000 (n = 42/17, 27/32, 71/65, 39/97); > 1000 (n = 16/7, 17/6, 18/20, 6/32). , Above average: predicted; ······, below average: predicted; ●, above average: observed; ○, below average: observed.

Figure 3.

Predicted relative importance of an individual's foraging efficiency and dominance in determining its intake rate at (a) low and (b) high competitor densities. Each symbol shows the foraging efficiency and dominance of an individual, and the symbol shading its intake rate relative to the mean for all individuals (○, below-average intake rate; ●, above-average intake rate). The predictions shown are restricted to stabbers, but very similar relationships were predicted for hammerers.

The predicted relative importance index (IE) was non-linearly related to competitor density in both stabbers and hammerers (Fig. 4). Up to about 100 birds ha-1 its value was approximately 1 because foraging efficiency was the sole determinant of variation in intake rate (Figs 2 and 3a). However, as density increased and interference intensified, the index decreased rapidly as the variation in interference suffered by individuals of differing dominance increased. At about 100–150 birds ha1 the index was about 0·5, indicating that foraging efficiency and dominance were equally important. Above 500 birds ha-1 its value approached 0 because dominance was the major determinant of intake rate (Figs 2 and 3b). The index calculated from empirical data for stabbers had a similar shape to that predicted, but did not reach zero at high competitor densities, indicating that dominance was slightly less important in reality than predicted by the model. In contrast, the observed relationship for hammerers was different to that predicted. The importance index did not decrease with increased density but remained close to 1, indicating that foraging efficiency was the major determinant of hammerer's intake rates regardless of competitor density. The unimportance of dominance in hammerers could have been a spurious result if the composition of birds changed with competitor density, with less subdominants occurring in the higher density ranges. However, this was not the case as the full ranges of both dominance and foraging efficiency in both hammerers and stabbers were represented in each competitor density range.

Figure 4.

Effect of competitor density on the relative importance of foraging efficiency (IE) in determining the intake rate of (a) stabbing and (b) hammering mussel-feeding oystercatchers. Two sets of simulations were run with oystercatcher populations comprised entirely of either stabbers or hammerers. The lines shows the predicted importance index and the symbols observed importance index of marked birds within the following competitor density ranges: 0–25 (n = 20 individuals for stabbers, n = 20 individuals for hammerers); 25–75 (n = 25, 31); 75–150 (n = 22, 30); 150–300 (n = 15, 21); 300–600 (n = 9, 15); 600–1000 (n = 9, 13). See text for the methods used to calculate the importance index. ●, Predicted; ○, observed.

A sensitivity analysis (Fig. 5) revealed that small changes in the duration of fights (TF) and avoidance (TA), the distance over which avoidance occurred (DA) and prey encounter rate (PC) had relatively little influence on the shape of the relationship between the importance index and competitor density. The relative importance of foraging efficiency was more sensitive to changes in handling time (TH), searching speed (v), the distance over which fights occurred (DF), the probability of an aggressor stealing prey after winning a fight (PS) and the extent of individual variation in foraging efficiency (ECV). The range of competitor densities over which foraging efficiency was the more important determinant of intake rate increased when foraging efficiency varied more between individuals, fights and handling time were shorter, searching speed was lower and aggressors were less likely to steal prey after winning a fight. Increased handling time, fighting distance and stealing probability increased the chances that a subdominant lost prey before consuming it. Increased searching speed had a similar effect because more rapidly searching dominants encountered subdominants and stole their prey at a higher rate.

Figure 5.

Sensitivity analysis of the model. The horizontal axis shows the competitor density at which foraging efficiency and dominance are equally important in determining an individual's intake rate. Each bar shows the effect of plus and minus 50% changes of a single parameter on the relative importance of foraging efficiency and dominance. The standard parameter values are based on mussel-feeding oystercatchers using the stabbing feeding method. □, 50% decrease; bsl00005, 50% increase.

It was not possible to fully parameterize the model for other predator–prey systems because few of its key parameters have been measured in other systems. Instead, the relative importance of foraging efficiency and dominance in determining variation in intake rate was predicted over a wide range of handling time and prey encounter, parameters which have been frequently measured in many predator–prey systems and vary considerably between different systems. All other parameters were set to their default value for stabbers. For each combination of prey encounter rate and handling time, the model was used to predict the relative importance of foraging efficiency at 150 birds ha-1 (i.e. the density at which foraging efficiency and dominance were predicted to be of equal importance in mussel-feeding oystercatchers) (Fig. 6). At this competitor density, foraging efficiency was predicted to be more important than dominance for nearly all combinations of the two parameters. Therefore, the competitor density at which foraging efficiency and dominance were of equal importance was greater than 150 birds ha-1 for all of these combinations of parameter values. When handling time was very short and encounter rate high, foraging efficiency was the sole determinant of intake rate, because under these conditions it was not profitable to steal prey and so no interference occurred. Within this range of parameter values, foraging efficiency remained the sole determinant of intake rate regardless of competitor density. The relative importance of foraging efficiency at 150 birds ha-1 was greater than in mussel-feeding oystercatchers for most other combinations of handling time and encounter rate, and so foraging efficiency remained the most important determinant of intake rate until competitor density was well in excess of 150 birds ha-1. Only when handling time was longer than in oystercatchers and encounter rate lower did the strength of interference, and so the importance of dominance, increase. Within this range of parameter values foraging efficiency and dominance were of equal importance at competitor densities less than 150 birds ha-1.

Figure 6.

Effect of handling time and prey encounter rate on the relative importance of foraging efficiency and dominance in determining intake rate at a competitor density of 150 birds ha-1. All other parameter values were those used to model mussel-feeding oystercatchers using the stabbing feeding method. The shaded part of the surface shows the combinations of handling time and prey encounter rate over which dominance is the more important determinant of intake rate at 150 birds ha-1.

Discussion

The model's predictions were derived from basic elements of foraging behaviour, such as the duration of handling, fights and avoidance and the distance over which interactions occur. None of the model's parameter values were chosen in order to force it to produce accurate predictions; all were estimated directly from field data. If the model had omitted important elements of the real system predictions could have diverged widely from observations, as was the case for the relative importance of foraging efficiency in hammerers. A close agreement between prediction and observation could also have arisen if more and more parameters were added until the difference between observation and prediction was minimized. Such a procedure would be possible for a well studied species such as the oystercatcher, but would make the model too oystercatcher-specific to be applied easily to other species. In fact the model was a considerable simplification of the real oystercatcher system and contains relatively few parameters, all of which could be measured in any species in which interference operates through prey stealing and avoidance.

The model predicted that foraging efficiency was the major determinant of the intake rate of both stabbing and hammering oystercatchers at competitor densities below 100–150 birds ha-1 but that dominance was the major determinant at higher densities. A similar shape of relationship was observed in stabbers, but in hammerers foraging efficiency remained the most important determinant over the full range of observed competitor densities. This happened even though the predicted strength of interference experienced by the average hammerer was similar to that observed. The discrepancy between observation and prediction in hammerers occurred because, in reality, the intake rate of dominant hammerers is only marginally higher than subdominants even at high competitor densities (Stillman et al. 1996). In contrast, dominant stabbers maintain or even increase their intake rate at high densities while subdominants have greatly reduced intake rates (Stillman et al. 1996).

The contrasting predictive power of the model for stabbers and hammerers suggests that it omits an important aspect of the real system, which influences the strength of interference experienced by dominant hammerers but not dominant stabbers. One possible limitation of the model is that it assumes that foragers always maximize their short-term intake rate, whereas in reality the maintenance of a territory or position in the dominance hierarchy may be important in maximizing intake over a longer time-scale. For example, in oystercatchers, the amount of time spent in aggressive displays, which are not directly related to immediate food consumption, increases with increased dominance (e.g. Ens & Goss-Custard 1986). Therefore, the apparently high susceptibility to interference observed in dominant hammerers at higher competitor densities may have been due to these individuals spending a higher proportion of the time engaged in disputes which were unrelated to short-term food intake. This type of behaviour was not modelled. A possible reason for such behaviour only being found in hammerers is that stabbers generally have a lower intake rate (e.g. Caldow et al. 1999) and will therefore need to feed for a higher proportion of the time in order to meet their energy requirements. As a consequence dominant stabbers will have less spare time in which to engage in activities other than foraging. However, at present this explanation is speculative and requires field testing. Nonetheless, it does raise the possibility that the decision rules used in such behaviour-based interference models may need to be based on principles other than short-term rate maximization.

The density at which oystercatchers forage changes throughout the tidal cycle, generally being higher at the extremes of the exposure period than at dead low water when intertidal areas are fully exposed (Ens & Cayford 1996). However, on the Exe estuary, where most of the model's parameters were estimated, the average density of mussel-feeders ranges between 100 and 250 birds ha-1 (J. D. Goss-Custard, unpublished). This is within the region in which, for stabbers, foraging efficiency and dominance are both predicted and observed to be of equal importance while, for hammerers, both are predicted to be of equal importance, but in reality, foraging efficiency is the major determinant of intake rate. These results are partially in accord with the findings of Goss-Custard & Durell (1988) and Caldow et al. (1999) who showed that foraging efficiency is more important than dominance in determining the competitive ability of mussel-feeding oystercatchers on the Exe, where a bird's competitive ability is measured by its lack of reliance on supplementary feeding habitats. Despite this empirical and model evidence for the importance of foraging efficiency at the densities at which oystercatchers generally forage, virtually all studies of individual variation in mussel-feeding oystercatchers have measured dominance (see Ens & Cayford 1996 for an overview). However, both the model and field data suggest that, at least in the case of stabbers, a slight increase in competitor density due, for example, to habitat loss or displacement from disturbed areas, would result in a large shift in the competitive advantage of dominant birds relative to efficient foragers. Under such stressed conditions, dominance may become the key determinant of intake rate.

The model assumed a strict dominance hierarchy in which the more dominant individual in a dispute always won, regardless of the absolute difference in dominance between the two competitors. While this assumption holds for oystercatchers (see for example Ens & Cayford 1996), this will not necessarily be the case in other species. If the model had assumed that the chances of winning a fight depended on the difference in dominance between two competitors, dominant individuals would have been less successful at stealing prey from subdominants. As a result the intake rates of dominant individuals would have been lower and, because they would lose less prey, the intake rates of subdominant individuals would have been higher than with a strict dominance hierarchy. Therefore, the relative importance of dominance in determining intake rate would have been lower than that predicted with a strict dominance hierarchy. The model also assumed, for simplicity, that fights were of fixed duration whereas, in reality, they may vary in length. For example, a subdominant individual which will ultimately lose a dispute could save time by fleeing immediately rather than disputing prey. Such behaviour would tend to increase the intake rates of all individuals, as all would waste less time in disputes.

Mussel-feeding oystercatchers have long handling times and low prey encounter rates and as a result subdominant individuals suffer considerable interference. In most other shorebird–prey systems handling times are shorter and encounter rates the same as, or greater than in oystercatchers and so interference is weaker or absent. Our model predicted that foraging efficiency would be relatively more important in such systems. This prediction relied on all parameters except handling time and encounter rate being the same as in mussel-feeding oystercatchers, because these other parameters have rarely been measured in other systems. The sensitivity analysis indicated that the key parameters needed to parameterize the model for other systems were (i) the distance at which aggressors start fights, (ii) the chances of an aggressor stealing prey, (iii) searching speed and (iv) the extent of individual variation in foraging efficiency. However, assuming that oystercatchers are not unusual in these respects, we predict that foraging efficiency is the more important determinant of intake rate in most other shorebird–prey systems. Since, the few studies of individual variation in other shorebirds have studied dominance (e.g. Ens, Esselink & Zwarts 1990), new studies will be required to test this prediction.

Acknowledgements

We are grateful to Tim Clutton-Brock and Will Cresswell for helpful comments on the model, and to Paul Dolman and an anonymous referee for useful comments on the manuscript. R.A.S. was funded by the Natural Environmental Research Council.

Received 14 May 1999;revisionreceived 1 November 1999

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