Contrasting interference functions and foraging dispersion in two species of shorebird (Charadrii)

Authors


Dr Michael G. Yates, Institute of Terrestrial Ecology, Monks Wood, Abbots Ripton, Huntingdon, PE17 2LS, UK.

Summary

1. Above a threshold density of ≈ 100 birds ha-1, strong interference occurred between redshank Tringa totanus (Linnaeus) feeding by sight on the amphipod crustacean Corophium volutator (Pallas). No aggressive interactions occurred between the birds and the probable cause was prey depression.

2. Redshank fed in a square metre of mud that had recently been exploited by another redshank much less often than would be expected by chance. By avoiding areas where prey would have been recently exploited, the feeding rate of redshank was up to three times faster than it would have been had they not avoided other foraging redshank.

3. Bar-tailed godwit fed in a square metre of mud that had been recently exploited by another godwit much more often than would be expected by chance in randomly moving birds. They tended to flock while foraging and showed no tendency to avoid areas where prey would have been recently exploited.

4. There was no evidence that interference occurred between bar-tailed godwit Limosa lapponica (Linnaeus) feeding on the polychaete lugworm Arenicola marina (Linnaeus) at densities below 300 birds ha-1, even though aggressive interactions occurred between birds.

Introduction

Interference, the decline in intake rate due to the presence of competitors ( Goss-Custard 1980; Sutherland 1983), is one of the major factors thought to influence the dispersion ( Goss-Custard 1970), distribution ( Goss-Custard 1977; Sutherland & Parker 1985; van der Meer & Ens 1997) and survival ( Goss-Custard & Sutherland 1997; Stillman et al. submitted) of foraging animals. Several mechanisms of interference have been postulated and observed in free-living vertebrates, including antipredator responses of the prey to the presence of the forager (‘prey depression’) ( Goss-Custard 1970), prey stealing among foragers (‘kleptoparasitism’) ( Ens & Goss-Custard 1984; Triplet, Stillman & Goss-Custard, 1999), competitive displacement from rich microsites ( Dolman 1995) and scanning for ( Cresswell 1997) or avoiding ( Goss-Custard 1970) potential aggressive attacks from other foragers. Interference therefore either occurs because of the risk of interactions between competing foragers or because of the effect of the foragers on the available density of their prey. Of these two classes of mechanisms, most attention has focused on behavioural interactions between the foragers in both theoretical (e.g. Ruxton, Gurney & de Roos 1992; Holmgren 1995; Moody & Houston 1995; Stillman, Goss-Custard & Caldow 1997) and empirical ( Ens & Goss-Custard 1984; Goss-Custard, Durell & Clarke 1984; Goss-Custard et al. 1988 ; Dolman 1995) studies. In contrast, interference through prey depression has received only limited attention ( Selman & Goss-Custard 1988; Ruxton 1995).

In prey depression, interference occurs because as forager density increases, the density of prey that can be both detected and caught, and so are available, decreases so that forager intake rate decreases. In vertebrates, field evidence of interference through prey depression has been acquired in redshank Tringa totanus, feeding on a burrow-dwelling crustacean Corophium volutator ( Selman & Goss-Custard 1988). The interference was very probably due to prey depression because (i) no discernible social interactions occurred between the redshank, (ii) the proximity of the subject bird to its nearest neighbour had no effect on the duration of its intercatch intervals ( Selman & Goss-Custard 1988) and (iii) Corophium retreat down their burrows when a redshank walks over the sediment surface ( Goss-Custard 1970).

However, in Selman & Goss-Custard’s study, intake rate was measured as the time elapsed between the consumption of successive prey while forager density was not measured directly but represented by the time elapsed between successive birds feeding in the same place. This makes comparison with the results of other studies problematic, since most theoretical and empirical studies of vertebrate foragers have portrayed interference as plots of intake rate, and not its reciprocal, against forager density. These have shown that, as forager density increases, intake rate may not decline until a certain threshold density of foragers has been reached, after which the gradient of decline may vary between feeding environments, classes of foragers and between individuals ( Dolman 1995; Stillman et al. 1997 ; Triplet, Stillman & Goss-Custard, 1999). At the present state of knowledge, we are still uncertain as to how much variation occurs within and between species in both the threshold forager densities at which interference begins and in the gradients of decline, yet both will have important implications for understanding forager dispersion, distribution and feeding competition. This study was carried out to enable the already acquired reciprocal measure of redshank intake rates to be related to forager density in order to allow comparison with other studies. In addition, similar data were obtained for bar-tailed godwit, Limosa lapponica, feeding on lugworms, Arenicola marina, in order to increase the number of empirical studies of this much neglected component of the mechanisms of feeding competition in vertebrates.

Methods

In the original redshank study ( Selman & Goss-Custard 1988), conducted between January and February 1983, a 1·41 × 1·41 m quadrat (2 m2) was marked out with bamboo canes within an area of upper mudflat situated at a level close to the mean high water of neap tides in Fremington Creek, on the Taw estuary. The single quadrat was considered to be representative of the adjoining mudflat which was very uniform in appearance and was occupied by feeding redshank throughout the winter. When a redshank walked through the quadrat, the time elapsed before a second bird entered the quadrat was measured by stopwatch; this is the ‘interbird interval’. The second bird’s feeding rate was measured by recording the time elapsing between the first and fourth prey capture using a second stopwatch. Four captures were timed to increase measuring accuracy in birds that consumed prey very rapidly. Later, the ‘intercatch interval’ was calculated by dividing by 3.

As prey size could not be measured, the numbers of prey consumed per unit time (the ‘feeding rate’) was used as the measure of intake rate (biomass or energy of prey consumed per unit time). It was calculated as the reciprocal of the intercatch interval. Interbird interval could not be converted retrospectively into redshank density so a new field study was carried out by MGY on the same mudflat used by Selman & Goss-Custard (1988). To measure redshank density, a series of contiguous 5 × 5 m squares of mudflat were marked out with bamboo canes. To measure interbird interval, 1·41 × 1·41 m quadrats were marked out in the centres of the larger squares. Over the low water period on two successive days in December 1993, interbird intervals were measured repeatedly in the quadrat, with the numbers of redshank in the larger, surrounding square being counted every 30 s between the times the first and second bird traversed the quadrat. Subsequently, the interbird intervals were plotted against the number of birds in the larger square, averaged over the time of the interbird interval, which varied in length a great deal. Fitting a curve to this relationship enabled the measurements of interbird interval made by Selman & Goss-Custard (1988) to be converted to redshank density. A potential problem with this approach is that the two data sets were collected 10 years apart, during which time redshank spacing behaviour may have changed. However, both data sets were collected in the same locality during the same season, and in both cases Corophium were abundant in the study area and the only prey consumed by redshank during foraging observations. We therefore have no reason to think that redshank spacing behaviour would have changed greatly, but cannot completely rule out this possibility.

Similar procedures as were adopted in these redshank studies were also used in the study of bar-tailed godwit carried out by M.G.Y. on the Wash, E. England, on the sandflats between Snettisham and Heacham at a level of the shore close to the mean low water mark of neap tides. Interbird and intercatch intervals were measured in February and March 1990 and 1991 while the interbird interval and bird density work was carried out in March 1993. Godwit flocks ranged widely throughout the study area and it was not possible to record interbird and intercatch intervals within a previously marked-out quadrat, as for redshank. Instead, each time a flock was located, these variables were recorded as birds passed within a fixed distance of an obvious feature (e.g. a piece of debris) on the sandflat. This distance had to be estimated by eye while the godwit were present, but on each occasion was later measured directly and had an average value of 0·5 m. Interbird and intercatch intervals were therefore measured as birds passed through a circle which, on average, had a diameter of 1 m. As a check on the possible influence of confounding variables on feeding rate, wind speed was measured by an anemometer placed at 0·5 m above ground-level, mud temperature was measured from a thermometer pushed 2 cm into the sediment and the stage of the tidal cycle was measured as time elapsed since high tide.

Results

Relationship between feeding rate and interbird interval

Spearman rank correlation on the data of Selman & Goss-Custard (1988) showed a significant positive relationship between redshank feeding rate and interbird interval (rs = 0·588; n = 40; P < 0·001). In contrast, no significant relationship was found for godwit between feeding rate and interbird interval (rs = 0·019; n = 248; P > 0·75), even though aggressive interactions between birds did occur occasionally. However, the feeding rate of bar-tailed godwit on lugworms is affected by strong winds, low temperatures and the wetness of the sediment ( Smith 1975), and therefore to how recently the receding tide uncovered the area. One, or a combination, of these variables may have masked any effect of interbird interval on feeding rate in godwit. Accordingly, feeding rate was also compared in a multiple regression analysis with wind speed, mud temperature and the time elapsed since high tide. Feeding rate was unrelated to wind speed and time elapsed since high tide but was positively and linearly related to mud temperature [feeding rate = 0·0293 (SE = 0·0071) + 0·0023 (SE = 0·0008) mud temperature; n = 248; P < 0·005]. However, after the effect of mud temperature had been removed, the residual feeding rate was still unrelated to interbird interval (rs = 0·096; n = 248; P > 0·1). Evidence for interference by prey depression was therefore found in redshank, but not in godwit.

In order to derive the shape of the interference function in redshank from our data, an equation was needed which related feeding rate to interbird interval. As these variables were non-linearly related, a linear regression with untransformed variables was not suitable. Either one, or both, variables could have been transformed until a linear relationship was found, but this would have involved some subjectivity in the choice of transformations. Instead, the expected relationship was derived by assuming that (i) individual prey are either vulnerable to, or immune from, attack; (ii) after a bird passes a fixed proportion of the prey are immune; (iii) immune prey become vulnerable again at a constant proportional rate; (iv) feeding rate is proportional to the density of vulnerable prey; and (v) handling time is very short in relation to intercatch interval. The derivation of the following equation is given in Appendix 1:

image(eqn 1)

where f = feeding rate, fmax = maximum feeding rate when all prey are vulnerable, p = proportion of prey that are immune after first bird passes, r = rate at which immune prey become vulnerable and t = interbird interval. The assumption that handling time is short relative to intercatch interval is true for redshank eating Corophium as handling time is 0·7 s ( Goss-Custard & Rothery 1976) and intercatch interval ranged from 1·1 to 9·4 s, with a mean of 4·5 s. Therefore, non-linear regression was used to calculate the least-squares estimates of fmax = 0·376, P = 0·827 and r = 0·00392. The fitted model accurately described the increase in feeding rate with increased interbird interval at intervals of less than 500 s, and the lack of such a relationship at longer intervals ( Fig. 1a). In order to assess the significance of this three-parameter, best fit model over the one-parameter, null model of constant feeding rate (f = fmax), the two-dimensional profile likelihood for p and r was calculated ( McCullagh & Nelder 1989, p. 254). This was performed by building a set of alternative models in which p and r had fixed values and using non-linear regression to calculate, for each combination of p* and r*, the least-squares estimate of fmax and the residual sum of squares (R(p*, r*)) around the fitted relationship. Assuming a normal error distribution, the residual sum of squares for each combination of p* and r* was compared with the residual sum of squares (Rmin) for the best fit model (i.e. fmax = 0·376, P = 0·827 and r = 0·00392) using an F-test.

Figure 1.

Relationships between feeding rate and interbird interval in redshank and bar-tailed godwit. The symbols show mean observed feeding rates with associated standard errors within ranges of interbird intervals. For redshank, the line shows the predictions of equation 1 with parameter values estimated by non-linear regression. For godwit, the line shows the mean feeding rate. Although grouped means are shown, the relationships were fitted to individual observations.

image

where m = number of parameters in best fit model (3), m– 1 = number of additional parameters in best-fit model compared with null model (2) and n = sample size (40). The 95% confidence sets for p and r were given by all combinations of p* and r* for which F(p*, r*) was less than the upper 95% value of an F distribution with (m-1) and (n-m) degrees of freedom (i.e. those combinations of p* and r* which did not yield a significantly worse fit than the least-squares estimates in the best fit model). The confidence sets did not include zero for either p or r, suggesting that the best fit model was a significant improvement over the null model. Equation 1 was not applied to godwit because no evidence of interference was found for this species. Instead, the value of fmax = 0·048 (SE = 0·003) was estimated as the mean feeding rate across all interbird intervals, and the value of p assumed to be zero ( Fig. 1b).

Relationship between interbird interval and density

Spearman rank correlation showed that, in both species, interbird interval decreased as the density of birds increased (redshank: rs = - 0·898; n = 68; P < 0·001; godwit: rs = − 0·634; n = 62; P < 0·001). These variables were strongly, non-linearly related, with interbird interval increasing very rapidly at low bird densities to the extent that, as bird density approaches zero, interbird interval would be expected to approach infinity.

These characteristics made it difficult to fit a regression equation to the data and so the reciprocal of interbird interval, termed ‘quadrat encounter rate’, was used in the analysis instead. Quadrat encounter rate measures the rate at which different birds passed through the quadrat, and so approached zero rather than infinity, as bird density decreased to zero. Initially, the expected relationship between quadrat encounter rate and bird density was calculated by assuming that birds followed random and independent search paths. The derivation of the following equation is given in Appendix 2:

image(eqn 2)

where q = quadrat encounter rate, v = walking speed, w = width of quadrat and D = bird density. In order to predict the quadrat encounter rate expected with random and independent foraging, v was set to 0·23 m s-1 for redshank ( Goss-Custard 1970) and 0·17 m s-1 for godwit (M.G. Yates, personal observation), and w to 1·58 m for redshank (the mean width of a 1·41 × 1·41 m quadrat when traversed through its centre from any direction) and 1 m for godwit. In order to fit the equation to the field data, v and w were combined into a single parameter b:

image(eqn 3)

Linear regression was used to estimate b as 0·124 (SE = 0·006) for redshank and 3·267 (SE = 0·506) for godwit. The fitted equation for redshank provided a good fit to the data across the full observed range of bird density ( Fig. 2a). Although statistically significant, the fit for godwit was poor due to the exceptionally high quadrat encounter rates observed at 100 birds ha-1 ( Fig. 2b).

Figure 2.

Relationships between quadrat encounter rate (i.e. the reciprocal of interbird interval) and competitor density in redshank and bar-tailed godwit. The symbols show mean quadrat encounter rate with associated standard errors within ranges of competitor density. The solid lines show the predictions of equation 3 with parameter values estimated by linear regression. The broken lines show relationships calculated by equation 2, assuming that birds followed random search paths. Although grouped means are shown, the relationships were fitted to individual observations.

The observed relationships were then compared with with those expected on the ‘null hypothesis’ assumption that the birds moved at random and independently of each other. For redshank, the observed quadrat encounter rate was consistently lower than that expected on the null hypothesis ( Fig. 2a) and the estimated b significantly lower than expected (observed = 0·124 (SE = 0·006); expected = 0·23 × 1·58 = 0·36; P < 0·001), while for godwit the observed rate was consistently higher than expected ( Fig. 2b) and the estimated b significantly higher (observed = 3·267 (SE = 0·506); expected = 0·17 × 1·00 = 0·17; P < 0·001). This means that redshank walked over areas that had previously been crossed by other birds less frequently than would be expected on the basis of random and independent movement, i.e. they were avoiding each other’s search paths. In contrast, successive godwit recrossed the area more frequently than would expected from random and independent movement, i.e. they were aggregated while foraging.

Shape of the interference function

In order to define the shape of the interference function in redshank, a relationship was required between feeding rate and bird density. Since quadrat encounter rate was the reciprocal of interbird interval, this was done by converting the result of equation 3 to interbird interval (t = 1/q = 10 000/(bD)), and substituting it into equation 1:

image(eqn 4)

This equation was then used to predict the feeding rate of redshank across the range of observed competitor densities.

Redshank feeding rate was unrelated to competitor density up to about 100 birds ha-1, after which it decreased rapidly ( Fig. 3a,b). The strength of interference is usually measured by m, the rate of change in log feeding rate with log competitor density. On a log scale, feeding rate decreased at a changing rate with competitor density, but within the range 500–1500 birds ha-1, had an average value of - 0·48 ( Fig. 3b). No interference was found in godwit and so feeding rate in this species had a constant value of 0·048 (fmax from equation 1) across the full range of competitor density.

Figure 3.

Relationship between feeding rate and competitor density in redshank. The two figures show the same relationship plotted on either a linear or log scale. The lines show the interference function predicted by equation 4 based on the parameter values estimated for equations 1 and 3.

Interference and avoidance in redshank

The avoidance of other birds by redshank ( Fig. 2) may have reduced the strength of the interference that occurred between them because, by avoiding each other’s search paths, they would have encountered a lower density of immune prey than if they had searched at random with respect to each other. To test the magnitude of this effect, the interference function of redshank was compared with that expected with random movement ( Fig. 4).

Figure 4.

Effect of avoidance behaviour on the shape of the interference function in redshank. The two figures show the same relationship plotted on either a linear or log scale. The lines show the interference function predicted by equation 4 based on either the observed relationship between interbird interval and bird density ( equation 3 with parameters estimated by linear regression) or that expected if redshank followed random search paths ( equation 2 with parameters based on redshank walking speed and the quadrat width).

Both the shape of the interference function and the strength of interference were strongly influenced by avoidance behaviour. With random movement, feeding rates were predicted to fall above a threshold competitor density of about 50 birds ha-1 instead of 100 birds ha-1. As the observed feeding rates of redshank were consistently higher than those expected on the null hypothesis of random movement – the difference being greatest between about 250–750 birds ha-1– mutual avoidance of search paths in redshank appeared to reduce the strength of interference substantially.

Discussion

The new fieldwork carried out on redshank in this study enabled feeding rate to be expressed as a function of competitor density, the standard relationship used in interference studies of birds. The relationship between feeding rate and bird density derived in this way showed that strong interference occurred between redshank feeding by sight on the amphipod crustacean C. volutator. One potential problem is that the observations used to derive this relationship were made at a single locality. However, we see no reason why redshank feeding on the same prey and sediment type elsewhere should behave differently. Another potential problem is that the two data sets were collected 10 years apart, and that any changes in the spacing behaviour of redshank during this time could have influenced the predicted strength of interference. Although this possibility cannot be completely ruled out, we do not believe that redshank behaviour had changed because both data sets were collected in the same locality during the same season, and in both cases Corophium were abundant in the study area and the only prey consumed by redshank during foraging observations. Further evidence against these potential problems is that studies along a predominantly muddy transect on the Ythan estuary, Scotland, showed that redshank feeding on Corophium spread out from the most preferred zone within the transect as the total numbers of birds in the transect increased ( Goss-Custard 1977). Also, discernible interactions occurred; the birds increasingly tended to avoid the areas of highest bird density as overall numbers increased.

The probable cause of the interference in redshank was prey depression. On both the Ythan and the Taw, discernible social interactions occurred between the redshank and, on the Taw, the proximity of the subject bird to its nearest neighbour had no effect on the duration of its intercatch intervals ( Selman & Goss-Custard 1988). Furthermore, Corophium are known to retreat down their burrows – presumably as an antipredator response – when a redshank walks over the sediment surface ( Goss-Custard 1970). None the less, it would be desirable to confirm these points by further studies on prey availability in relation to both prey and predator abundance.

Interference through prey depression in redshank contrasts with the three other field studies made of interference in foraging birds in which aggressive interactions were thought to be responsible for interference. In another shorebird, the oystercatcher Haematopus ostralegus Linnaeus, feeding on mussels, Mytilus edulis Linnaeus, interference seems to occur because, as bird density increases, subdominant individuals lose an increasing number of the prey they find in kleptoparasitic attacks from dominants and, in particular, may be less successful at finding vulnerable mussels because of being distracted by the threat of being attacked ( Ens & Cayford 1996). Similarly, Cresswell (1997) argued that the interference he recorded in blackbirds, Turdus merula Linnaeus, feeding on immobile artificial prey arose because of the increased cost of monitoring competitors in order to avoid aggressive interactions at higher bird densities. Finally, Dolman (1995) argued that snow buntings, Plectrophenox nivalis (Linnaeus), eating artificial prey, experienced interference as bird density increased because of the increasing frequency with which birds were displaced from rich microfeeding sites by other birds. In the present study, bar-tailed godwit did steal feeding sites and prey items from each other but, over the range of bird densities studied, there was no evidence that this resulted in interference.

Despite the differences in mechanisms underlying interference in these bird species, the general shapes of the interference functions are rather similar across species. In redshank (this study), oystercatchers eating both mussels ( Stillman et al. 1996 ) and cockles, Cerastoderma edule (Linnaeus) ( Triplet et al. 1999 ) and snow buntings ( Dolman 1995), interference only occurs above a threshold density of competitors. This suggests that there are underlying similarities in the nature of interference in these systems, despite their differing natural histories. Similarly, Ruxton (1995) showed that a behaviour-based interference model based on prey depression was structurally identical to another ( Ruxton et al. 1992 ) based on competitive interactions. Such similarities between cases with such different natural histories does suggest that it will be possible eventually to select a function shape that applies widely across foraging vertebrates.

Indeed, in the two species of shorebirds in which interference among free-living birds eating natural prey has been studied, the values of the parameters describing the interference function are rather similar. The two parameters are (i) the threshold bird density at which interference begins, and (ii) the strength of interference above the threshold. Interference in redshank only started once bird density had reached ≈ 100 birds ha-1, a value quite similar to the range of 50–150 birds ha-1 found in oystercatchers eating mussels ( Stillman et al. 1996 ) and cockles ( Triplet et al. 1999 ). Once the threshold predator density has been reached, the subsequent strength of interference is usually measured by m, the rate of change in log feeding rate with log competitor density. In the redshank, feeding rate decreased at a changing rate with competitor density, but within the range 500–1500 birds ha-1 had an average value of − 0·48 ( Fig. 3b). This lies within the high part of the range found in oystercatchers feeding on mussels ( Stillman et al. 1996 ) and cockles ( Triplet et al. 1999 ). In this species, interference was strongest among subdominant birds that opened mussels by stabbing, the maximum values of the slopes recorded being − 0·44. However, for birds of average dominance, the slope in birds that opened their mussels by stabbing was − 0·08 while in hammerers it was − 0·22. This suggests that interference is stronger in redshank than in oystercatchers. Whether interference is generally stronger in species where it is caused by prey depression than in species where it is caused by social interactions – so that it falls on only a minority of the population – remains to be seen. Regrettably, there have been very few empirical studies of interference functions in foraging vertebrates, an omission that badly needs redressing ( Dolman 1995; Sutherland 1996). In addition, most theoretical studies of interference omit aspects of the biology of the systems, such as avoidance of competitors and dominance hierarchies, that may have a large influence on the value of the threshold predator density at which interference begins.

There was, however, no evidence that interference occurred between bar-tailed godwit feeding on the polychaete lugworm at densities below 300 birds ha-1, even though aggressive interactions occurred between birds. This may either suggest that the threshold density at which interference begins is indeed higher than in the other two shorebird species studied or that the absence of data from high densities of birds prevented the threshold from being estimated. The birds’ own behaviour suggests that the former possibility may be the more likely. Bar-tailed godwit fed in a square metre of mud that had been recently exploited by another godwit much more often than would be expected by chance in randomly moving birds. They tended to flock while foraging and showed no tendency to avoid areas where prey would have been recently exploited. In contrast, redshank fed in a square metre of mud that had recently been exploited by another redshank much less often than would be expected by chance. By avoiding areas where prey would have been recently exploited, the feeding rate of redshank was up to three times faster than it would have been had they not avoided other foraging redshank. Had interference been present at higher bird densities, we would have expected godwits to have avoided each other, as did redshank. In the absence of any reason to think that godwit have a stronger incentive to aggregate while foraging than redshank, any interference in godwit either starts at a much higher threshold than in redshank or oystercatchers or is very weak above the threshold.

Acknowledgements

We are very grateful to Ralph Clarke for helpful statistical advice and to Richard Caldow for valuable comments on the manuscript. Bar-tailed godwit observations were made as part of a study commissioned by the Department of the Environment. R.A.S. was funded by the Natural Environmental Research Council.

Received 18 December 1998;revisionreceived 16 June 1999

Appendices

Appendix 1

Expected relationship between feeding rate and interbird interval

Individual prey are assumed to be either vulnerable to or immune from predation and so the total prey density (N) is directly related to the density of vulnerable (V) and immune (I) prey.

image(eqn 5)

It is assumed that after a bird passes, prey respond so that a fixed proportion (p) are immune, while the remainder (1–p) remain vulnerable. After the bird leaves, vulnerable prey are divided into those that did not respond to the bird ((1–p) N) and those that became immune but have since returned to being vulnerable (R).

image(eqn 6)

Substituting V into equation 5 gives

image(eqn 7)

At each point in time after the bird has passed, individual, immune prey are assumed to have a constant probability of switching back to the vulnerable state. Therefore, the rate of change in the density of vulnerable prey is proportional to the density of immune prey.

image(eqn 8)

where t = time after bird passed (interbird interval) and r = instantaneous rate at which immune prey become vulnerable. By rearranging equation 7 (I=N−(1−p)NR), I can be substituted to give

image(eqn 9)

which can be simplified to

image(eqn 10)

Integration then gives

image(eqn 11)

The total density of vulnerable prey is found by substituting R into equation 6

image(eqn 12)

which can be simplified to

image(eqn 13)

The prey encounter rate (λ) of the following bird is assumed to be directly proportional to the density of vulnerable prey.

image(eqn 14)

where a = instantaneous area of discovery. Handling time is assumed to be very short in relation to intercatch interval and so feeding rate (f) is also given by equation 14.

image(eqn 15)

The term aN is the maximum feeding rate when all prey are vulnerable (fmax), and so can be substituted to give

image(eqn 16)

Appendix 2

Expected relationship between interbird interval and bird density

Birds are assumed to walk at constant speed (v; ms-1) along random search paths, within a patch that contains a single quadrat, which has a mean width (w; m) when approached from any direction. As an individual bird moves through the patch, it is assumed to enter the quadrat if it passes within 0·5w of the quadrat’s centre. Birds are assumed to follow independent search paths and so the area of discovery of the population is

image(eqn 17)

where D = population density (ha-1) and 1/10000 converts the ha-1 unit of D to the m unit of v and w. By ignoring the time spent in the quadrat, the rate at which the population encounters the quadrat is also given by equation 18.

image(eqn 18)

where q = quadrat encounter rate. The length of time between two birds entering the quadrat is the reciprocal of q

image(eqn 19)

where t = interbird interval.

Ancillary