using behaviour-based models

Transcription

1 Ecology , Predicting the strength of interference more quickly Blackwell Science, Ltd using behaviour-based models RICHARD A. STILLMAN, ALISON E. POOLE, JOHN D. GOSS-CUSTARD, RICHARD W. G. CALDOW, MICHAEL G. YATES* and PATRICK TRIPLET Centre for Ecology and Hydrology Dorset, Winfrith Technology Centre, Winfrith Newburgh, Dorchester, Dorset DT2 8ZD, UK; *Centre for Ecology and Hydrology Monks Wood, Abbots Ripton, Huntingdon, PE17 2LS, UK; and RN Baie de Somme, SMACOPI, 1 Place Amiral Courbet, F Abbeville, France Summary 1. Interference between foraging animals can be quantified directly only through intensive studies. A quicker alternative is to predict the strength of interference using behaviour-based models. We describe a field method to parameterize an interference model for shorebirds, Charadrii. 2. Kleptoparasitic attack distance is the main factor affecting the strength of interference but has rarely been measured. Attack distance is related to handling time, a frequently measured parameter, allowing the model to be parameterized for systems in which attack distance has not been measured. 3. The model accurately predicts the strength of interference between oystercatchers Haematopus ostralegus L. feeding on cockles Cerastoderma edule L. and the absence of interference between bar-tailed godwits Limosa lapponica L. feeding on lugworms Arenicola marina L. at low competitor densities. 4. We predict the strength of interference in black-tailed godwit Limosa limosa L. and oystercatcher systems in which it has not been measured previously. The strength of interference is almost entirely determined by attack distance; interference is stronger in systems with longer attacks. Interference is usually weaker in black-tailed godwits because handling time is generally shorter and this limits attack distance. 5. The interference model can be parameterized much more quickly than the alternative of measuring interference directly. Behaviour-based models have the potential to be a valuable tool for predicting the strength of interference. Key-words: attack distance, handling time, kleptoparasitism, pace length, shorebirds Charadrii. Ecology (2002) 71, Ecological Society Introduction Interference is a major component of food competition but is difficult to measure in natural animal populations (Ens & Cayford 1996). Empirical studies of vertebrates have usually taken a decline in intake rate with increasing competitor density as evidence for interference (e.g. Zwarts & Drent 1981; Ens & Goss-Custard 1984; Goss-Custard & Durell 1988; Dolman 1995; Stillman et al. 1996; Cresswell 1997; Triplet, Stillman & Goss-Custard 1999), but there is no guarantee that any decline is due to interference, as other confounding Correspondence: Richard Stillman, Centre for Ecology and Hydrology Dorset, Winfrith Technology Centre, Winfrith Newburgh, Dorchester, Dorset DT2 8ZD, UK. Tel: rast@ceh.ac.uk factors may exist (Ens & Cayford 1996). Alternatively, the lack of a relationship does not necessarily mean that interference is absent, only that it could not be detected. Additionally, empirical results may only apply to the situation for which the data were collected, as the strength of interference may vary with environmental conditions (e.g. Stillman et al. 1996; Cresswell 1998; Triplet et al. 1999). To provide convincing, direct evidence for interference, long-term studies are required, and even then, the results may not apply to new situations. An alternative, potentially more rapid and general approach, is to predict the strength of interference using behaviour-based models (e.g. Beddington 1975; Ruxton, Gurney & de Roos 1992; Holmgren 1995; Moody & Houston 1995; Stillman, Goss-Custard & Caldow 1997). These models have made successful

2 533 Predicting the strength of interference qualitative predictions (e.g. Dolman 1995; Stillman et al. 1996; Cresswell 1998; Triplet et al. 1999), but only one has been tested quantitatively and for only one system (Stillman et al. 1997, 2000). The mechanism of interference in most models is direct interaction between animals, such as prey stealing (kleptoparasitism). One key factor determining the strength of interference is the distance over which kleptoparasitic attacks occur (Stillman et al. 1997). However, attack distances have rarely been measured, and are more difficult to measure than parameters such as handling time. In this paper we show how a behaviour-based model can predict the strength of interference more quickly than measuring interference directly. We describe a field method to parameterize the model. Ideally, all parameters should be measured directly, but this may not always be possible. Therefore, we show how kleptoparasitic attack distance may be predicted from handling time, which is more easily and frequently measured, to show how the model can be applied when attack distance is unknown or cannot be measured. To determine the reliability of our approach, we quantitatively test the model for two new systems. We then make predictions for several systems in which interference has not been measured directly and show why the strength of interference varies between these systems. We study three shorebirds, the oystercatcher Haematopus ostralegus L., black-tailed godwit Limosa limosa L. and bar-tailed godwit Limosa Lapponica L. The model The model is described by Stillman et al. (1997, 2000). It progresses in discrete time steps and follows the location and behaviour of each animal within a population as they encounter prey and competitors. Animals can be searching for or handling prey, fighting over prey or avoiding competitors (Fig. 1). They either find prey independently (Fig. 1a) or steal prey from a competitor (Fig. 1b). Interference occurs when animals waste time avoiding (Fig. 1c) and fighting (Fig. 1b), and when prey is stolen (Fig. 1b). The responses to competitors are calculated using intake rate-maximizing decision rules. The model assumes a strict dominance hierarchy, so that the more dominant individual in a dispute always wins regardless of the absolute difference in dominance. We used a model assuming a dominance hierarchy and optimal decision rules because its predictions were more accurate than simpler models (Stillman et al. 1997). Previous model versions contained the following parameters: prey encounter rate, prey handling time, kleptoparasitic attack distance, kleptoparasitic dispute duration, the probability of stealing prey, searching speed, competitor avoidance distance, avoidance time and the amount of individual variation in foraging efficiency. To simplify parameterization, we assumed that avoidance distance equalled attack distance, and that avoidance time was one time step. Avoidance and Fig. 1. How model animals change behaviour (t = time, d = distance between animals). (a) Prey capture a searching animal encounters prey, handles it for T H seconds, consumes it and then resumes searching. (b) Prey stealing a searching animal approaches a less dominant animal which is handling prey to within the attack distance (D A ). It initiates a fight over the prey which lasts for T K seconds, after which (in this example) it wins and starts handling the prey. The loser changes direction and starts searching directly away from the winner. (c) Competitor avoidance two animals approach each other to within the attack distance. The less dominant animal starts avoiding by changing direction and moving directly away from the more dominant animal (for 1 s in this example) until separated by more than the attack distance, after which it resumes searching. The more dominant animal does not change its behaviour. Adapted from Stillman et al. (1997). attack distances were equal in Stillman et al. (1997) and similar in Stillman et al. (2000), and avoidance time was previously two time steps, so the changes to avoidance distance and time were not great. Additionally, the predicted strength of interference is relatively insensitive to these parameters (Stillman et al. 1997, 2000). The changes meant that animals continued to avoid until there were no dominant neighbours within the attack distance (Fig. 1c). We could not measure individual variation in foraging efficiency and so, as in Stillman et al. (1997), this parameter was excluded. The model predicts the intake rate of each individual within the model population. In this paper, we present the average intake rate of all individuals because the observed intake rates used to test the model were also population averages. Stillman et al. (2000) shows how predicted intake rate varies among individuals.

3 534 R.A. Stillman et al. Methods Groups of birds, feeding close enough to each other so that kleptoparasitism would be possible, were located. They were then videoed either until they became too dispersed or until enough data had been recorded. The video field of view was set wide enough to allow several birds to be videoed simultaneously (to increase the chance of observing kleptoparasitism), but not so wide that their behaviour could not be determined. As far as possible, the camera was held still so that the movement of birds could be measured relative to the fixed field of view (see below). The following parameters were recorded. PREY ENCOUNTER RATE (λ) Prey encounter rate was measured by following individual birds from when they appeared on the screen or finished handling prey, until they walked off the screen or located prey. We recorded the amount of time each bird spent searching for prey (excluding time attacking or avoiding others) and whether it found prey. The prey encounter rate for each system was the total number of prey located by all birds observed (N P ) divided by the total time spent searching by all birds (T S ). λ = N P. T eqn 1 In one system, oystercatchers fought over patches of prey, not individual prey. Within a patch, birds fed while standing still or moving slowly, but moved more rapidly between patches. In this case, we recorded the time each bird spent searching for patches and whether it found a patch. N P was the number of patches found by all birds and T S the total time spent searching for patches. PREY HANDLING TIME (T H ) Handling time was measured as the time between a bird s bill first making contact with the prey, or the substrate in the case of buried prey, and the last piece of prey being swallowed. Handling time was only measured for prey captured independently (i.e. not stolen) and over which no kleptoparasitic attacks occurred. In the system in which oystercatchers fought over patches, handling time was the time a bird spent feeding in a patch. KLEPTOPARASITIC DISPUTE TIME (T K ) The duration of kleptoparasitic disputes was calculated as the average time spent by the aggressor moving towards the victim (T A ) and by the loser (nearly always the victim) avoiding the winner (T V ), plus the time spent by the two birds in any escalated fight over the prey (T F ). T K S TA + T = 2 V + T. F eqn 2 The average of attack and avoidance time was calculated to obtain a single estimate the time spent by both the aggressor and victim. Only disputes in which the aggressor reached the victim and the full duration of avoidance was observed were included in calculations. ATTACK DISTANCE (D A ) Birds made kleptoparasitic attacks by moving rapidly towards a stationary victim. In nearly all cases, birds attacked by running, not flying (100% of black-tailed godwit fights, 90% of oystercatcher fights), allowing attack distance to be estimated from the number of paces and pace length. Distance was calculated from either (i) the number of paces taken by aggressors which reached the victim or (ii) where the victim was not reached, from the number of paces taken plus an estimate of the extra number needed to reach the victim. D A = L A (N A + N X ), eqn 3 where L A = attacking pace length, N A = number of paces made by aggressor moving towards victim and N X = number of extra paces needed for aggressor to reach victim. The aggressor usually reached the victim (95% of godwit fights, 90% of oystercatcher fights) and so the average values of N X (0 1 in godwits, 0 3 in oystercatchers) were small compared to N A (5 8 in godwits, 11 3 in oystercatchers). Attacking pace lengths were measured from attacks made directly across the screen. The aggressor s maximum body length, near the start and end of the attack, and attack distance were measured in screen pixels. Body length was calculated as the maximum length from the tail tip to the base of the bill, plus bill length. Two measures were combined because the body and bill seldom formed a straight line. We estimated maximum body length because this approximated most closely the lengths of dead birds (see below). Attack distance was converted from pixels to bird-lengths by dividing the number of pixels covered by the attack by the average length of the bird measured in pixels. This distance was converted from bird-lengths to m by multiplying the number of bird-lengths covered by the attack by the average length of an individual of the appropriate species. Observed bird lengths were the mid-point of the range of lengths given by Hayman, Marchant & Prater (1986) for dead birds: black-tailed godwit = 0 40 m; oystercatcher = 0 43 m. The length of each pace was calculated by dividing the distance covered by the number of paces. SEARCHING SPEED (S S ) We measured searching speed from birds which were constantly searching for prey and not involved in aggressive disputes. Searching speed was calculated from the time spent moving and the number and length of paces.

4 535 Predicting the strength of interference Table 1. Pace lengths and rates (mean ± standard deviation (sample size)) of black-tailed godwits and oystercatchers searching for prey and attacking a competitor. Sample size is the number of searching bouts or attacks from which pace lengths and rates were estimated. To increase sample size, data from different sites have been merged. Pace lengths from three previous studies are also shown Present study Previous studies Species and behaviour Pace rate (s 1 ) Pace length (m) (1) (2) (3) Black-tailed godwit Searching 2 1 ± 1 0 (20) 0 13 ± 0 03 (20) 0 12 Attacking 4 0 ± 1 2 (17) 0 17 ± 0 03 (17) Oystercatcher Searching 2 3 ± 1 0 (20) 0 12 ± 0 03 (20) Attacking 5 9 ± 2 0 (54) 0 18 ± 0 05 (54) (1) Mean value in Table 3 of Goss-Custard (1970), calculated from footprints of birds walking at low pace rates (black-tailed godwit = 0 8 s 1 ; oystercatcher = 1 6 s 1 ). (2) Derived from pace rate using relationships in Table 1 of Speakman & Bryant (1993), calculated from birds walking over a known distance. (3) Derived from pace rate using relationship on page 64 of Sitters (2000), calculated from birds walking over a known distance. S S LN S = T eqn 4 where L S = searching pace length, N S = number of paces while searching and T M = time spent moving. Searching pace lengths were estimated from birds searching directly across the screen using the same approach as for attacking pace length. PROBABILITY OF STEALING PREY AFTER DISPLACING A VICTIM (P S ) An aggressor was recorded as displacing a victim if the victim retreated as the aggressor approached or after a short escalated dispute. The probability of displacing a victim was calculated as the number of disputes in which victims were displaced divided by the total number of disputes. Even if an aggressor displaced a victim it did not necessarily obtain the prey if the victim escaped with the prey or if neither bird relocated the prey. The probability of stealing prey, after displacing a victim, was calculated as the number of disputes in which an aggressor displaced a victim and captured the prey, divided by the total number of disputes in which a victim was displaced. Results M S ACCURACY OF ATTACKING AND SEARCHING PACE LENGTHS Attack distance and searching speed were calculated from the pace lengths of attacking and searching birds. Due to the importance of these parameters (Stillman et al. 1997), we compared our estimates of pace length with previous estimates (Table 1). Pace lengths have not been related previously to behaviour, but have been related to pace rate. Pace rate was more rapid in attacking than in searching birds (Table 1) and so we compared attacking and searching pace lengths with previous estimates from birds with either high or low pacing rates. Previous estimates of pace length in oystercatchers encompassed the values measured in this study: pace lengths in slowly pacing birds ranged from 0 09 to 0 13 m, compared to our figure of 0 12 m for searching birds; and pace lengths in rapidly pacing birds ranged from 0 16 to 0 21 m, compared to our figure of 0 18 m for attacking birds. The previous estimate of pace length in slowly pacing black-tailed godwits, 0 12 m, was also close to our figure for searching birds, 0 13 m. No data were available to compare with attacking pace length in black-tailed godwit, so we compared the attacking pace lengths of godwits with oystercatchers. Attacking godwits had pacing rates 1 9 and pace lengths 1 3 times greater than searching godwits. The equivalent values in oystercatchers were 2 6 and 1 5. These comparisons gave us no reason to suspect the value of attacking pace length in godwits; when attacking, godwits increased both their pacing rates and pace lengths slightly less than oystercatchers. We therefore considered that our estimates of attacking and searching pace lengths were reliable. MECHANISM OF INTERFERENCE IN OYSTERCATCHERS AND BLACK-TAILED GODWITS Despite short handling times in several systems (Table 2), limiting the opportunity for kleptoparasitic attacks, prey stealing was observed in all systems. Attacks were similar in the two species. Attack distance varied from 0 7 to 3 6 m (Table 2). Aggressors nearly always successfully displaced their victims; 91% in black-tailed godwits and 92% in oystercatchers. In virtually all cases, victims immediately retreated, rather than escalating the dispute by standing their ground. Consequently, the overall duration of disputes

5 536 Table 2. Parameter values (mean ± standard deviation (sample size)) for the model. Standard deviations were not calculated for λ and P S. Sample size for R.A. λ is the Stillman number of et prey al. located and for P S the number of fights in which a victim was displaced. Birds fought over prey items in all systems except 8, in which they fought over prey patches; in this system, T H is patch handling time and λ patch encounter rate. Data for systems 4 and 7 were combined to calculate P S, as no prey were stolen successfully in system 7 System Site and habitat T H (s) λ (s 1 ) S S (ms 1 ) D A (m) P S T K (s) Black-tailed Godwit 1. Earthworms Lumbricus sp. Exe Estuary, England (wet grassland) ±4 1 (23) (25) ±0 08 (54) ±0 3 (25) (31) ±0 9 (29) 2. Ragworms Hediste diversicolor Exe Estuary, England (mudflat) ±4 9 (26) (21) ±0 07 (58) ±0 7 (15) (11) ±0 5 (11) 3. Scrobicularia plana Exe Estuary, England (mudflat) ±4 3 (48) (61) ±0 05 (70) ±0 3 (9) (16) ±0 7 (16) Oystercatcher 4. Small Polychaetes Bangor Flats, Wales (mussel bed) ±2 0 (76) (48) ±0 03 (60) ±1 0 (6) (17) ±0 6 (6) 5. Earthworms Lumbricus sp. Exe Estuary, England (grassland) ±3 1 (63) (34) ±0 02 (52) ±0 7 (73) (78) ±0 9 (76) 6. Small Spisula solida Exe Estuary, England (sandflat) ±3 7 (10) (35) ±0 01 (54) ±0 9 (42) (52) ±0 8 (49) 7. Polychaetes Bangor Flats, Wales (mudflat) ±8 1 (51) (61) ±0 03 (53) ±0 6 (10) (17) ±1 5 (11) 8. Diptera larvae patches Kimmeridge Bay, England (strandline) ±8 4 (30) (33) ±0 07 (42) ±1 0 (5) (9) ±0 8 (5) 9. Large Spisula solida Exe Estuary, England (sandflat) ±7 9 (18) (28) ±0 16 (61) ±2 0 (9) (9) ±1 3 (9) 10. Small mussels Mytilus edulis Seine Estuary, France (mussel bed) ±26 6 (30) (33) ±0 01 (55) ±1 2 (65) (86) ±1 7 (84) 11. Cockles Cerastoderma edule Exe Estuary, England (mudflat) ±27 2 (30) (28) ±0 01 (58) ±1 4 (19) (27) ±2 1 (27) 12. Large mussels Mytilus edulis Exe Estuary, England (mussel bed) ±115 8 (59) (31) ±0 01 (52) ±1 4 (53) (67) ±2 0 (67) was relatively short, ranging from 1 5 to 3 2 s. Having displaced a victim, the aggressor successfully stole the prey in 37% of black-tailed godwit fights and 32% of oystercatcher fights. Prey stealing was profitable in all systems; a bird could increase its intake rate by attempting to steal prey from a less dominant bird rather than searching independently (Fig. 2). This was because the time to steal prey was shorter than the time to find prey independently. In contrast, being attacked reduced the intake rate of victims as they wasted time in the dispute and sometimes lost their prey. One potential mechanism of interference was therefore present in all systems. RELATIONSHIP BETWEEN ATTACK DISTANCE AND HANDLING TIME To allow the model to be parameterized for systems in which attack distance cannot or has not been measured, we related attack distance to handling time. We anticipated that attack distance and handling time were related because, unless aggressors can anticipate when a potential victim will start to handle prey, which seems unlikely for the shorebirds systems studied, they have only the duration of handling time in which to complete an attack. Handling time therefore imposes the upper limit on the duration, and hence distance of attacks. Attack distance (D A ) was related non-linearly to handling time (Fig. 3); in the nine systems with handling times of 15 s or less, attack distance was related positively to handling time (linear regression of individual attack distances against system handling times: D A = T H ; P < ; n = 193), whereas there was no relationship in the five systems with handling times longer than 10 s (D A = T H ; P > 0 05; n = 151). We fitted a hyperbolic function to describe this relationship. D A D = T TH, + T MAX 50 H eqn 5 where D MAX = maximum attack distance and T 50 = handling time at which attack distance is 50% of its maximum value. We used this function because (i) it predicts an attack distance of zero when handling time is zero (as handling time imposes the upper limit on the duration of attacks, attack distance must approach zero as handling time approaches zero) and (ii) the linear regressions showed that attack distance approached an asymptotic value as handling time increased. Nonlinear regression (NLIN procedure of SAS version 8) of individual attack distances against system handling times was used to estimate parameter values: D MAX = 2 74 m (95% confidence limits = ) and T 50 = 5 1 s (95% confidence limits = ) (Fig. 3). This model explained 17% of the among-system variation in attack distance compared with 8% explained by a linear regression (D A = T H ; P < ; n = 330). Both equation 5 and the regression equation

6 537 Predicting the strength of interference Fig. 3. Observed relationship between kleptoparasitic attack distance (±95% confidence interval) and handling time in black-tailed godwits (open symbols) and oystercatchers (closed symbols). The line shows equation 5 fitted to the data (see text for parameter values). Fig. 2. Profitability of prey stealing in black-tailed godwits and oystercatchers. For each system, the graph shows the intake rate obtained through independent foraging (1/[(1/ λ) + T H ]; Holling 1959) and the profitability for a bird stealing prey from a subdominant competitor (1/[(T K /P S ) + (T H /2)]; Stillman et al. 1997). had the same number of parameters and so this was further evidence of a nonlinear relationship. TEST OF THE INTERFERENCE MODEL FOR OYSTERCATCHERS AND BAR-TAILED GODWITS Except for mussel Mytilus edulis L.-feeding oystercatchers, the strength of interference has not been measured directly in any of the systems in Table 2, and so predictions could not be tested for these systems. Therefore, we tested the model for the only systems for which data were available. We have shown previously that the model predicts accurately the strength of interference in mussel-feeding oystercatchers (Stillman et al. 1997, 2000). As two further tests, we tested predictions for cockle Cerastoderma edule L.-feeding oystercatchers (Triplet et al. 1999) and lugworm Arenicola marina L.-feeding bar-tailed godwits (Yates, Stillman & Goss-Custard 2000). Attack distances have not been measured in these systems, and so we used equation 5 to predict attack distances from handling times. Table 3 lists the parameter values used. The model predicted accurately the shape of the relationship between intake rate and competitor density (Fig. 4) in cockle-feeding oystercatchers; intake rate varied little with density up to about birds ha 1, after which it declined with increasing density. The model predicted that the intake rate of bar-tailed godwits should not be related to competitor density below about 150 birds ha 1. The observed intake rate was also unrelated to competitor density within this range, but unfortunately no data were available to test the model s predictions above 150 birds ha 1. Predictions overlapped the observed 95% confidence intervals of five of the seven competitor density ranges in oystercatchers and seven of the eight ranges in godwits. These and the previous tests suggest that the predictions for the remaining systems, although untestable, are likely to be similar to the true interference relationships. Table 3. Parameter values used to predict the strength of interference between cockle-feeding oystercatchers in the Baie de Somme, France and lugworm-feeding bar-tailed godwits in The Wash, England Parameter Oystercatcher Bar-tailed godwit Prey encounter rate (λ) s 1 (1) s 1 (2) Prey handling time (T H ) 14 s (1) 12 s (3) Searching speed (S S ) 0 22 ms 1 (4) 0 17 ms 1 (2) Attack distance (D A ) 2 0 m (5) 1 9 m (5) Kleptoparasitic dispute time (T K ) 3 s (4) 2 s (6) Probability of stealing prey (P S ) 0 19 (4) 0 37 (6) (1) Measured from the data used by Triplet et al. (1999). (2) Measured from the data used by Yates et al. (2000). (3) M. G. Yates, personal observation. (4) Measured for cockle-feeding oystercatchers in this study (Table 2). (5) Predicted from handling time using Equation 5. (6) Average value measured for black-tailed godwits in this study (Table 2).

7 538 R.A. Stillman et al. Fig. 4. Predicted and observed strength of interference in (a) cockle-feeding oystercatchers in the Baie de Somme, France and (b) lugworm-feeding bar-tailed godwits in The Wash, England. The lines show the average intake rates of birds within the model. The symbols show the observed intake rates (mean ± 95% confidence limits) of birds feeding in different competitor density ranges. The oystercatcher data are from Fig. 3b of Triplet et al. (1999) and the godwit data are a combination of the data from Figs 1b and 2b of Yates et al. (2000). Intake rates are expressed as a percentage of those achieved in the absence of interference from competitors. See Table 3 for parameter values. PREDICTED STRENGTH OF INTERFERENCE IN OYSTERCATCHERS AND BLACK-TAILED GODWITS Figure 5 shows examples of the interference relationships derived from the parameter values in Table 2. Figure 6 shows two summary statistics for each relationship; D 95, the competitor density at which intake rate is 95% of that in the absence of interference and m 1000, the strength of interference at 1000 birds ha 1, measured as the proportional decrease in intake rate with a proportional change in density. Interference was predicted to reduce intake rate in all systems. The general shape of the interference relationship was the same in all cases (Fig. 5); interference was insignificant at low competitor densities, but increased steadily in intensity as competitor density increased. However, the values of D 95 and m 1000 varied widely among different systems (Fig. 6). In order to determine the causes of this variation, we regressed D 95 and m 1000 against various combinations of the model s parameters. We log 10 transformed D 95 because trials showed Fig. 5. Examples of predicted interference relationships in (a) black-tailed godwits and (b) oystercatchers. These examples were chosen to highlight the range of interference relationships predicted in the two species. The relationships are the average intake rates of birds within the model. Intake rates are expressed as a percentage of those achieved in the absence interference from competitors. See Table 2 for parameter values. that this increased the amount of its variation explained. Attack distance explained the greatest proportion of among-system variation in both variables (Fig. 7): log 10 (D 95 ) = D A, P < 0 001, r 2 = 73 1%; m 1000 = D A, P < 0 001, r 2 = 98 1%. Of the remaining parameters, searching speed was related significantly to both D 95 and m 1000 when regressed in combination with attack distance: log 10 (D 95 ) = D A 1 03 S S, P DA < 0 001, P SS < 0 05, r 2 = 83 2%; m 1000 = D A S S, P DA < 0 001, P SS < 0 05, r 2 = 99 0%. D 95 was related negatively, and m 1000 related positively to both attack distance and searching speed. No other variables were related significantly to either variable when regressed in combination with attack distance. Interference was weaker in godwits than in most oystercatcher systems (Fig. 5); D 95 was larger and m 1000 smaller (Fig. 6). This was because godwit handling times were generally shorter (Table 2), causing attack distances to be shorter (Fig. 3) and hence less interference to be predicted (Fig. 7). Discussion Obtaining direct estimates of the strength of interference in natural animal populations is a difficult

8 539 Predicting the strength of interference Fig. 6. Summary statistics for interference relationships predicted in black-tailed godwits and oystercatchers. (a) Competitor density at which intake rate is 95% of that in the absence of interference (D 95 ). (b) Strength of interference at 1000 birds ha 1 (m 1000 ). Fig. 7. Relationships between kleptoparasitic attack distance and (a) the competitor density at which intake rate is 95% of that in the absence of interference (D 95 ) and (b) the strength of interference at 1000 birds ha 1 (m 1000 ). and time-consuming process. As interference is a key component of food competition, knowing its strength is key to understanding and predicting the dynamics of animal populations (e.g. Goss-Custard & Sutherland 1997). The aim of this study was to show that a behaviour-based interference model can predict accurately the strength of interference much more rapidly. The methods used to estimate the model s parameters were simple. Importantly, we were able to estimate the speed and distances travelled by birds, parameters to which the strength of interference is most sensitive, without the use of reference markers in the habitat (see Moody et al. (1997) for an approach using markers). Therefore, we could collect data rapidly; if a suitable flock of birds was located, they could be filmed and the strength of interference predicted. If we had relied on reference markers, a suitable flock would need to feed within a marked area. As the location of flocks was variable, far fewer data could have been recorded using this approach. By deriving a relationship between attack distance and handling time, we showed how attack distance can be predicted for oystercatcher and godwit systems in which it cannot or has not been measured directly. We have confidence in the shape of this relationship because a hyperbolic function explained a higher proportion of the variation in attack distance than a linear regression, providing evidence that attack distance was related positively to short handling times, but unrelated to longer handling times. Additionally, the interference model accurately predicted the strength of interference in cockle-feeding oystercatchers, and the absence of interference among low-density bar-tailed godwits, when parameterized using the relationship between attack distance and handling time. The interference model could not have been applied to these systems without this relationship. Despite these accurate predictions we believe that, whenever possible, the best approach is to measure each of the model s parameters directly in each new system to which it is applied.

9 540 R.A. Stillman et al. That handling time is an upper limit on attack duration, and hence distance, seems inevitable. Especially when handling time is short, it seems likely that it will be the major factor limiting attack distance, explaining the positive relationship between these variables for short handling times. It is more likely that other factors limit attack distance when handling time is long, for example, whether a victim detects an attacker approaching. The shorebirds studied could usually escape with their prey if they detected an attack, because the prey were relatively small compared to the size of the birds. The vigilance of potential victims could therefore limit attack distance. However, this will not be true in systems in which prey are larger and so cannot be moved or only moved slowly. Additionally, attacking shorebirds never pursued a victim that escaped with its prey. In situations in which such pursuits do occur (e.g. Thompson 1986) the chances of escaping with prey, even if an aggressor is detected will be smaller. Another factor which may limit attack distance is the profitability of kleptoparasitism compared to searching independently (e.g. Brockmann & Barnard 1979; Thompson 1986; Stillman et al. 1997; Broom & Ruxton 1998). Longer attacks are more time-consuming and are less likely to be successful because victims are more likely to finish handling or detect the attack. Therefore, whether it is profitable to attack will depend on the distance over which the attack is to be made. If an aggressor is a long way from a potential victim, the chances of success may be so low and the time cost so great, that it is more profitable to continue searching independently. As this distance decreases, the chances of success will increase and the time cost decrease, and below a certain distance it may become more profitable to attack than search independently. This distance will depend on the profitability of searching independently, which will be higher when prey are more abundant. Therefore, the threshold attack distance will be shorter when prey are more abundant. Because the strength of interference increases with increased attack distance, this will mean that interference will be stronger when prey are rare. A negative relationship between the strength of interference and prey abundance has been identified before (Brockmann & Barnard 1979) and predicted by behaviour-based models (e.g. Moody & Ruxton 1996). Further studies will be required to test the prediction that attack distance decreases as prey abundance increases. Behaviour-based interference models have made a number of successful qualitative predictions and have aided understanding of the basic process of interference (e.g. Ruxton et al. 1992; Holmgren 1995; Moody & Houston 1995). However, to be of applied value they must also produce accurate quantitative predictions. This can be determined only by testing their predictions for a wide range of predator prey systems, but previously only one model has been tested quantitatively and for just one system (Stillman et al. 1997, 2000). The current study therefore increases the number of tests of this model, and provides further evidence that accurate predictions can be made. It is important to note that these were true tests of predictive ability, as no parameters were chosen in order to make the model fit the data. While encouraging, the number of tests is still small and more are needed. However, in the context of the current study, these successful tests increase confidence in the predictions made for systems in which interference had not been measured previously. We have shown previously that the strength of interference is most sensitive to attack distance and that searching speed is the second most important parameter (Stillman et al. 1997). It was not surprising, therefore, that the same result was found in this study, and that variation in the strength of interference among systems was only related significantly to these parameters. Differences in attack distance explained why the predicted strength of interference was usually weaker in godwits than oystercatchers. Interference was usually weaker in godwits because attack distance was usually shorter in this species. Attack distance was probably shorter in godwits because handling time was also shorter, and handling time limited attack distance. Although influencing interference through its relationship with attack distance, handling time itself was not related to the strength of interference, once the influence of attack distance had been removed. Handling time has little influence on the predicted strength of interference because it has two opposing effects (Stillman et al. 1997). When handling time is long, animals handling prey are exposed to the risk of kleptoparasitism for longer (tending to increase the strength of interference), but potential aggressors are also more likely to be handling prey themselves, rather than searching for victims to attack (tending to decrease the strength of interference). Although attack distance and searching speed determine the predicted strength of interference, the other parameters are not redundant. For example, prey encounter rate and handling time determine intake rate in the absence of interference. The datasets with which we have tested the interference model were derived from intensive field studies. In contrast, the data needed to parameterize the model can be collected relatively quickly or can be obtained from the literature. If further tests also support the predictive power of behaviour-based interference models, they will provide a valuable means of predicting the strength of interference. Acknowledgements We are very grateful to two anonymous referees whose comments greatly improved the manuscript. R.A.S. was funded by the Natural Environment Research Council.

10 541 Predicting the strength of interference References Beddington, J.R. (1975) Mutual interference between parasites or predators and its effect on searching efficiency. Ecology, 44, Brockmann, H.J. & Barnard, C.J. (1979) Kleptoparasitism in birds. Animal Behaviour, 27, Broom, M. & Ruxton, G.D. (1998) Evolutionarily stable stealing: game theory applied to kleptoparasitism. Behavioral Ecology, 9, Cresswell, W. (1997) Interference competition at low competitor densities in blackbirds Turdus merula. Journal of Animal Ecology, 66, Cresswell, W. (1998) Variation in the strength of interference with resource density in blackbirds, Turdus merula. Oikos, 81, Dolman, P.M. (1995) The intensity of interference varies with resource density: evidence from a field study with snow buntings, Plectrophenox nivalis. Oecologia, 102, Ens, B.J. & Cayford, J.T. (1996) Feeding with other Oystercatchers. The Oystercatcher: from Individuals to Populations (ed. J.D. Goss-Custard), pp Oxford University Press, Oxford. Ens, B.J. & Goss-Custard, J.D. (1984) Interference among oystercatchers, Haematopus ostralegus, feeding on mussels, Mytilus edulis, on the Exe estuary. Ecology, 53, Goss-Custard, J.D. (1970) Feeding dispersion in some overwintering wading birds. Social Behaviour in Birds and Mammals (ed. J.H. Crook), pp Academic Press, London. Goss-Custard, J.D. & Durell, S.E.A. le V. dit (1988) The effect of dominance and feeding method on the intake rates of oystercatchers, Haematopus ostralegus, feeding on mussels. Ecology, 57, Goss-Custard, J.D. & Sutherland, W.J. (1997) Individual behaviour, populations and conservation. Behavioural Ecology: an Evolutionary Approach, 4th edn (eds J.R. Krebs & N.B. Davies), pp Blackwell Science, Oxford. Hayman, P., Marchant, J. & Prater, A.J. (1986) Shorebirds: an Identification Guide to the Waders of the World. Christopher Helm, London. Holling, C.S. (1959) The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Canadian Entomologist, 91, Holmgren, N. (1995) The ideal free distribution of unequal competitors: predictions from a behaviour-based functional response. Ecology, 64, Moody, A.L. & Houston, A.I. (1995) Interference and the ideal free distribution. Animal Behaviour, 49, Moody, A.L. & Ruxton, G.D. (1996) The intensity of interference varies with food density: support for behaviour-based models of interference. Oecologia, 108, Moody, A.L., Thompson, W.A., de Bruijn, B., Houston, A.I. & Goss-Custard, J.D. (1997) The analysis of the spacing of animals, with an example based on oystercatchers during the tidal cycle. Ecology, 66, Ruxton, G.D., Gurney, W.S.C. & de Roos, A.M. (1992) Interference and generation cycles. Theoretical Population Biology, 42, Sitters, H.P. (2000) The role of night-feeding in shorebirds in an estuarine environment with specific reference to musselfeeding oystercatchers. PhD Thesis, University of Oxford. Speakman, J.R. & Bryant, D.M. (1993) The searching speeds of foraging shorebirds: redshank (Tringa totanus) and oystercatcher (Haematopus ostralegus). American Naturalist, 142, Stillman, R.A., Caldow, R.W.G., Goss-Custard, J.D. & Alexander, M.J. (2000) Individual variation in intake rate: the relative importance of foraging efficiency and susceptibility to interference. Ecology, 69, Stillman, R.A., Goss-Custard, J.D. & Caldow, R.W.D. (1997) Modelling interference from basic foraging behaviour. Ecology, 66, Stillman, R.A., Goss-Custard, J.D., Clarke, R.T. & Durell, S.E.A. V. le dit (1996) Shape of the interference function in a foraging vertebrate. Ecology, 65, Thompson, D.B.A. (1986) The economics of kleptoparasitism, optimal foraging, host and prey selection by gulls. Animal Behaviour, 34, Triplet, P., Stillman, R.A. & Goss-Custard, J.D. (1999) Prey abundance and the strength of interference in a foraging shorebird. Ecology, 68, Yates, M.G., Stillman, R.A. & Goss-Custard, J.D. (2000) Contrasting interference functions and foraging dispersion in two species of shorebird (Charadrii). Ecology, 69, Zwarts, L. & Drent, R.H. (1981) Prey depletion and the regulation of predator density: oystercatchers (Haematopus ostralegus) feeding on mussels (Mytilus edulis). Feeding and Survival Strategies of Estuarine Organisms (eds N.V. Jones & W.J. Wolff ), pp Plenum Press, New York. Received 6 July 2001; revision received 5 February 2002