Within predator-prey systems, interference has been defined

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1 Behavioral Ecology Vol. 11 No. 6: Predator search pattern and the strength of interference through prey depression Richard A. Stillman, John D. Goss-Custard, and Matthew J. Alexander Centre for Ecology and Hydrology, Dorset, Winfrith Technology Centre, Winfrith Newburgh, Dorchester, Dorset DT2 8DE, United Kingdom We develop a model of predators foraging within a single patch, on prey that become temporarily immune to predation (depressed) after detecting a predator. Interference through prey depression occurs because the proportion of vulnerable prey (and hence intake rate) decreases as predator density increases. Predators in our model are not forced to move randomly within the patch, as is the case in other similar models, but can avoid areas of depressed prey and so preferentially forage over vulnerable prey. We compare the extent to which different avoidance rules (e.g., move more quickly over depressed prey or turn if approaching depressed prey) influence the amount of time spent foraging over depressed and vulnerable prey, and how this influences the strength of interference. Although based on a different mechanism, our model produces two similar general predictions to interference models based on direct interactions between predators: the strength of interference increases with (1) increased competitor density and (2) decreased prey encounter rate. This suggests that there are underlying similarities in the nature of interference even when it arises through different processes. Not surprisingly, avoidance of depressed prey can substantially reduce the strength of interference compared with random foraging. However, we identify the region of the model s parameter space in which this reduction is particularly large and show that the only system for which suitable data are available, redshank Tringa totanus feeding on Corophium volutator, falls within this region. The model shows that, by adjusting its search path to avoid areas of depressed prey, a predator can substantially reduce the amount of the interference it experiences and that this applies over a wide range of parameter space, including the region occupied by a real system. This suggests that behavior-based interference models should consider predator search pattern if they are to accurately predict the strength of the interference. Key words: foraging behavior, interference competition, prey refuge, redshank Tringa totanus, search path avoidance. [Behav Ecol 11: (2000)] Within predator-prey systems, interference has been defined as the short-term, reversible decline in intake rate due to the presence of competitors (Goss-Custard, 1980; Sutherland, 1983) and is thought to be one of the major factors influencing the distribution (e.g., Goss-Custard, 1970; Meer and Ens, 1997; Parker and Sutherland, 1986) and survival (e.g., Goss-Custard and Sutherland, 1997) of foraging animals. Several mechanisms of interference have been observed, including prey stealing (kleptoparasitism) (Ens and Goss-Custard, 1984; Goss-Custard et al., 1984; Triplet et al., 1999; Whitehouse and Lubin, 1999), intraspecific aggression (Mansour and Lipcius, 1991), displacement from rich microsites (Bautista et al., 1998; Dolman, 1995), the time-cost of avoiding (Ens and Cayford, 1996) or scanning for potential aggressors (Cresswell, 1997) and prevention of access to prey (Whitehouse and Lubin, 1999). Interference in these systems occurs because of the presence of or risk of interactions between competitors, and several behavioral models have been based on similar mechanisms (e.g., Holmgren, 1995; Moody and Houston, 1995; Ruxton et al., 1992; Stillman et al., 1997, 2000). In contrast, interference through prey depression has received relatively little attention. Prey depression occurs when prey are able to respond to the close proximity of predators (e.g., by retreating down a burrow) and temporarily become immune to predation (e.g., until they re-emerge from their burrow). Interference occurs because, as predator density increases, the proportion of depressed prey also increases and so the intake rate of predators decreases. Interference through prey depression has been recorded in redshank Trin- Address correspondence to R. A. Stillman. rast@ceh.ac.uk. Received 25 August 1999; revised 3 February 2000; accepted 10 April International Society for Behavioral Ecology ga totanus L., feeding on a burrow-digging, amphipod crustacean Corophium volutator (Pallas) (Selman and Goss-Custard, 1988; Yates et al., 2000), which retreats into a burrow after detecting a predator (Goss-Custard, 1970). It has also been portrayed in a mathematical behavior-based model (Ruxton, 1995). Behavior-based models have successfully predicted the shape of the interference function, the relationship between intake rate and competitor density, in shorebirds (Stillman et al., 1996, 1997), and the increased strength of interference, the proportional change in intake rate caused by a proportional change in competitor density, at low prey densities (Cresswell, 1998; Dolman, 1995; Triplet et al., 1999). Although promising, these models do have limitations. In particular, for tractability, mathematical behavior-based models (e.g., Holmgren, 1995; Moody and Houston, 1995; Ruxton, 1995; Ruxton et al., 1992) assume that animals follow random search paths, where in reality, they may actually avoid each other. The potential importance of avoidance in reducing the strength of interference has been stressed by Goss-Custard (1970) and Vines (1980). Recently its importance has been re-emphasized by Norris and Johnstone (1998) who suggest that the avoidance behavior of oystercatchers Haematopus ostralegus L. feeding on cockles Cerastoderma edule L. reduces kleptoparasitism rates below those expected with random searching. These results suggest that behavior-based models may need to incorporate details of avoidance behavior if they are to accurately predict the strength of interference. In this article we present a model of interference through prey depression, which is a development of that previously used to model interference through kleptoparasitism and avoidance (Stillman et al., 1997, 2000). Our model differs from previous mathematical models in that it does not need to assume that predators forage randomly. Instead, by using

2 598 Behavioral Ecology Vol. 11 No. 6 simple rules to alter their search path, they can avoid areas of depressed prey and so forage preferentially over areas of vulnerable prey. We model local avoidance within a single patch; predators do not have the option of moving to other patches. We compare the extent to which different movement rules cause predators to avoid depressed prey. While it is self evident that the avoidance of depressed prey will decrease the strength of interference, the key issues are: (1) in which region of the model s parameter space is the decrease in the strength of interference particularly large, and (2) do any real predator-prey systems fall within this sensitive region. Therefore, we determine the influence of movement rules on the strength of interference throughout the model s parameter space, and parameterize the model for the only system for which data are available, redshank feeding on Corophium. We conclude that, if real systems fall within the region in which the strength of interference is sensitive to search patterns, interference models will need to incorporate these patterns if they are to produce accurate predictions. THE MODEL The model simulates the foraging behavior of a predator population and the predator avoidance behavior of a prey population within a two-dimensional square patch. Prey are uniformly distributed across the patch within a square grid of cm cells. Prey cannot move across the patch but can respond to the presence of a predator and so make themselves temporarily immune to predation. All prey within a single cell respond at the same time to the presence of a predator. Each individual in the predator population is followed continuously as it moves across the patch searching for prey. To remove edge effects, the patch has wrap-around margins, so that a predator moving off of one side reappears on the opposite and prey close to one edge may respond to the presence of a predator close to the opposite. Prey are not depleted during the course of simulations. Simulations progress in discrete time steps. Prey behavior At any point in time, each prey cell is in one of two vulnerability states: (1) vulnerable to attack or (2) depressed and hence immune to attack. Vulnerable prey respond to the presence of a predator when it is within the response distance (D R )of the center of the prey cell. Prey show the same response, regardless of the behavioral state of the predator. They spend a fixed amount of time responding to the predator (T R ) during which they may still be attacked, but after the response time has elapsed, prey become depressed and immune to attack. If the model had assumed that prey became depressed as soon as they detected a predator, rather than only after the response time had elapsed, predators would never have been able to feed; they would always have been moving into prey cells that had already detected them and become depressed. The response time means, in effect, that prey cells do not become depressed until after a predator has passed. After becoming depressed, during each subsequent time interval, each prey cell has a fixed probability (P V ) of returning to the vulnerable state. This assumption means that depressed prey return to the vulnerable state at a constant proportional rate. At the start of simulations all prey are vulnerable. Predator behavior During each point in time, each predator occupies one of two behavioral states: (1) searching for prey or (2) handling and consuming prey. There are no aggressive interactions between predators. Searching predators walk in a straight line in search of prey. If a predator is within a prey cell in which prey are vulnerable, it moves at a fixed speed (S V ) and has a fixed probability ( ) of capturing a prey item during a single time step. Although the model does not specify the actual density of prey within each cell, it assumes that is positively related to prey density (i.e., increases in occur because of increases in prey density). When a predator captures prey it turns into the handling state. Handling predators are stationary and spend a fixed amount of time consuming the prey (T H ) before continuing to search in their previous searching direction. When predators are searching in prey cells in which prey are depressed, they move at a fixed speed (S D ) and are unable to capture prey until they leave the cell. At the start of simulations each predator is set to the searching state, positioned at a random location within the patch and given a random searching direction. Predator movement rules Predators in the model may avoid the search paths of others and reduce the amount of time they spend foraging over depressed prey. The different avoidance rules used in the model differ according to whether predators can detect the vulnerability of the prey and in the distance over which prey vulnerability can be detected. No knowledge: random foraging Predators are unaware of the vulnerability of the prey and search at the same speed (S D S V ) and in the same direction, regardless of whether they are foraging over vulnerable or depressed prey. Local knowledge: change in search speed and fixed turning Predators know the vulnerability of the prey cell they are currently in and the one they are about to move into, but are unaware of the states of more distant cells. Predators may move more quickly over areas of depressed prey than over vulnerable prey (S D S V ), or turn at a fixed angle (a), at random to either the left or right, if about to move from a vulnerable prey cell to one which is depressed. Distant knowledge: optimal turning Predators know the vulnerability of all cells up to the perfect knowledge distance (D K ) from their current location. If about to enter a depressed cell, predators change direction and walk directly towards the center of the nearest vulnerable prey cell within the perfect knowledge distance. If none are available within this distance, they continue walking in their current direction. Mechanism of interference The only mechanism of interference in the model is prey depression. Therefore, the more time a predator spends over vulnerable prey, the higher its intake rate, or conversely, the more time spent over depressed prey, the greater the strength of interference. Predators spend more time over depressed prey when a higher proportion of prey cells are depressed, but can reduce this time if they avoid depressed prey cells and preferentially forage over vulnerable cells. Model parameter values: general predator-prey systems In order to investigate the general properties of the model, we ran simulations over a wide range of parameter values. It was not possible to run simulations based on all possible combinations of parameter values and so each parameter was var-

3 Stillman et al. Search pattern and interference 599 Table 1 Parameter values used to model general predator-prey systems and the redshank-corophium system Parameter Competitor density Prey encounter rate ( ) Handling time (T H ) Searching speed over vulnerable prey (S V ) Searching speed over depressed prey (S D ) Response distance (D R ) Response time (T R ) Recovery rate (P V ) Fixed turning angle (a) Perfect knowledge distance (D K ) General predator-prey systems 1000 (1 1000) 0.05 (0 1) time step 1 25 (0 100) time steps 0.5 (0 1) time step (0 2.5) time step 1 1 (0 5) 10 (0 20) time steps 0.01 (0 0.1) time step 1 0 (0 180) 0 (0 2) Redshank-Corophium system 1000 (1 1000) birds ha (0 1) s (0 1.4) s 0.23 (0 0.46) ms ms (0 1.2) m 5 (0 60) s ( ) s (0 2) m The table shows the default value and range of values for each parameter. ied individually with all others set to their default value. Table 1 shows the default value and range of values used for each parameter. Model parameter values: Redshank-Corophium system The following parameter values were used to model redshank feeding on Corophium volutator (see Table 1). Goss-Custard and Rothery (1976) measured handling time in this system (T H 0.7 (95% c.i. 0.1) s). As handling time is so short, the length of one time step in the model was set to 0.1 s for simulations of this system. Prey encounter rate was calculated from the data used by Yates et al. (2000) ( 0.50 (95% c.i. 0.1) s 1 ). Redshank searching rate over vulnerable prey (S V ) and, unless otherwise stated, over depressed prey (S D ) was calculated from Table 4 of Goss-Custard (1970) (S V S D 0.23 (95% c.i. 0.02) ms 1 ). To ensure that prey cells did not become depressed until a redshank had passed through them, the response time (T R ) was set to 5 s (cells detected redshank when the distance between the bird and the center of the cell was less than 0.6 m (see below); the maximum distance between the center of a m cell and its edge is 0.35 m; therefore, the maximum distance a redshank needs to travel so that it has passed through a cell that has detected it m; it takes a redshank traveling at 0.23 ms 1 5 s to exceed this distance). The range of T R was 0 60 s, the upper limited calculated from Figure 4 of Goss-Custard (1970), which shows that Corophium respond to a passing redshank in less than 60 s, the frequency with which counts were made. Although no quantitative measurements have been made, Corophium actually respond very quickly to vibrations on the substrate surface (Goss-Custard JD, personal observation.), and so the upper limit of the response time is a large overestimate. The rate at which depressed prey returned to the vulnerable state (P V ) was estimated using Figure 4 of Goss- Custard (1970), which plots the increasing numbers of Corophium appearing at the mud surface per min against the time after a redshank had passed over the mud. We used non-linear regression to fit an asymptotic exponential function (N a(1 e bt ); where N number of Corophium appearing at mud surface per min; t time (s) after redshank has passed over mud; a maximum number of Corophium appearing at mud surface; b instantaneous rate at which the numbers of Corophium appearing at the surface increases with time after redshank passes) to each of the three trials performed by Goss- Custard (1970). The following parameter values were estimated for the three trials respectively: a 23.6, 11.4 and 10.9; b , and The values of b were converted from instantaneous rates of change to rates of change during 0.1 second time steps (b 0.1 )(b e 0.1b , , ) and the mean value of b 0.1 was taken as the probability of a prey cell changing from the depressed to vulnerable state during a time step (P V (95% c.i ) 0.1s 1 ). For presentation, P V is expressed per second, rather than per 0.1 second (P V s 1 ). Yates et al. (2000) showed how the feeding rate of redshank in an area of mudflat depended on the time since another redshank had passed through and depressed Corophium within the quadrat. They fitted an asymptotic exponential function to their data and estimated the instantaneous rate of increase in feeding rate with time after the first bird (i.e., the equivalent parameter to b) as s 1. This is another, similar measure of the rate at which Corophium return to the mud surface after being depressed by a redshank, which suggests that the value of P V used in the model is a reliable estimate. The distance over which Corophium detect and respond to redshank (D R ) is unknown, but Yates et al. (2000) showed that, on average, the feeding rate of redshank within a quadrat was 80% lower than its interference-free rate if another redshank had just passed through the quadrat. This interference was due to depression of Corophium within the search path of the first redshank (Goss-Custard, 1970; Yates et al., 2000), suggesting that Corophium were depressed over 80% of the square. A simple simulation model was developed to determine the distance over which Corophium would need to respond to redshank in order to produce this result. In the model redshank were assumed to follow random search paths at 0.23 ms 1 through a m quadrat and depress all prey (which were arranged in a grid of cells) within a fixed distance. Prey returned to the vulnerable state during each time step with probability The response distance was adjusted in different simulations until 80% of prey in the quadrat were depressed. If redshank passed through the quadrat very infrequently, all prey returned to being vulnerable between birds, and the response distance needed to be 1 m for, on average, 80% of the prey to be depressed after a bird passed through. If a sequence of birds passed through the quadrat, one entering as soon as the previous one left, most prey were still depressed when a new bird entered the quadrat. In this case, the response distance only needed to be 0.1 m for the equilibrium percentage of depressed prey to be 80%. The actual rate at which a series of redshank passed through the quadrat was not measured in the field study, but was probably somewhere between these extremes. Therefore, as no other data were available, the average response distance (D R ) of 0.6 m (range m) was used in the model.

4 600 Behavioral Ecology Vol. 11 No. 6 RESULTS Comparison of depressed prey avoidance rules Simple avoidance rules, in which predators had very restricted knowledge of the vulnerability of the prey, had little effect on the proportion of time over depressed prey, and hence the predicted strength of interference (see Figure 1a,b). The proportion of time spent foraging over vulnerable prey was very similar to the proportion of vulnerable prey throughout the patch. The strength of interference actually increased slightly when predators accelerated across depressed prey cells (see Figure 1a) and only decreased slightly when predators changed direction before entering depressed prey cells (see Figure 1b). Increased movement speed had little effect on the strength of interference because it had two opposing actions. By accelerating over depressed prey, predators more rapidly moved into areas of vulnerable prey, but at the same time they also depressed prey close to their search path at a higher rate, thus increasing the overall proportion of depressed prey. Turning at a fixed angle when entering depressed prey had little effect on the strength of interference because predators usually encountered depressed prey even after turning. The strength of interference was decreased to a much greater extent by optimal turning, in which predators had perfect knowledge of prey up to a fixed distance and moved towards the nearest vulnerable prey cell (see Figure 1c). This happened because predators spent a higher proportion of the time searching over vulnerable prey than if they had searched randomly or used more simple forms of avoidance. The percentage of time foraging over vulnerable prey was up to twice as high as the proportion of vulnerable prey throughout the patch. All further simulations compare random foraging with optimal turning. Effect of search pattern on the strength of interference Predators in the model followed independent search paths and so the total proportion of the prey that were depressed increased with increased competitor density. As a result, the intake rate of predators decreased with increased competitor density, both when predators searched at random and when they avoided depressed prey (see Figure 2). For both types of search pattern, log intake rate decreased at an accelerating rate with increased log competitor density, indicating that the strength of interference, the proportional decrease in intake rate due to a proportional increase in density, increased with increased competitor density. Although, the general relationship between competitor density and intake rate was uninfluenced by search pattern, the strength of interference increased more slowly with increased competitor density when predators avoided depressed prey than when they searched randomly. The general relationships between parameter values and the strength of interference were unaffected by search pattern (see Figure 3). Increases in either prey encounter rate or handling time caused predators to spend more time stationary and handling prey rather than moving and searching for prey. Stationary predators depressed fewer prey cells than moving predators, and so increases in either encounter rate (see Figure 3a) or handling time (see Figure 3b) decreased the predicted strength of interference. Increased searching speed increased the chance that a predator had passed through a prey cell before it became depressed (which acted to decrease the strength of interference), but also increased the rate at which predators encountered and so depressed other prey cells (which acted to increase the strength of interference). The overall effect of changes in searching speed depended on the speed from which changes occurred. When searching speed Figure 1 Effect of depressed prey avoidance rules on the strength of interference between predators (solid circles), percentage of time spent by predators over vulnerable prey (squares) and percentage of vulnerable prey (open circles): (a) changes in search speed over depressed prey; (b) fixed turning away from depressed prey; (c) optimal turning towards vulnerable prey. Note that intake rate is expressed as a percentage of that achieved in the absence of competitors. Each graph shows the effect of varying one avoidance parameter with all other parameters set to the default values given in Table 1. Predators do not respond to the prey and so follow random search paths when S D S V 0.5 time step 1, a 0 and D K 0.

5 Stillman et al. Search pattern and interference 601 Figure 2 Predicted shape of the interference function when predators either search at random (squares) or avoid depressed prey by optimal turning with a perfect knowledge distance of 1(circles). Two examples are shown in which the recovery rate of depressed prey is either high (p V 0.01 time step 1 ; open symbols) or low (p V time step 1 ; solid symbols). All other parameters are set to the default values given in Table 1. For consistency with previous studies, both relative intake rate and competitor density are plotted on log scales. was initially low, the main effect of increased speed was that predators were more likely to pass through cells before they became depressed, hence increased speed reduced the strength of interference (see Figure 3c). In contrast, when searching speed was high enough that predators always passed through cells before they became depressed, further increases in speed depressed prey at a higher rate, hence increased speed increased the strength of interference (see Figure 3c). The proportion of depressed prey was increased if prey responded to predators at a greater distance or if they remained depressed for longer. Decreased prey response time decreased the chance that predators had passed through prey cells before they became depressed. Hence, the strength of interference was increased by increased prey response distance (see Figure 3d), decreased response time (see Figure 3e) or decreased recovery rate (see Figure 3f). Although, the general effects of each parameter on the strength of interference were not influenced by search pattern, for most parameter values tested, the actual strength of interference predicted by the model was much lower when animals avoided depressed prey than when they searched at random (see Figure 3). Only when prey encounter rate (see Figure 3a) or recovery rate (see Figure 3f) were very high, or predator searching speed (see Figure 3c), prey response distance (see Figure 3d), response time (see Figure 3e) or recovery rate (see Figure 3f) very low, was the strength of interference relatively unaffected by search pattern. In general, search pattern had little influence on the strength of interference when interference was absent or very weak (high encounter rate, short response distance or rapid recovery rate), or when it was so intense that intake rates were virtually zero (low searching speed, rapid response time, or slow recovery rate). Therefore, the strength of interference was sensitive to the search pattern of predators throughout most of the model s parameter space. Effect of search pattern on the strength of interference in the redshank-corophium system Real redshank do not search at random, but avoid depressed prey with the search paths of competitors (Yates et al., 2000). This suggests that a model incorporating depressed prey avoidance would more closely mimic the behavior of real birds than one based on random searching, even though the actual mechanism of avoidance in redshank is unknown. However, this level of detail would be unnecessary if the redshank-corophium system fell in the area of parameter space within which predator search pattern had little effect on the strength of interference. Therefore, we tested the effect of depressed prey avoidance on the predicted strength of interference in this system, and compared which type of search pattern most accurately predicted the observed strength of interference. As it is unclear how real redshank avoid each others search paths, simulations were run to determine the extent to which the predicted strength of interference depended on searching speed over depressed prey, fixed or optimal turning when about to enter a depressed prey cell. As in the previous simulations, only optimal turning had any substantial effect on the predicted strength of interference, and so the results presented are restricted to this avoidance rule. The general shape of interference function predicted by the model was uninfluenced by search pattern and similar to that observed (see Figure 4a). Both the observed and predicted strengths of interference increased with increased competitor density. However, when redshank searched at random, the predicted strength of interference was greater than that observed. In contrast, optimal turning decreased the strength of interference and matched more closely its observed strength (see Figure 4a). The predicted strength of interference decreased with increased perfect knowledge distance, and when assuming a perfect knowledge distance of 0.75 m, the model accurately predicted intake rate at 1000 birds ha 1 (see Figure 4b). Even though the strength of interference was decreased, the precise shape of interference function predicted by the model was still different to that observed. In reality, interference remained insignificant up to higher competitor densities, but then increased in intensity at a higher rate with increased density (i.e., the observed interference function was more curved than that predicted). These simulations showed that the predicted strength of interference was sensitive to search pattern within the region of parameter space occupied by the redshank-corophium system, and that the discrepancy between observation and prediction could be reduced by allowing redshank to avoid areas of depressed prey as real birds do. However, not all of the model s parameters were measured directly and it could have been errors in parameter estimates that caused the discrepancy rather than search pattern. Therefore, we examined the model s sensitivity to changes in its parameter values. The model was relatively insensitive to changes in prey encounter rate (see Figure 5a), handling time (see Figure 5b) and searching speed (see Figure 5c). These parameter values also had relatively small confidence limits, within which the model always over-predicted the strength of interference when based on random searching, but more accurately predicted the strength of interference when based on avoidance. The model was more sensitive to changes in the prey response time (see Figure 5e) and recovery rate (see Figure 5f), but again, when based on random searching, the strength of interference was over-predicted throughout the confidence limits of these parameters. Avoidance of depressed prey, substantially reduced the strength of interference throughout these limits, except when response time was very low. The model was most sensitive to changes in the prey response distance (see Figure 5d), which has not been measured directly in the field, and could only be estimated within relatively large confidence limits. As a result, within the confidence limits of this parameter, the random search version of the model could either under or overestimate the observed strength of interference. Neverthe-

6 602 Behavioral Ecology Vol. 11 No. 6 Figure 3 Effect of each parameter on the predicted strength of interference when predators either search at random (open circles) or avoid depressed prey by optimal turning with a perfect knowledge distance of 1 (solid circles). Each graph shows the effect of varying one parameter with all others set to the default values given in Table 1. less, within these limits, the avoidance of depressed prey still substantially reduced the strength of interference. In summary, the strength of interference remained sensitive to search pattern throughout the confidence limits of each parameter, except for low response times, and the strength of interference was consistently overestimated by the random search version of the model, except towards the lower confidence limit of response distance. DISCUSSION Many models dealing with ecologically relevant behavioral processes are not able to explore whether the main conclusions of the model apply within the region of parameter space occupied by real animals. In an attempt to avoid this limitation, we parameterized the model for the only system for which most of the necessary data were available. In this way, we were able to show that our conclusion derived from the theoretical model that predator avoidance of depressed prey is likely to have an important influence on the predicted strength of interference seems likely to apply to at least one system. Because there are no reasons to regard that system as unusual among predators, it is probable that the conclusion holds for many more predator-systems than the one for which we are currently able to parameterize the model. Even though we showed that avoidance of depressed prey can substantially reduce the strength of interference, it is unknown whether the mechanisms of avoidance used in the model are the same as those adopted by real animals. In the case of redshank, Yates et al. (2000) showed that real birds crossed each others search paths less frequently than if they had searched randomly, providing evidence for the avoidance of depressed prey, but did not identify the mechanism of this avoidance. The most efficient avoidance mechanism tested, optimal turning, relied on birds being able to detect the vulnerability of prey within 1 m. Whereas redshank can undoubtedly detect the presence of vulnerable Corophium within a very short distance, because they hunt visually for this prey (Goss-Custard, 1970), the maximum distance over which they can do this is uncertain. The other mechanisms of avoidance used in the model, accelerating over depressed prey and fixed turning, only relied on redshank detecting the vulnerability of prey in their immediate vicinity. Similar behavior has been observed in real animals (e.g., Pienkowski, 1983; Smith, 1974) and can cause aggregation within favorable patches (e.g., Stillman and Sutherland, 1990), but in the current model these

7 Stillman et al. Search pattern and interference 603 Figure 4 Effect of depressed prey avoidance on the predicted strength of interference between redshank feeding on Corophium: (a) shape of the interference function when redshank either search at random (open circles) or avoid depressed prey by optimal turning with a perfect knowledge distance of 1 m (solid circles); (b) effect of the perfect knowledge distance at a competitor density of 1000 birds ha 1. The observed strength of interference is shown by the squares in (a) and the broken line in (b) (Yates et al., 2000). For comparison with previous studies, in (a), both intake rate and competitor density are plotted on a log scale. See Table 1 for parameter values. rules did not cause much avoidance of depressed prey. Alternatively, other mechanisms of avoidance could be employed that have the same effect on the birds search paths, but do not rely on the direct detection of vulnerable prey. For example, the absence of other birds from an area could be used as a cue to whether prey are likely to be vulnerable. In this way, animals would only need to monitor the current and recent locations of competitors in order to concentrate their foraging effort in areas that have been unexploited for some time. Avoidance behavior is also frequent in animals in which interference occurs through direct competitive interactions, and is thought to reduce the strength of interference in these systems as well (e.g., Goss-Custard, 1970; Norris and Johnstone, 1998; Vines, 1980). Studies of the precise ways in which predators avoid each other, or each others search paths, are required in order to fully parameterize behavior-based interference models. The observed and predicted interference functions had approximately similar shapes, but the observed function was more curved; interference remained low up to higher competitor densities, but then increased at a higher rate with further increases in density. A possible explanation for this discrepancy could be that the model s parameters were fixed across the full range of competitor densities, whereas in reality they may be density-dependent. For example, prey were assumed to respond in the same manner, regardless of the density of redshank. If the response distance increased with predator density, the strength of interference would increase more rapidly as redshank density increased. Alternatively, if prey became habituated to redshank, the response distance would decrease with increased redshank density. This would lead to the strength of interference increasing less rapidly as redshank density increased. The model s predictions were highly sensitive to the response distance and so even relatively small density-dependent changes could have a large effect on the predicted interference function. Another possibility is that redshank avoidance behavior changes with density. For example, at low densities, they may be able to detect likely areas of vulnerable prey at a long distance, simply from the absence of other birds, but, at high densities, may have to use rules similar to those in the model. Further studies are required to test these ideas. The current model produced two general predictions which are also produced by interference models based on direct competitive interactions, such as kleptoparasitism or avoidance (e.g., Ruxton et al., 1992; Stillman et al., 1997); the strength of interference increased both with increased competitor density and decreased prey encounter rate. Similarly, Ruxton (1995) stressed that his mathematical model of interference through prey depression was structurally identical to another model based on competitive interactions (Ruxton et al., 1992). Field studies of species in which interference occurs through competitive interactions have given support for the predicted shape of interference function (Dolman, 1995; Stillman et al., 1996; Triplet et al., 1999) and the association between the strength of interference and prey abundance (Cresswell, 1998; Dolman, 1995; Triplet et al., 1999). However, further studies are required to test the prediction that the strength of interference caused by prey depression is related to prey abundance. Both types of interference model also predict that the strength of interference is most sensitive to parameters expressing the distance over which interactions occur. In the current model, interference was most sensitive to the response distance of prey to predators, whereas the previous version, based on direct interactions, was most sensitive to the distance over which predators initiate fights (Stillman et al., 2000). This suggests that the predictive power of behavior-based interference models will rely on the accurate measurement of the distance over which interactions occur. These similarities between model predictions suggest that there are underlying similarities in the nature of interference, even when it operates through different mechanisms. It is important to realize that applying the model to redshank should not be regarded as a test of the model. We could not test and validate the model because one important parameter value, the response distance of Corophium to redshank, has not yet been measured, and it is unknown whether the mechanisms of depressed prey avoidance used in the model are the same as those used by real birds. In common with many other theoretical models, formal testing cannot yet be attempted in our case. But what the model does show is that, by avoiding areas of depressed prey, a predator can very substantially reduce the magnitude of the interference it experiences and that this applies over a wide range of parameter space, including the region occupied by the one system for

8 604 Behavioral Ecology Vol. 11 No. 6 Figure 5 Sensitivity analysis of the redshank-corophium model at 1000 birds ha 1. The broken lines show the observed relative intake rate (Yates et al., 2000) and the symbols show changes in predicted relative intake rate caused by changes in each parameter when redshank either search randomly (squares) or avoid depressed prey by optimal turning with a perfect knowledge distance of 1 m (circles). The open symbols indicate default parameter values and the shaded area indicates each parameter s 95% confidence interval (, T H,S V,S D and P V ) or range (D R and T R ). In each graph, one parameter is varied with the others set to the default values given in Table 1. which most of the necessary data are available. This suggests that other modeling attempts should consider this aspect of the system if the models are to provide realistic predictions of the strength of the interference in the real world. We are very grateful to Richard Caldow for many useful discussions and to three anonymous referees who provided valuable comments on the manuscript. R.A.S. and M.J.A. were funded by the Natural Environment Research Council. REFERENCES Bautista LM, Alonso JC, Alonso JA, Foraging site displacement in common crane flocks. Anim Behav 56:

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