Behavioral Ecology Vol. 11 No. 6: 597-605
© 2000 International Society for Behavioral Ecology
Predator search pattern and the strength of interference through prey depression
Centre for Ecology and Hydrology, Dorset, Winfrith Technology Centre, Winfrith Newburgh, Dorchester, Dorset DT2 8DE, United Kingdom
Address correspondence to R. A. Stillman. E-mail: rast{at}ceh.ac.uk .
Received 25 August 1999; revised 3 February 2000; accepted 10 April 2000.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHE MODELRESULTSDISCUSSIONREFERENCES |
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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.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTSDISCUSSIONREFERENCES |
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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 Tringa 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 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.
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THE MODEL |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTSDISCUSSIONREFERENCES |
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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 50 x 50 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 (DR) 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
(TR) 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 (PV) 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
(SV) 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 (TH) 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
(SD) 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 (SD = SV) 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 (SD >
SV), 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 (DK) 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 varied individually with all others set to their default value.
Table 1 shows the default value
and range of values used for each parameter.
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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 (TH = 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
(SV) and, unless otherwise stated, over depressed prey
(SD) was calculated from Table 4 of Goss-Custard
(1970) (SV
= SD = 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 (TR) 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 0.5 x 0.5 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 = 0.6 + 0.35 = 0.95 m; it takes a redshank traveling at 0.23
ms-1 5 s to exceed this distance). The range of
TR 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 (PV) 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 = 0.0037, 0.0041 and 0.0045. The
values of b were converted from instantaneous rates of change to
rates of change during 0.1 second time steps (b0.1)
(b0.1 = 1 - e-0.1b =
0.0003699, 0.0004099, 0.0004499) and the mean value of
b0.1 was taken as the probability of a prey cell changing
from the depressed to vulnerable state during a time step
(PV = 0.0004099 (95% c.i. = 0.0000453) 0.1s-1).
For presentation, PV is expressed per second, rather than
per 0.1 second (PV = 0.004 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 0.004
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 PV used in the
model is a reliable estimate. The distance over which Corophium
detect and respond to redshank (DR) is unknown, but Yates
et al. (2000) showed that, on
average, the feeding rate of redshank within a 1.41 x 1.41 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 1.41 x 1.41 m quadrat and depress all
prey (which were arranged in a 100 x 100 grid of cells) within a fixed
distance. Prey returned to the vulnerable state during each time step with
probability 0.0004099. 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
(DR) of 0.6 m (range = 0.1-1 m) was used in the model.
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RESULTS |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTSDISCUSSIONREFERENCES |
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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.
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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.
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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 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).
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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. Nevertheless, 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.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTSDISCUSSIONREFERENCES |
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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 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 though 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 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.
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ACKNOWLEDGEMENTS |
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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.
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