Behavioral Ecology Vol. 10 No. 3: 213-219
© 1999 International Society for Behavioral Ecology
Avian prey-dropping behavior. I. The effects of prey characteristics and prey loss
a Department of Zoology, Eastern Illinois University, Charleston, IL 61920, USA b Department of Biology, The College of William and Mary, Williamsburg, VA 23187-8795, USA
Address correspondence to D. A. Cristol. E-mail: dacrifs{at}facstaff.wm.edu
Received 30 September 1997; accepted 20 April 1998.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHE MODELRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Numerous species of birds break hard-shelled prey items by dropping them from a height. This intriguing prey-extraction method provides an excellent opportunity for studying foraging behavior because a single, easily measurable quantity—height of drop—may be influenced by a wide variety of identifiable characteristics of the prey (e.g., breakability, weight) and social environment (e.g., alone or in the presence of kleptoparasites). Using a dynamic, state variable modeling approach, this paper presents the first theoretical framework for avian prey-dropping systems that incorporates the diversity of prey characteristics and social situations. The model yielded a series of qualitative predictions about prey-dropping behavior that can be tested readily in any prey-dropping system. In particular, the results indicated that quantitative and qualitative differences in item breakability and potential kleptoparasitism should have a significant effect on the height and pattern of prey dropping.
Key words: dynamic state variable model, foraging behavior, kleptoparasitism, prey dropping.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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One of the classic examples of optimal foraging behavior is the breaking of whelks by northwestern crows, Corvus caurinus (Zach, 1978
,
1979). By dropping a
hard-shelled whelk repeatedly onto a hard surface, these crows crack the
whelk's shell and expose its edible interior. Northwestern crows dropped
whelks from near the height that minimized the total height required to break
a whelk, behavior that minimized energy output and supported the predictions
of optimal foraging theory (Zach,
1979; but see Plowright et
al., 1989).
Avian prey-dropping is an ideal behavior for investigating foraging
questions. The foraging task is well defined: the individual must find an
appropriate prey item, locate a suitable site for dropping, and choose a
height that is sufficient for breaking the prey item
(Richardson and Verbeek, 1986,
1987;
Wunderle, 1991;
Zach and Smith, 1981). Also,
researchers can easily measure both the behavior (i.e., the height of drop)
and the factors that influence the behavior (e.g., breakability of the prey).
Further, prey-dropping behavior is widespread. Twenty-three species of birds,
representing three different orders, have been reported to break open a wide
range of hard-shelled prey by dropping (reviewed in
Cristol and Switzer, 1999).
In the northwestern crow-whelk system, the birds fed alone on defended
breeding territories, dropped a very narrow range of prey sizes of a single
species of whelk, and always used flat rocks for a breaking substrate
(Zach, 1978,
1979). Other prey-dropping
systems differ considerably. For instance, many prey-dropping birds forage in
groups, thereby facing potential kleptoparasitism
(Brockmann and Barnard, 1979),
which is not a factor when solitary individuals drop prey. Also, although most
of the prey items birds drop share the feature of having a hard, but
breakable, outer shell (Cristol and
Switzer, 1999), different types of prey may differ in how easily
the outer shell breaks. In addition to such quantitative differences in
breakability, prey types may differ qualitatively in how they break: some prey
items become increasingly more likely to break with subsequent drops
(Cristol and Switzer, 1999),
while others have the same probability of breaking regardless of how many
times they have been dropped (Whiteley et
al., 1990; Zach,
1979). Further, different habitats will provide different
substrates of varying hardness, and even within the same foraging area,
individuals may have several substrates available for dropping
(Barash et al., 1975;
Beck, 1980;
Zach and Smith, 1981). For
example, carrion crows (Corvus corone) at one site in Scotland
dropped mussels on sand, mussel beds, and a concrete airport runway
(Whiteley et al., 1990).
Clearly, how easily a prey item breaks will depend on the hardness and texture
of the substrate.
Because prey dropping occurs in many bird species and habitats, with a
variety of food items, and under different social conditions, it is difficult
to predict intuitively how a specific factor should affect prey-dropping
behavior. For example, should a bird drop a large and sturdy prey item from a
greater height, or would this unduly increase the probability of losing
valuable prey through kleptoparasitism? Constructing a mathematical model
provides a solution to this problem; with a model, one can manipulate single
factors or groups of factors to generate testable predictions. Previous models
of avian prey-dropping (Plowright et al.,
1989; Zach, 1979)
have focused only on a single species and prey item—northwestern crows
dropping whelks. To further our understanding of this intriguing behavior, we
must develop a theoretical framework general enough to encompass the large
range of conditions found both within and among prey-dropping systems. We
present a dynamic, state variable model that extends earlier investigations by
incorporating differences in the hardness of the substrate and the risk of
losing dropped items (e.g., through kleptoparasitism), as well as including a
variety of characteristics of the prey. The results of the model provide a
series a qualitative predictions that can be tested in any prey-dropping
system. In a companion paper we test the predictions of this model using a
newly described avian prey-dropping system, that of American crows, Corvus
brachyrhynchos hesperis, dropping walnuts, Juglans regia and
Juglans hindsii (Cristol and
Switzer, 1999).
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THE MODEL |
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General modeling approach
We chose to use a dynamic, state variable model (Mangel and Clark, 1988
) to
investigate food-dropping behavior. A dynamic model allowed us to investigate
the probabilistic events (i.e., breaking and/or losing the food item) and
sequences of behaviors (i.e., drops) that are inherent in prey-dropping
systems. Using the model, we investigated differences in prey items, habitats, and social environments. Prey items may differ with respect to the energy they contain, their physical dimensions and weight, how easily they break, and whether the number of previous drops affects the probability of breaking. Our model allowed us to vary the energy value and the probability that different food items would break. To examine the effects of different sizes and weights of food items, we assumed that birds carrying larger or heavier items incurred greater flight costs. Items dropped on hard surfaces are more likely to break than those dropped on soft surfaces, so we examined the effect of surface hardness by varying the probability that a food item broke from a given height. We considered loss of food items due to kleptoparasitism or other causes by including a probability that a food item would not be retrieved. This probability of loss is likely to vary depending on the cleanliness of drop areas or the presence of kleptoparasites.
Model description
We have divided our description of the model into two sections,
distinguished by whether the probability of the food item breaking is
independent of or dependent on the number of times the item has been dropped
previously. Within each of these sections, we consider three situations with
regard to the probability of loss: no loss, fixed loss, and height-dependent
loss (see below).
Probability of breaking independent of previous drops
- No loss case. In the simplest situation, an individual has a food item that
it cannot lose, and the probability of breaking open the food item is
independent of the number of times it has been dropped previously. As a
necessary simplification for the model, we assumed that the individual has a
set number of drops, D, in which to break the food item open. If the
food item is not open after D, the maximum number of drops, the bird
gets no reward from the food. Hence, the expected energy value from the food
item at D is
(1)
In most situations, birds would not have a finite number of drops to break open a food item, so we set D very high to separate any effects of this assumption from the initial drop behavior in which we were interested (see Results).
For all drops prior to D, an individual may drop its food item from any height, h, with h ranging from 0 to hmax. A food item dropped from h will break with probability Pb (h), with Pb(h) obtained from the following equation (Plowright et al., 1989):
(2)
where k is a constant that includes the acceleration due to gravity and the mass of the food item. Equation 2 assumes that there is an increasing chance of the item breaking with increasing momentum and that air resistance has a negligible effect on the velocity of the dropped food item. This equation provides an excellent description for the probability of breaking whelks reported by Zach (1979; see also
Plowright et al., 1989). The
individual drops at the height that maximizes the expected energy function
(3)
If the food item breaks, the individual gets a net energy amount
(4)
where Ef refers to the total energy from the food item, a is the energetic cost per unit height of flight, and s is the energetic cost of finding and examining the food item after a drop (Plowright et al., 1989). If
the food item does not break, it receives
(5)
Therefore, for each drop, the food item may break with probability Pb (h), with the individual receiving the net energy Ebreak. Alternatively, the food item may not break [with probability 1 — Pb(h)], and the individual receives the net energy of Eintact plus the maximum energy it can expect from having the food item from drop d + 1 to D. Thus, the dynamic programming equation is
(6)
The results consist of a single optimal height, h*, that satisfies Equation 6 for each d.
- Fixed loss case. The fixed loss case treats the situation where the food
item may be lost between drops, but the loss is independent of the height from
which the item was dropped. For this model, we use logic and equations similar
to the no loss case, with the added probability, PL, of
losing the food item on each drop. Therefore, Equations 1-5 remain the same;
however, the dynamic programming equation (Equation 6) needs to be modified to
incorporate PL.
When the food item is lost, we assume that the individual pays the cost of the drop, but not the cost of handling the food item between drops (i.e., s). Note that this differs from the intact and break situations, which both include locating and examining the food item. By eliminating s in prey loss situations, our model more closely describes loss due to kleptoparasitism (where the item is quickly and visibly stolen after the drop) than loss due to a "cluttered area" (which may require more searching). Therefore, the net energy value to an individual that loses its food item is
(7)
We tested the effect of our kleptoparasitism emphasis by incorporating s into Equation 7 and found no qualitative differences (with values to > 10s) or quantitative differences (up to 2s for fixed loss, > 10s for height-dependent loss). This insensitivity is due to the minimal impact of this handling cost on the results (see Results) and suggests that our emphasis on loss through kleptoparasitism did not affect our results.
For fixed loss, three possibilities exist for an individual dropping a food item: (1) the food item breaks and is not lost; (2) the food item does not break and is not lost; (3) the food item is lost whether it breaks or not. These three possibilities are incorporated in the dynamic programming equation for fixed loss (i.e., each line of the equation below corresponds to one of the possibilities). Thus, the dynamic programming equation for fixed loss becomes
(8)
- Height-dependent loss case. The height-dependent loss case is similar to
the fixed loss case, but PL is now assumed to increase
linearly with the height of the drop. Therefore, the probability of losing in
the variable loss case is:
where Lmax is the maximum probability of losing the food item and hmax is the maximum height from which a food item can be dropped. The dynamic programming equation for the height-dependent loss case matches that of the fixed loss case (Equation 8), with PL replaced by PL (h).
(9)
Probability of breaking dependent on number of previous drops
The previous equations considered food items with probabilities of breaking
that were independent of the number of drops. Below, we modify the no loss,
fixed loss, and height-dependent loss cases to model food items for which the
probability of breaking may be affected by the number of previous drops.
The key change to the equations is modifying the probability of breaking to
include the effect of previous drops. Because the probability of breaking
increases with the height of the drop, simply considering the number of
previous drops may be inappropriate. That is, a previous drop of 20 m will
probably have more of an effect on the subsequent probability of breaking than
a previous drop of 1 m. Therefore, we assumed that the previous drop heights
combined to create a new state variable: the cumulative effect of previous
drops, C. Because previous drops likely will affect types of prey
items differently, as a general simplification we assumed that the increase in
the probability of breaking that resulted from a previous drop was related
linearly to the height of that drop (i.e., a previous drop of 1 m increased
Pb by 0.01). We set a maximum amount,
Cmax, that could be added to the breaking probability and
assumed that the resulting probability of breaking could not exceed 0.99.
Thus,
 | (10) |
and
 | (11) |
With this new state variable, the dynamic programming equation for the no
loss case becomes:
 | (12) |
Similarly, the dynamic programming equation for the fixed loss case is
 | (13) |
Again, modifying Equation 13 for the height-dependent loss case requires simply replacing PL with PL (h).
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RESULTS |
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Probability of breaking independent of previous drops
Using baseline parameters (Table 1), the results predict that drop height will be constant over successive drops but will differ depending on the likelihood of loss and whether the probability of loss is dependent on height (Figure 1). If loss varied with height (height-dependent loss), drop heights were lower than when there was no possibility of loss (no loss); if loss were independent of drop height (fixed loss), drop heights were higher relative to the no loss case. This last, somewhat counterintuitive result of an increased drop height for the fixed loss case can be explained by noting that the more times the item is dropped, the greater the chance of loss, and increasing the drop height decreases the number of drops required. The quantitative prediction for the optimal height for the no loss case was 16 m, equivalent to the optimal height determined by Plowright et al. (1989
).
Additionally, our sensitivity analyses indicated that the qualitative
predictions for relative height among the three loss cases were robust to
changes in the energy value of the food item (Eintact;
range: 0-10.1), the handling cost between drops (s; range: 0-0.195),
the flight cost (a; range: 0-0.019), and the probability of breaking
the food item at a given height (value from Equation 2 incremented; range:
±0.3; see below).
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Using sensitivity analyses, we first examined the effects on drop height of varying a single characteristic. To investigate how the probability of breaking affected drop height, we either added or subtracted a constant amount from the probability function (Equation 2; e.g., + 0.1 added to the probability of breaking for each height; tested range ± 0.3) and limited the maximum probability of breaking to 0.99. Not surprisingly, drop height decreased as the probability of breaking increased (Figure 2). Thus, the results predict that an individual will use relatively low drop heights when dropping items that are easier to break or when dropping items onto harder surfaces.
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We varied the cost of flying per unit height (a) to examine the effect of the weight or size of the food item on drop height (Figure 3). For the no loss and fixed loss cases, drop height decreased with increasing flight cost. For the height-dependent loss case, however, the predicted drop height did not change with changes in flight cost. Having the probability of losing the item vary with height apparently overcame the energetic effects of the size of the food item. Therefore, item weight or size may affect drop height, but this depends on the loss case for the study system.
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The effect of the energy content of the prey item (prey value) on drop height also differs among loss cases (Figure 4). Each loss case had a certain energy value below which dropping was not profitable. Above this value, drop height remained constant across energy content for the no loss case. For fixed loss, the increase in energy value made the increase in height profitable; this decreased the number of drops necessary and hence the likelihood of losing the prey item. For height-dependent loss, individuals dropped from lower heights to decrease the loss risk. Although dropping lower required more drops to break the prey item, the extra energy value of the prey made this worthwhile; thus drop height decreased with an increase in item energy.
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We examined the effect of varying degrees of risk of losing the prey item (Lmax) on drop height (Figure 5). As the probability of losing the food item increased, the predicted drop height decreased for the height-dependent loss case. Hence, the greater the chance of losing the prey item (e.g., having it stolen), the lower the predicted drop height for this case. For fixed loss, drop heights were relatively greater when a risk of loss was present. Again, this result can be explained by the benefit of minimizing the number of drops required. Interestingly, however, for fixed loss the drop height first increased with Lmax, and then decreased. This relationship occurred because the expected energy gain from a prey item decreases as the constant probability of loss increases. That is, if the item is likely to be lost, the expected energy for the item is low. At high probabilities of loss, the expected energy became low enough that the cost of flying (a) was relatively big enough to be influential; therefore, drop heights decreased.
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Although these univariate sensitivity analyses predict the impact of changes in single characteristics of avian prey-dropping systems, multivariate approaches can provide more comprehensive predictions in many situations. For example, larger items are likely to increase both flight cost and energy content. Thus, we varied energy content and flight cost simultaneously to see whether the relationship between the two factors would affect relative drop heights for large and small prey items (Figure 6). For this analysis, we assumed that the breakability of small and large prey items was equal (this assumption is relaxed below). For the no loss case, a higher flight cost always resulted in the individual dropping from a lower height, regardless of the energy content of the prey item. With height-dependent loss, the difference in predicted drop height between the two extremes of flight cost was slight over all energy values. Under height-dependent loss, therefore, neither energy content nor flight cost had a significant impact on drop height. Under fixed loss, however, if larger prey items contain significantly more energy, they may be dropped from the same height as smaller prey items of lesser energy content. Thus, under some situations, the increase in energy gained from a large prey item may balance the extra flight cost incurred.
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The above analysis assumed that the probability of breaking was not
dependent on item size. However, depending on the prey item, this assumption
may not be valid. For some mollusk species, the larger individuals break more
easily (Siegfried, 1977,
Whiteley et al., 1990,
Zach, 1979), whereas in
others, either smaller individuals break more easily
(Maron, 1982) or there is no
relationship between item size and breakability
(Richardson and Verbeek,
1986). The results described above test the latter situation,
where breakability is independent of item size. By using different
combinations of flight cost and increment in the probability of breaking
(i.e., the amount added to Equation 2), we were able to test the two remaining
size-breakability relationships in which larger items break more easily or
less easily than smaller items. The results yielded qualitative drop height
predictions for each situation (Figure
7). As might be expected from previous results, when heavier prey
items were easier to break than lighter items (affecting both flight cost and
probability of breaking in the same direction), drop heights were lower for
heavier prey items. When heavier prey items were more difficult to break, drop
heights were higher for heavier prey items. These predictions were robust to
changes in the relative energy of the two food items. The relative qualitative
predictions for drop heights (see above) remain unchanged for the no loss,
fixed loss, and height-dependent loss cases. The robustness of these
predictions was primarily due to the influence of the probability of breaking
and the probability of losing the food item, which overwhelmed energetic
rewards and flight costs in most circumstances.
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Probability of breaking dependent on number of previous drops
If the probability of breaking is not affected by the number of drops, the
individual was predicted to drop from the same height for all the drops
(Figure 1). However, when the
probability of breaking increases with the number and height of previous
drops, drop height decreased with the number of drops
(Figure 8). Varying
Cmax, the maximum amount that could be added to the
breaking probability, did not change the presence of the decline; however, the
steepness of the slope was affected. That is, as Cmax
decreased, the number of drops until reaching the minimum drop height
decreased. As with previous results, the qualitative predictions concerning
relative drop height among the no loss, fixed loss, and height-dependent loss
cases were not affected by the change in the breaking probability.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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To date, most studies of avian prey-dropping have either been anecdotal or descriptive and have lacked a theoretical framework. Our model provides both the framework for studying prey-dropping behavior and a series of qualitative predictions that are readily testable in any prey dropping system. Avian prey-dropping behavior is extremely widespread and involves a large number of bird and prey species (Cristol and Switzer, 1999
). The results of our model indicate that this diversity in
predator and prey characteristics should lead to dramatic differences in
dropping behavior both within and among systems. Table 2 summarizes many of the results of the model, listing potential characteristics of different hypothetical study systems and the qualtative predictions that follow from these characteristics. Two of the most robust results were the overriding influences on drop height of both risk of loss and the probability of breaking. The strength of their influence suggests that in many cases these two factors are more responsible for shaping dropping behavior than differences in the energy value of the prey item or the cost of flying. Thus, future studies of dropping behavior should emphasize prey loss and breakability.
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Indeed, the results of the model indicate that when armed with two key pieces of information (prey characteristics, loss case), generating and testing the appropriate predictions is a relatively straightforward process. First, one needs information on the characteristics of the prey species that are naturally dropped by the birds. By dropping prey from set heights onto appropriate substrates, one can obtain the probability of breaking as a function of drop height, assess differences in breakability among substrates and prey types (species, size, or weight), and determine whether the probability of breaking changes with subsequent drops. Second, determining the loss case that applies to a particular system requires recording the drop height, kleptoparasitism pressure (if the loss is due to theft), and percentage of prey items lost.
The qualitative nature of the predictions generated by our model
facilitates both correlational and experimental tests. Rather than trying to
match a specific, predicted drop height with observed behavior (sensu
Zach 1978,
1979), one can simply compare
drop heights under different conditions. The easiest method for testing these
qualitative predictions is to observe and compare naturally occurring prey
choice and drop behavior. However, specific predictions may also be tested by
experimentally presenting the birds with prey items of known characteristics
and comparing drop behavior (e.g., Cristol
and Switzer, 1999; Maron,
1982).
The predictions of the model concern variation both between distinct avian
prey-dropping systems and within the same system. Many of the bird species
that drop prey are omnivores (e.g., gulls and crows), and the same individual
may be foraging on multiple types of hard-shelled prey in areas with multiple
substrates available. Developing appropriate drop behavior under these
circumstances is a complicated task. Previous authors have suggested that
learning (e.g., operant conditioning and cultural learning) may play a role in
dropping behavior (Barash et al.,
1975; Beck, 1980;
Ingolfson and Estrella, 1978;
Maron, 1982;
Richardson and Verbeek, 1987;
Siegfried, 1977). However, the
role of learning in shaping dropping behavior in either simple or more complex
situations remains virtually unstudied (but see
Beck, 1980). Although our model
does not address the ontogeny of dropping behavior, dynamic models can be used
to investigate learning behavior. For example, future studies may relax the
assumption that the individual knows the value of k in the
probability of breaking equation (Equation 2). Instead, the individual may
start with a possible distribution of k and use their dropping
experience to modify their estimate of the parameter for that food item
(Hilborn and Mangel, 1997;
Luttbeg 1996;
Mangel 1990). Our current model
does provide the predicted benchmark for fully developed dropping behavior
under specified circumstances. Researchers interested in learning can use this
benchmark to aid them when investigating the development of inexperienced
birds into proficient, prey-dropping foragers.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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We thank M. Mangel, B. Luttbeg, J. Stamps, I. Cristol, I. Switzer, J. Switzer, and an anonymous reviewer for discussions and comments on the model and manuscript. Partial support during the development of this research was provided by Eastern Illinois University and M. Mangel to P.V.S. and by the Faculty Research Committee at the College of William and Mary and National Science Foundation DIR 9113287 from the University of California-Davis to D.A.C.
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  D. A. Cristol and P. V. Switzer Avian prey-dropping behavior. II. American crows and walnuts Behav. Ecol., May 1, 1999; 10(3): 220 - 226. [Abstract] [Full Text] [PDF]
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