Behavioral Ecology Vol. 12 No. 2: 246-260
© 2001 International Society for Behavioral Ecology
A reevaluation of the logic of pilferage effects, predation risk, and environmental variability on avian energy regulation: the critical role of time budgets
a Department of Biological Sciences, Purdue University, West Lafayette, IN 47907, USA b Wabash College, Department of Biology, Crawfordsville, IN 47933, USA
Address correspondence to J. Lucas. E-mail: jlucas{at}bilbo.bio.purdue.edu . V.V. Pravosudov is now at Section of Neurobiology, Physiology, and Behavior, University of California Davis, Briggs Hall, One Shields Ave., Davis, CA 95616-8519, USA.
Received 2 December 1999; revised 12 June 2000; accepted 11 September 2000.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHE MODELRESULTSDISCUSSIONREFERENCES |
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We studied the effect of pilferage rates, variation in food encounter rate, and predation risk on cache and fat-storage regulation using dynamic programming. Previous predictions that small birds facing increased pilferage rates should cache less and store more body fat are not generally supported. Instead, cache investment (caching rate or percent of food cached) is predicted to be unimodal, peaking at intermediate pilferage rates. This pattern is determined, in part, by pilferage-induced changes in time budgets: at low pilferage rates, a marginal increase in pilferage rates can be offset by an increase in cache investment. However, increased caching increases time allocated to both caching and foraging. The increased foraging is caused by the energetic costs of caching and by the loss of energy from the cache. Increased time spent caching and foraging in turn decreases time spent resting under low predation risk. Above some threshold pilferage rate, the marginal value of resting exceeds the marginal value of caching, and cache investment declines with further increasing pilferage rates. These patterns hold for three levels of variation in food encounter rate: time-invariant, between-day, and within-day variation; they also hold across different mean rates of food encounter. We show that previous predictions concerning decreased energy-storage levels with increased food abundance are not supported when there is between-day variation in mean food encounter rates and food abundance increases only on "good" days. Finally, predation risk affects the predictions described above in two ways. First, these trends assume that the birds can rest in a predator-free refuge. If the refuge is not available, birds are predicted to cache less at higher pilferage rates irrespective of the absolute level of pilferage. With the refuge in place, levels of predation risk affect the skew in the pilferage-rate/caching function. As a result, the relative effect of predation risk on caching intensity varies with pilfer rate. At very low pilfer rates, lowered predation risk causes more caching, but lowered predation risk under high pilferage rates can lower caching intensity, contrary to previous predictions. Surprisingly, predation risk has an appreciable effect on body mass only when the bird is predicted to cease caching (i.e., at the highest pilfer rates); otherwise a change of two orders of magnitude in the probability of encountering predators has little effect on body mass. Our results suggest that the tradeoffs associated with the joint regulation of internal energy stores and externally cached stores are more complicated than previous literature would indicate. Our results also show that we have underestimated the role that time budgets play in patterns of energy regulation.
Key words: dynamic optimization, dynamic programming, caching, chickadee, energy regulation, fat regulation, paridae, parus, pilferage, predation risk, poecile, time budgets.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTSDISCUSSIONREFERENCES |
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The study of the adaptive significance of foraging behavior has focused on the diversity of trade-offs associated with behavioral alternatives (Stephens and Krebs, 1986
).
One critical tradeoff made by foragers who themselves are subject to predation
is the balance between starvation risk and predation risk. This is because a
reduction in starvation risk generally requires increased levels of energy
storage (Pravosudov and Grubb,
1997a; Witter and Cuthill,
1993). However, increased levels of energy storage can increase
metabolic expenditures and thereby increase the time required for foraging, a
behavior considered to put the animal at enhanced risk of predation
(Lima, 1986;
McNamara and Houston, 1990).
Also, the extra mass associated with fat storage may itself reduce the ability
of the forager to avoid predators (Ekman
and Hake, 1990; Lima,
1986), although the validity of this cost has received equivocal
support (as discussed in the Model section below).
These tradeoffs are complicated further by the fact that animals can store
energy in several forms. For example, many animals cache food externally in
addition to storing energy internally as fat
(Vander Wall, 1990). One
fundamental tradeoff faced by animals that cache is the allocation of time and
energy to cache maintenance versus internal fat storage. Indeed, cached food
and fat are widely seen as substitutable forms of energy storage
(Brodin and Clark, 1997;
Ekman and Lilliendahl, 1993;
Hurly, 1992;
Källander
and Smith, 1990; Lucas,
1994; Lucas and Walter,
1991; McNamara et al.,
1990; see Pravosudov and Grubb,
1997b,
1998, for an alternative
view). However, while cached food and fat are both forms of stored energy,
there are a number of important differences that may affect the degree to
which either form is preferred. There is a fairly strict limit on how much fat
an animal can store (Blem and Pagels,
1984), although an advantage to fat storage is that there is
little variation in the animal's estimate of the size of the fat reserves.
There are also additional factors that limit the value of carrying extra mass,
such as increased metabolic rates and increased flight costs
(Lima, 1986;
Lucas and Walter, 1991;
Witter and Cuthill, 1993). In
contrast, the size of a bird's cache is not nearly as constrained as fat
stores (Vander Wall, 1990).
However, cached food represents a less certain source of energy due to
pilferage and the potential for forgotten cache locations
(Lucas and Zielinski, 1998).
Additional costs unique to the use of cached food as a storage mechanism are
the time and energy required to store then retrieve food, and pilferage and
memory constraints that can limit the length of time the food can profitably
be stored.
A logical extension of the concept of a tradeoff between cache and fat
stores is that a reduction in the value of caches, such as through increased
cache pilferage, should cause a reduction in caching rates and a concomitant
increase in fat reserves (Ekman and
Lilliendahl, 1993; Lucas and
Walter, 1991; McNamara et al.,
1990; Sherry,
1985). We will refer to this idea as the "prevailing
prediction."
From a practical perspective, cache pilferage is an important component of
energy regulation tactics, because it provides a means by which the value of
energy storage can easily be manipulated. For example, a cacher's response to
experimentally induced cache pilferage can provide a critical insight into the
degree to which animals regulate energy stores
(Lucas and Zielinski, 1998).
However, the only current test of the prevailing prediction generated
contradictory results: Carolina chickadees (Poecile carolinensis)
cached more seeds and cached a higher percentage of encountered seeds when
cached seeds were pilfered, compared to conditions in which cached seeds were
left in place (Lucas and Zielinski,
1998). Lucas and Zielinski
(1998) suggested that these
results imply that the general theoretical approach to cache/fat tradeoffs may
need to be reevaluated (also see
Pravosudov and Grubb, 1998).
However, there are alternative explanations for this failure of the prevailing
hypothesis. One is that the birds' response to cache pilferage was an artifact
of extreme environmental conditions presented in the aviaries
(Lucas and Zielinski, 1998).
Another possibility is that previous models incorporated appropriate tradeoffs
between the maintenance of a cache or fat stores (e.g.
Brodin and Clark, 1997;
Hurly, 1992;
Lucas and Walter, 1991;
McNamara et al., 1990), but
the effect of pilferage was not fully explored. For example, Lucas and Walter
(1991) described the
theoretical effect of cache pilferage on cache size and body mass, but not the
effect of pilferage on caching behavior (e.g., percent of seeds cached or
caching rate). To explore this second alternative, we analyzed an updated
version of previous dynamic programs
(Brodin, 2000;
Lucas and Walter, 1991;
McNamara et al., 1990). Here
we specifically ask whether the predicted negative correlation between cache
intensity and pilferage rate is robust in the context of the
"standard" approach to the dynamic regulation of energy stores
described in these papers. In effect, we ask whether the general theoretical
approach taken by previous modelers really does need to be reevaluated, as
implied in Lucas and Zielinski
(1998).
Three of the most important environmental conditions that should affect
regulation of energy storage of any form are variability in the access to
food, the level of predation risk, and the mean amount of food available
(e.g., McNamara and Houston,
1990). Our analysis of pilferage effects on energy regulation
(both fat and cache stores) will incorporate all three conditions.
Many papers focus on the effect of variability in food access on energy
regulation (e.g., Kacelnik and Bateson,
1996; McNamara and Houston,
1992). One of the most important advantages of energy storage is
that it provides an energy source when foraging is dangerous or food is
otherwise difficult to find (Pond,
1978). The presence of predators imposes an obvious immediate
mortality risk of foraging, but in addition may reduce the frequency and
duration of foraging bouts (Lucas,
1985). Indeed, resources themselves vary over different time
scales, from within days to between seasons
(Vander Wall, 1990). The scale
over which resources vary is a potentially critical component of the ability
of an animal to use energy storage to adapt to variation in resource
variability (Lucas et at.,
1993). Obviously a response to short-term variation in resource
abundance (e.g., variation in the rate at which prey are found in a patch)
requires very different energy storage strategies than the response to a
weather-induced reduction in food availability over the course of days.
Nonetheless, much of the literature on energy regulation uses only a single
time scale. Here we consider three different environments that vary in the
scale of resource variation: one in which mean food encounter rates are
constant throughout the day and all days are uniform, a second in which mean
food encounter rates vary between days, and a third where food encounter rates
vary within days (as might happen in response to interruptions). The effect of
pilferage rate on energy regulation is evaluated for each of these
environments. We also vary the effect of mean food encounter rates for the
first two environments and the effect of the transition probabilities from
high to low food encounter rates when resources vary within days. These
manipulations allow us to determine how animals should simultaneously manage
internal fat reserves and cached food under a variety of foraging
conditions.
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THE MODEL |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTSDISCUSSIONREFERENCES |
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We model caching and mass regulation in a small passerine in response to different levels of pilferage, food availability, and predation risk, under temperate zone, winter conditions. The model is an extension of several previous models. We use a more realistic description of cache dynamics than found in McNamara et al. (1990
), who assumed that all
cached food could be retrieved in any given 30 min period during the day, and
that all cached food was pilfered overnight. We use a more realistic
description of food encounter rate than found in Lucas and Walter
(1991) who assumed that
constant food encounter rates occurred over fixed intervals every 2 h during
the day, with no food available at other times. We extend McNamara et al.'s
(1994) results on foraging
interruptions in a non-caching bird to evaluate effects of interruptions (or
more generally, within-day variation in access to resources) on a caching
bird. We extend a model by Pravosudov and Lucas
(in press) who considered only
a single, short-term level of variation in food encounter rate. Here we
consider two additional temporal scales of variation in access to food:
withinday variance (e.g., that might result from short-term interruptions in
foraging) and between-day variation (that might result from weather-induced
reduction in food access). We extend Brodin's
(2000) study of short-term
caching. Brodin treated short-term prey encounter rates as fixed
(non-stochastic) but allowed for variation in encounter rates from morning to
midday to evening. Here we consider a broader range of temporal scales over
which food encounter rates might vary. Finally, our model provides a more
synoptic view of pilferage effects than found in any of these previous
papers.
Brodin and Clark (1997) also
modeled between-day variation in food encounter rates, in addition to
fall/winter variation (the latter is not addressed with our model). However,
their approach differs substantially from ours. They assume that caches whose
location the forager forgets are nevertheless available to the forager and
that these forgotten items increase the local encounter rate of food (also see
Smulders, 1998). In contrast,
we assume that forgotten caches are lost and have no effect on subsequent food
encounter rates. The two approaches address different life-histories. Ours is
likely to be more appropriate for short-term cachers: birds that occupy
low-latitudes or high-latitude, winter-caching birds that cache relatively
little food. These birds are unlikely to find cached food by chance. Long-term
hoarding models are likely to be more appropriate for fall caching of
high-latitude species that can cache more than 10,000 seeds in a single season
(Brodin, 1994;
Pravosudov, 1985). Our model
is also designed to evaluate temporal trends on a finer scale than in the
Brodin and Clark (1997)
model.
We equate cache loss with pilferage throughout the paper, but pilferage is
only one of two sources of cache loss. The other is the forgetting of cache
locations. Both sources of loss may be functionally equivalent if a bird
attempted to retrieve a cache but failed because the item was no longer
present or because the cache location was misjudged. Alternatively, if the
bird's memory loss results in its never attempting to retrieve the item, then
the energetic consequences of pilferage and forgetting would differ. The fine
details of cache retrieval are not understood well enough for us to provide a
realistic distinction between these alternative sources of cache loss
(Lucas and Zielinski, 1998).
Therefore, we will assume that "cache loss" is governed by a
single process analogous to cache pilferage. Our model addresses how birds
should respond to differences in cache-loss rates across different habitats
that vary in pilferage rates as well as in the overall probability of finding
food. As such, our results appropriately reflect the effect of pilferage rates
on energy regulation.
The model is a stochastic dynamic program
(Mangel and Clark, 1988), in
which body mass and cache size are treated as state variables. We assume that
the bird chooses among four alternative behaviors (search for food and eat,
search for food and cache, retrieve cached food, or rest) and that the
behavior chosen will maximize the probability of over-winter survival.
Survival is affected by two sources of mortality, starvation and predation
risk.
As with any dynamic program, we divide state and time into discrete
intervals. We assumed that day length was 7 h (i.e., day length of Stockholm,
Sweden, in late November or Edinburgh, Scotland, in late December). Each 7-h
day is divided into 20-min intervals (shortening this to 10 min has no effect
on the results). The maximum cache size is 300 food items (increasing this to
400 has no effect on the results), and body mass is assumed to range from 8-12
g divided into 100 increments (increasing this to 150 increments has no
effect). This variation in body mass is assumed to be primarily due to changes
in internal fat reserves (Blem,
1990). Linear interpolation was used to estimate survival
consequences of fractional increments of both body mass and cache size.
"Night" was treated as a single 17-hour time interval. Cache
pilferage rates were constant throughout a 24-hour day, assuming that caches
are susceptible to diurnal and nocturnal seed predators. We ran each dynamic
program for the higher of 65 days or twice the cache half-life (i.e., time
until there is a 50% chance of a cached seed being lost); our simulations
indicate that this amount of time ensures that the data we are describing
represent equilibrium conditions.
Mass changes and concomitant changes in starvation risk are a function of the behavior-dependent metabolic rates, food encounter rates, and the longevity of cached food items. Each of these elements is discussed below, followed by a description of predation risk.
Metabolic rate
Mass-dependent metabolic rates were taken from Lucas and Walter
(1991) and scaled for
20-minute intervals. Basal metabolism was:
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The metabolic rate of the four alternative behaviors and nighttime rate was
taken as multiples of BMR:
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The value for resting metabolic rate is from Buttemer et al.
(1986). The values for active
foraging (cache, eat, or retrieve) are based on the fact that short flights
(which are typical of the mode of foraging in the Paridae) cost 12 x
nighttime BMR (Carlson and Moreno,
1992). Thus, we assume the cost of active foraging is intermediate
between the costs of rest and short flight.
Food encounter rate
Three behaviors result in the encounter of food items: retrieval, searching
and eating encountered food, and searching and caching encountered food. We
assume that the number of food items encountered per 20 min when engaged in
any of these behaviors can be described using a truncated normal probability
distribution. The approximations of food encounter rates we used are derived
from data on small parids in Pravosudov
(1983,
1985) and Brodin
(1994), converted to fat
equivalents using estimates from Lucas and Walter
(1991). We also assume that
food encounter rates are independent of the frequency of caching exhibited by
other birds (see Smulders,
1998, for an alternative approach).
Retrieval
We assume that food encounter rate while retrieving increases with the
number of food items in the cache. The rationale is that a higher density of
caches allows for a shorter mean distance that a bird will have to fly to the
nearest cache site, and therefore a more rapid retrieval of those caches
compared to a condition with a lower cache density. We also assume that
retrieval encounter rate is not affected by the abundance of uncached food.
Thus, encounter rate while retrieving (
R), measured in g
mass gained/20 min, was:
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The interpretation of Equation 1 is as follows: The birds retrieve a baseline of 0.26 ± 0.04 g/20 min. As cache size increases, the actual encounter rate asymptotes at 0.31 ± 0.04 g (i.e., 1.2 x 0.26). Equation 2 states that retrieval rate is constrained to be no more than the body fat derived from all the food currently in the bird's cache (each item represents a gain of 0.04 g fat).
Search
We assume that mean food encounter rates while searching are fixed
irrespective of whether the birds ultimately cache or eat the food. We
simulated three types of environments, each of which is characterized by a
different level of variability in food encounter rate.
(1) Search—time-invariant variance
In this type of environment we assume that the mean encounter rate and its
variance are fixed both within and across days. We simulated three different
time-invariant-variance environments by testing three mean search encounter
rates (0.17, 0.19, and 0.21 g/20 min). The standard deviation in search
encounter rate was 0.08 for all simulations. Note in all cases that mean
search encounter rates are lower than mean retrieval encounter rates and that
the variance in encounter rates is higher for search than for retrieval.
(2) Search—between-day variance
In this environment type, we assume that there are two types of day, good
and bad. This would simulate, for example, weather patterns that limit a
bird's capacity to locate noncached food on "bad" days. The days
differ in food encounter rate while searching (as opposed to retrieving
caches). Mean encounter rate within each day type is assumed to be constant.
We assume that the arrival of good and bad days is a Markov process in which
the probability that any given day is good is PG =.9 (thus
the probability of a bad day is PB =.1.) We ran three
simulated environments that varied in food encounter rate on good days (0.17,
0.19, and 0.22 g/20 min) with a fixed encounter rate on bad days (0.12 g/20
min). We also ran three simulated environments that varied in food encounter
rate on bad days (0.12, 0.15, and 0.17 g/20 min) with a fixed encounter rate
on good days (0.22 g/20 min). The standard deviation of the encounter rates
was 0.08 for all environments and both day types.
(3) Search—within-day variance
In this environment, we assume that all days are the same. However, there
is a fixed probability that prey encounter rates vary between two levels (high
and low) from one time interval to the next. Conceptually, this within-day
variation could be viewed as stochastic variation in food-patch quality (note:
here patches do not deplete), or it could be viewed as an interruption of
foraging activity (e.g., by a dominant bird or by a predator) that restricts
access to food in a homogeneous environment. If the bird is in a poor-quality
patch (or is interrupted), search encounter rates
(SL) are approximately one half of the search
encounter rates when the bird is in a high-quality patch (or is not
interrupted) (SH):
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We assume that access to cached food is unaffected by this variation in
prey encounter rates. We model the transition between patch types with a 2
x 2 transition matrix that describes the probability of being in a patch
of either type in any 20-min interval given that the bird was in either patch
type in the last 20-min interval (Table
1). We simulated two different environments for these conditions:
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Thus, for example in environment I1, if the forager is in a low-quality patch in a previous interval then there is a 40% chance that it will stay in a low-quality patch in the current interval (PLL =.4). There is also an 80% chance that a forager in a high-quality patch in the previous interval stays in that patch in the current interval (PHH =.8).
In the first interruption environment, (I1), the average duration of a block of continuously uninterrupted foraging is 100 min, and the average duration of a block of continuously interrupted foraging is 33 min (note: the minimum duration in our model is 20 min, the length of a single time interval). In the second interruption environment, (I2), the average duration of a block of uninterrupted foraging is 200 min, and the average duration of a block of interrupted foraging is 25 min. Heuristically, if patch quality results from interruptions then longer interruptions in environment I1 could be caused by the arrival of a top predator (e.g., sharp-shinned hawk, Accipter striatus), whereas shorter interruptions in I2 could be caused by the arrival of a dominant conspecific. If patch quality is caused by variation in prey abundance, then environment I1 would have larger (or more) low-quality patches and smaller (or fewer) high-quality patches than environment I2.
Cache loss (pilferage)
We model pilferage as a fractional loss of the cache in each 20-min
interval. For any given simulation, pilferage rates are treated as constant
(i.e., with neither between-day nor within-day variation). We varied pilferage
rates (range: 0.86 to 100.00 percent lost per day) to test for the effect of
pilferage rate on energy regulation patterns. Pilferage rates are described as
the percent of items lost per day, although pilferage loss was imposed in each
20-min interval throughout the day.
Starvation risk
We have described the mass-dependent metabolic loss and the
environment-dependent increase in mass that results from retrieving or
searching for food items. The net change in mass resulting from these two
processes affects survival through mass-dependent starvation risk. Following
Lucas and Walter (1991), we
assume that the risk of starvation for birds above 8.4 g is zero. An
incomplete beta function was used to model the starvation risk for birds
ranging from 8 g (Pstarve = 1.0) to 8.4 g
(Pstarve =.0). The incomplete beta function,
(Ix (a,b)), is similar to a cumulative normal
distribution, although it has the realism of finite tails over the interval 0
x 1. The arguments (a,b) determine the relative
shape of the curve. In our simulations, a = b = 3.3, as in
Lucas and Walter (1991). Thus:
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Predation risk
Following Lima (1986) and
Lucas and Walter (1991), we
modeled predation as a two-step process, including the probability that a
predator is encountered in a given 20-min interval
(Pencounter) and the probability that the bird is killed
conditional upon encounter (Pkill|encounter). For our
baseline condition, we assume that the bird is not at risk of predation
(Pencounter = 0) while resting or while roosting
overnight, and that predation risk while retrieving or searching for food
(i.e., foraging) is Pencounter = 0.67 x
10-3. This latter figure is in the middle of the range used in
Lucas and Walter (1991). We
evaluate the effect of levels of predation risk on energy regulation tactics
by varying Pencounter. Three alternative conditions are
considered. For two of these we increase or decrease foraging
Pencounter by an order of magnitude compared to the
baseline condition (0.67 x 10-2 and 0.67 x
10-4, respectively). For the third condition, we let the predation
risk while resting be equivalent to the baseline predation risk while
foraging, in effect eliminating any predator-safe refuge.
Lima (1986) and Lucas and
Walter (1991) modeled the
conditional capture probability as a quadratic function of body mass; however,
over the range of body mass we are simulating, the quadratic function is
nearly linear. Current empirical evidence
(Kullberg, 1998) indicates
that the escape response is not mass dependent for birds carrying low to
moderate fat loads, although birds with very high fat loads may suffer
increased predation risk (also see van der
Veen and Lindstrom, 2000). Thus escape probabilities are likely to
be an accelerating (non-linear) function of mass. We therefore chose an
arbitrary function with predation-risk values similar to those of Lima's
(1986) model at the extreme
mass values in our model (Pkill|encounter =.078 at mass =
8 g; Pkill|encounter =.173 at mass = 12 g), but that also
included an accelerating risk with an increase in mass.
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The probability of depredation is:
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RESULTS |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTSDISCUSSIONREFERENCES |
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Our model results are described separately for each environment type. In each case, we consider the effect of pilferage on cache size, body mass, percent of food items cached, and time budgets (i.e., the proportion of the day spent resting, eating, caching food, or retrieving food). Table 2 provides a synopsis of the main results.
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Time-invariant variance
If the mean and variance of the food encounter rate are constant, cache
size is predicted to decline with increasing pilferage rates
(Figure 1A). The birds are
predicted to increase body mass to compensate for a reduction in cache size,
but the compensation does not occur at the highest food density we simulated
(0.21 g/20 min in Figure 1B).
These results are consistent with the prevailing prediction (see
Introduction). However contrary to the prevailing prediction, the percent of
encountered food items that are cached is maximal at intermediate (rather than
at lowest) pilferage rates (Figure
2A).
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:
0.19 g/20 min; 
: 0.21 g/20 min. (C) Dawn mass as a function of pilferage
rate for three levels of the probability of encountering a predator
(Pencounter): 
:.00006; [UNK]:.0006; 
:.006.
For (A) and (B), Pencounter =.0006. For (C), mean food
encounter rate is 0.17 g/20 min. Error bars are SD. derived from a forward
simulation of the decision matrix generated by the dynamic program (see text).
These relationships are for "time-invariant variance" (see
text).
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The percent of food items cached and caching rate (number of items cached per unit time) are alternative ways of measuring the investment a bird makes in its cache. Both caching rate and percent food items cached are predicted to increase with an increase in pilferage rate until some threshold, and at pilferage rates above this threshold cache investment then declines sharply with continued increase in pilferage (data not shown). In fact, the percent of food items cached and caching rate exhibit similar patterns for all three environments. To simplify the presentation of our data, we will show only percent cached.
The unimodal pattern in cache investment (as a function of pilferage rate)
can be understood based on the effect of pilferage rate on the time budget
(Figure 3). Time budgets were
estimated using a forward simulation from the decision matrix generated by the
dynamic program (as in McNamara et al.,
1990). In essence, the forward simulation calculates the
probability that a given forager is engaged in any of the four alternative
behaviors at any given time interval throughout the day. These probability
distributions are then averaged over the day to get a daily mean proportion of
time spent in each behavior. An alternative way of viewing these results is
that they represent the mean fraction of a population of birds that would be
expected to engage in any of the four behavior patterns over the course of the
day.
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The results from the forward simulation underscore the fundamental role that time budgets, and explicitly tradeoffs between resting in a predator-safe refuge versus foraging, play in energy regulation. From low to intermediate pilferage rates, the bird increases the time invested in caching to compensate for a loss of cached food. This increased caching incurs an energetic cost, which must be compensated for by a further increase in time spent eating. The increase in metabolic expenditure with increased pilferage rate is exacerbated by an increase in body mass (Figure 1B), which also increases time spent eating. The combination of increased caching and eating time comes at the expense of a reduction in time spent resting in a predator-safe refuge (Figure 3). At some threshold pilferage rate, the marginal value of rest exceeds the marginal value of caching. Beyond this threshold, caching rates decline and eventually drop to zero when pilferage rates are so high that birds simply eat any encountered food.
Predation risk can influence energy regulation patterns in two important ways. First, if there is no predator-safe refuge (i.e., Pencounter is the same for resting and foraging), then caching investment declines monotonically with an increase in pilferage rate (Figure 2B). Here, the birds never rest (data not shown), and energy regulation reflects a simple tradeoff between eating and caching. Second, the shape of the percentcached/pilfer-rate function is relatively unaffected by a change in predation risk incurred while foraging (Figure 2C). Nonetheless, predation risk has a substantial effect on cache regulation: caching increases with a reduction in predation risk at any given fixed level of pilferage rate, and increased predation shifts the peak caching rate to higher pilferage rates (Figure 2C). Surprisingly, predation risk has an appreciable effect on body mass only when the bird is predicted to cease caching (i.e., at the highest pilfer rates); otherwise a change of two orders of magnitude in the probability of encountering predators has little effect on body mass (Figure 1C).
Note that, over the entire range of pilferage rates, the birds are never predicted to overcompensate for pilferage by increasing cache size (Figure 1A). Also, the mean food encounter rate will determine, in part, the shape of the percentcached/pilferage-rate function. Increased food availability decreases overall caching rates and shifts the maximal point of the function to lower pilferage rates (Figure 2A). Increasing food encounter rate also causes an overall reduction in energy storage: cache size decreases with encounter rate at low pilferage rates (Figure 1A) and body mass decreases with encounter rate at high pilferage rates (Figure 1B).
Between-day variance
For this version of the model, we keep mean and variance in food encounter
rates constant within days but allow for variation between days. If we vary
food encounter rates on good days and keep the encounter rates on bad days
constant, our results are broadly similar to those described above: cache size
declines (Figure 4A) and body
mass increases (Figure 4B) with
increasing pilferage rates, and the percent of food items cached is generally
a unimodal function of pilferage rates
(Figure 5A; note that there is
no caching on bad days). However, the occurrence of "bad" days is
expected to cause birds to maintain substantially larger caches relative to
environments with constant food encounter rates (compare
Figure 4A with
Figure 1A).
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Similar to the results for time-invariant variance, tradeoffs between resting and foraging are a critical component of the relationship between cache intensity and pilferage rate. On good days with relatively low food encounter rates (0.17 ± 0.08 g/20 min), the percent of time spent resting in a predator-safe refuge is expected to drop to zero when caching rates are at their peak (Figure 6A). Similar results are generated for bad days (Figure 6B). The lack of rest will constrain caching rates, because the bird will be incapable of increasing search (for both cached and eaten food) by reducing time spent resting. However, this constraint is not universally characteristic of between-day variation. For example, with higher mean food encounter rates on good days (0.22 ± 0.08 g/20 min), the birds are able to rest at all pilferage rates (Figure 6C), even on bad days (Figure 6D).
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Increased mean food encounter rates on good days induce more caching overall and a shift in the peak of the percentcache/pilferage-rate function to higher pilferage rates (Figure 5A). Surprisingly, this result is completely opposite to the predicted decrease in caching intensity with increased food encounter rate seen with time-invariant variance. An additional, potentially counterintuitive, result is that increased food encounter rates on bad days induce less caching overall and a shift in the peak to lower pilferage rates (Figure 5B). Thus, changes in food encounter rates on good versus bad days have opposite effects on caching intensities.
Interestingly, tradeoffs associated with time budgets expressed on both good and bad days can also generate a bimodal percent-cached/pilferage-rate function (Figure 5B). On good days, the tradeoff between time invested in resting versus time invested in caching plus eating (see Figure 7A at pilferage rates >4% per day; the tradeoff is also clearly shown in Figure 6C for a different set of parameter values) results in the higher peak in Figure 5B (see the middle function, 0.15 g/20 min encounter rates). The lower peak in Figure 5B results from a different tradeoff between the use of cached food and the use of encountered (and uncached) food, a tradeoff that drives changes in time budgets primarily exhibited on bad days. On bad days, this tradeoff is expressed as a predicted shift from cache retrieval at very low pilferage rates to eating uncached food at somewhat higher pilferage rates (see Figure 7B at pilferage rates <4% per day; also see Figure 6D for a different example). In effect, increased pilferage rates decrease cache size, that in turn forces the bird to search more for uncached food. The former tradeoff (rest versus cache/eat) is a characteristic of all simulations we ran. The latter tradeoff is not shown under high food encounter rates on bad days (see Figure 7C,D for time budgets) when the birds rely primarily on locating uncached food items. These tradeoffs also affect total cache size, causing somewhat different shapes in the pilferage-rate/cache-size function (Figure 4A).
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If we eliminate the predator-safe refuge (i.e., set Pencounter the same for resting and foraging), then the birds never rest (data not shown) and cache investment simply decreases with an increase in pilferage (data not shown). This is the same result we found with time-invariant variance (see above) and underscores the relative importance of rest on our results. If the refuge is retained, the birds are generally predicted to increase cache investment with a decrease in predation risk (Figure 5C). This trend is also similar to that reported above for time-invariant variance (Figure 2C), although cache intensity is much higher with between-day variation in resources than with time-invariant variation. However, there is an exception to the cache-investment relationship at moderately high pilferage rates where decreased predation risk actually causes a decrease in cache investment (Figure 5C). We address this exception below where we discuss time budgets.
Caching requires an overall increase in time invested in foraging and an increase in predation risk generally reduces the relative value of foraging compared to resting (Figure 5C). However, consider the effect of pilferage rates on this relationship. As pilferage increases, the birds must compensate for cache pilferage by reducing time spent resting. At moderately high levels of predation risk (here Pencounter = 6 x 10-4), there is some level of pilferage above which the bird should reduce caching intensity and instead spend more time at rest in a predator-free refuge (Figure 6C). However, if predation risk while foraging is low enough (here Pencounter = 6 x 10-5), then the bird can compensate for cache loss by eating instead of resting on good days (compare Figure 6C and 6E), and foraging (eating and retrieving) instead of resting on bad days (compare Figure 6D and 6F). Thus predation levels have a direct effect on the allocation of time to resting or foraging, and also on the investment in energy storage patterns.
Finally, if the birds cache appreciable amounts of food (i.e., at low pilferage rates; Figure 4A) then mean dawn mass levels are relatively unaffected by changes in either food encounter rates on good days or by changes in the level of predation risk (Figure 4C). This trend is similar to that seen with time-invariant variation (Figure 1C).
Within-day variance
The patterns observed with within-day variation in prey encounter rates are
similar to those observed with time-invariant variation and between-day
variation: cache size is predicted to generally decline with increases in the
pilferage rate (Figure 8A), and
the reduction in cache size should be offset by an increase in mass
(Figure 8B). In addition, the
percent of food cached should increase with an increase in pilferage rate
until some threshold pilferage rate, and decline above the threshold
(Figure 9A). Similar to the
results for environments with between-day resource variation, overall cache
sizes are predicted to be larger under conditions of within-day variance
compared to conditions of time-invariant variance (compare
Figure 1A,B with
Figure 8A,B).
|
PLL =.2, PHH =.9; +:
PLL =.4, PHH =.8. See text for
description of environment and for definitions of variables. (C) Dawn mass as
a function of pilferage rate for three levels of the probability of
encountering a predator (Pencounter):
|
For the simulations with time-invariant variation and with between-day variation, at any fixed pilferage rate the percent of seeds cached per day either decreased (Figure 2 and Figure 5B) or increased (Figure 5A) monotonically with an increase in the level of food availability. Within-day variation does not generate the same pattern. For example, at low and high pilferage rates, short infrequent periods of low food availability (e.g., interruptions) induce a higher percent food cached than longer more frequent periods of low food availability. However, this relationship is reversed at intermediate pilferage rates (Figure 9A).
The role of time budgets for environments with within-day variance is similar to that described for the other two environments we simulated: pilferage induces increased caching, which induces increased eating, which in turn decreases rest time (Figure 10A,B). Under some conditions, caching rates when the birds are interrupted appear to be constrained by a lack of resting time (e.g., Figure 10A,C). Birds will generally rest through the interruption (Figure 10B,D), with retrieval used only by very low-weight birds (data not shown).
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The effect of predation risk on energy regulation under cache pilferage is similar to that seen with between-day variation. Generally, increased predation risk is predicted to cause a reduction in cache intensity, but not under all conditions (Figure 9B). The exception to this rule is caused by the birds eating more encountered food in "good" patches under the joint conditions of high pilferage rates and low predation risk (Figure 10E), instead of resting more under high pilferage rates and high predation risk (Figure 10C). Levels of predation risk have little effect on time budgets in "bad" patches; here birds should generally rest unless they are near starvation, in which case they will retrieve food (Figure 10D,F).
Finally, mean dawn mass levels are relatively unaffected by changes in the level of predation risk if the birds cache appreciable amounts of food then; otherwise body mass decreases with an increased predation risk (Figure 8C).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTSDISCUSSIONREFERENCES |
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Earlier models of caching behavior predicted that cache investment would decrease monotonically as pilferage increased. The results from our model do not support this prediction, irrespective of the environment type and food encounter rate we simulated. Instead, our results suggest that cache investment will peak at intermediate pilferage rates, even though cache size does decline monotonically with increased pilferage rate (Table 2). This statement holds whether cache investment is measured as the percent of food items cached per day or as the absolute rate of caching. Lucas and Zielinski (1998
) suggested that the
falsification of the prevailing prediction implied that we should reconsider
the fundamental assumptions built into our dynamic models of caching behavior.
However, the theoretical results described here suggest a less extreme
conclusion: the previous models may not have incorporated sufficient detail to
provide a robust understanding of the effect of pilferage rates on energy
regulation patterns.
Our analyses suggest that we have been too simplistic in developing the
logic of energy regulation. In particular, the joint regulation of the cache
and internal energy stores includes tradeoffs between a number of variables.
Fat storage should decrease starvation risk but increase metabolic rates and
increase predation risk; this tradeoff is the focus of a large literature
(Bednekoff and Krebs, 1995;
Ekman and Hake, 1990;
Houston et al., 1997;
Lilliendahl et al., 1996;
Lima, 1986;
Pravosudov and Grubb 1997a;
Witter and Cuthill, 1993).
Cache storage involves similar starvation/predation risk tradeoffs
(Källander
and Smith, 1990; Lucas and
Walter, 1991; McNamara et al.,
1990; Pravosudov and Grubb,
1997a). The similarity in tradeoffs associated with these two
forms of storage presumably led to the prediction that cache pilferage should
cause a reduction in the reliance on the cache while concomitantly increasing
reliance on fat reserves (Ekman and
Lilliendahl, 1993; Lucas and
Walter, 1991; McNamara et al.,
1990; Sherry,
1985). However, cache storage incurs metabolic expenditures that
in turn affect tradeoffs associated with the time budget, particularly with
respect resting time (Table 2).
Cache pilferage can induce increased caching rates that offset the loss to
pilferage. This increase in time invested in caching comes at the cost of a
reduction in time spent resting, and this shift in time budget is exacerbated
by the increased energy expended while caching, which induces an increase in
feeding rates. King and Murphy
(1985) suggested that time
budget manipulation can play a key role in energy regulation. Our results
underscore the validity of this conclusion: time-budget tradeoffs are an
important element in the adaptive response to cache pilferage but have been
under-appreciated to date.
Increased metabolic costs associated with pilferage-induced caching are
exacerbated by the effects of pilferage on fat regulation: fat reserves should
increase to offset the increase in starvation risk caused by a
pilferage-induced reduction in cache size (see
Lucas and Walter, 1991). The
increase in fat reserves will generate an increase in metabolic expenditure
beyond the expense incurred from caching. At generally low pilferage rates, a
marginal loss due to pilferage can profitably be offset by an increase in
caching intensity at the expense of a reduction in resting time. However,
beyond some threshold pilferage rate, the marginal value of rest becomes
greater than the marginal value of caching behavior. The result is an increase
in rest and a reduction in caching intensity with a further increase in
pilferage rate. Similar results have been observed in rats and pigeons when
the cost of obtaining food is increased
(Schrader and Green,
1990).
Characteristics of the environment other than pilferage will also influence energetic tradeoffs. For example, if the bird faces between-day variation in food abundance, on "bad" days it can choose to rely primarily on retrieving cached food, or it can rely primarily on the search for uncached food. Obviously, cached food is more valuable at low pilferage rates and low food-encounter rates. Under some circumstances, this tradeoff may complicate the relationship between caching intensity and pilferage rates on "good" days (e.g., Figure 5B).
Similarly, the level of predation risk incurred while foraging or resting can complicate tradeoffs associated with time budgets and energy regulation patterns. For example, the presence of a unimodal relationship between cache intensity and pilferage rate (Table 2) assumes that there is a predator-safe refuge for resting birds. Instead if there is no refuge, then the bird will never rest, and energy regulation is the result of a simple tradeoff between caching and eating: as pilferage rate increases, birds are predicted to shift from a preference for caching to a preference for eating. Another example is the effect of varying levels of predation risk on the cache-intensity/pilferage-rate function. If predation risk while foraging is high, then birds are predicted to reduce caching and increase resting at high pilferage rates. In effect, the increased survival from predation derived from resting is greater than the increased survival derived from foraging at high pilferage rates. However, if predation risk while foraging is low, then birds are predicted to reduce caching and increase foraging at high pilferage rates. Here, foraging to eat is more valuable than rest at high pilferage rates. These effects of predation risk on the time budget will in turn complicate the relationship between predation risk and energy regulation patterns (e.g., Figure 5C and Figure 9B).
Our results can be used to address several aspects of a broader issue
related to the tradeoff between cache and fat storage. Experimental evidence
indicates that the physiological mechanisms of mass regulation and hoarding
seem to share at least some neurophysiological components (e.g.,
Herberg and Blundell, 1970;
VanderWall, 1990). This
physiological linkage between forms of energy storage provides a mechanism for
the general perception that cache regulation is an alternative to fat
regulation (McNamara et al.,
1990; Sherry,
1985). Indeed, the result of this joint regulation of cached food
and fat reserves can be logically simple. For example, in rats and tufted
titmice, caching rates decrease with an increase in body mass
(Fantino and Cabanac, 1980;
Lucas et al., 1993, but see
Pravosudov and Grubb, 1997b,
for the opposite trend in tufted titmice). However, the sign of the
correlation between mass and caching intensity changes with overall levels of
food availability in Carolina chickadees
(Lucas, 1994). The correlation
is negative when food is relatively abundant, because future energy reserves
are devalued for relatively heavy birds foraging in "good"
environments. However, the correlation is positive when food is scarce,
because future energy reserves shift to become more valuable for relatively
heavy birds foraging in harsh environments and shift to become less valuable
for relatively lightweight birds. Clearly, the tradeoffs between the different
forms of energy are environment dependent. The results described here provide
another example of this point: as pilferage rates increase, mass should
increase monotonically, but caching intensity should first increase then
decrease. One consequence of this is that the relationship between body mass
and caching intensity (measured across environments that differ in pilferage
rates) should change sign with a change in pilferage rate.
Similar complications arise with expected energy regulation patterns under
increasing mean food abundance. We noted earlier that under relatively
abundant food supplies, increases in mean availability of food (holding
variance constant) were predicted to cause an overall reduction in both cache
size and fat reserves (Lucas and Walter,
1991; McNamara et al.,
1990): an increase in food encounter rates decreases starvation
risk which in turn reduces the need to store energy in any form. This general
pattern has been empirically demonstrated for several species of parids (e.g.,
Lucas, 1994;
Lucas et al., 1993;
Pravosudov and Grubb, 1997b).
This prediction seems robust for time-invariant resource variance (i.e., no
variability between days and constant diurnal levels of variance). Not
surprisingly, both Lucas and Walter
(1991) and McNamara et al.
(1990) simulated this type of
resource variability. In contrast, if birds experience between-day variation
in resource abundance, increased food availability on "good" days
should trigger higher caching intensities, whereas no substantial changes in
body mass were predicted. In effect, high food encounter rates on
"good" days provide excess cachable food for "bad"
days. At the same time, in contrast to time-invariant resource variance
scenario, it may be disadvantageous to reduce fat reserves in these conditions
because there is some probability of encountering a "bad" day with
low food availability. On the other hand, increased food availability on
"bad" days should trigger lower caching intensities, because the
value of caching on "good" days diminishes when starvation risk on
"bad" days is low.
A result from our model is that regulated fat levels may, under some
circumstances, be insensitive to mean food encounter rates (holding pilferage
rates fixed). Indeed, cache intensity and cache size appear to be more plastic
than fat regulation under a number of conditions we simulated (e.g., changes
in pilferage rate, food encounter rate, or level of predation risk). Hurly
(1992) found this trend in an
experiment on marsh tits (Parus palustris): an increase in food
encounter rate (and concomitant decrease in variability) generated lower
caching rates but no change in body mass. In another empirical study,
Pravosudov and Grubb (1997b)
found that tufted titmice increased both fat reserves and cache size in
response to decreased food availability, thus underscoring the complexity of
energy regulation tradeoffs.
In conclusion, the tradeoffs associated with energy regulation in caching
animals can be quite subtle. Our model underscores the fact that we have much
to learn about the nature of the tradeoffs faced by foraging animals making
multidimensional decisions. For example, while a large literature has
addressed the problem of foraging decisions in response to variability in the
food supply (Kacelnik and Bateson,
1996), we show that the temporal scale of this variation can be
critical in some aspects of energy regulation (e.g., overall cache size), but
may have little effect in others (e.g., qualitative relationship between cache
intensity and pilferage rate). In addition, these tradeoffs may be
particularly complicated in cases where animals jointly regulate several types
of energy stores (here the cache and body mass) that have fundamentally
different dynamic properties. Further, our results may apply broadly to other
classes of allocation decisions (e.g., parental resource allocation,
Clutton-Brock, 1991, or mate
provisioning, Moore et al.,
2000) that have fitness consequences associated with the
regulation of internal versus external energy demands.
Finally, the role that time budgets play in the behavioral ecology of
energy regulation appears to have been under-appreciated. A detailed analysis
of time budgets provides critical insight into number of the predicted
relationships generated by our model (e.g., the unimodal relationship between
caching intensity and pilferage rates), yet few studies of energy regulation
include such analyses (e.g., see van der
Veen, 1999). Our results suggest that the effort needed to include
time budgets in these experiments is warranted.
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ACKNOWLEDGEMENTS |
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This work was supported by the National Science Foundation under a Postdoctoral Research Fellowship in Biosciences Related to the Environment awarded in 1997 to V.V.P. We thank Peter Waser, Todd Freeberg, Jim Kellam, Karen Munroe, and Adam Boyko for their critiques of the manuscript. We would also like to thank David Westneat for an excellent job as editor of the manuscript.
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