Behavioral Ecology VoL 8 No. 2; 178-185
Long-term hoarding in the Paridae: a
dynamic model
Anders Brodin and Colin W. dark
Department of Mathematics, #121-1984 Mathematics Road, University of British Columbia,
Vancouver, British Columbia V6T 1Z2, Canada
Using stochastic dynamic programming we modeled the hoarding and foraging behavior of tits and chickadees, Paridtu, that
are resident in the boreal forest at high latitudes. Here autumns have a rich supply of seeds and temperatures are relatively
mild, while winters are cold with short days and a low food supply. We assumed that parids have a memory of limited duration
and that forgotten seeds accumulate in a bank that adds to the general food supply in the hoarder's territory. Our model
predicts that birds should start "high-intensity" hoarding in early autumn, but not before that. Because of mass-dependent costs
the birds will keep their fat levels low during the autumn. When winter arrives they will carry more body fat, both for the long
winter nights and to hedge against the large effects of weather variations in winter. After increasing the fat level at the start of
winter, fat should gradually increase even more, to compensate for the diminishing food supply. Most hoarding occurs in qi'tiiT"".
as a way of building up the supply of long-term stores. Remembered, or short-term caches, may hedge against stochastic events
in the environment. Even though conditions are not beneficial for hoarding in winter, the birds still stored in winter to maintain
larger short and long-term hoards if environmental variation increased. Almost all time in winter that not was spent foraging
was spent perching, mainly to avoid predation. Key words: Boreal climate, dynamic modeling, food hoarding, long-term hoarding,
Panda*. [Bthav Ecol 8:178-185 (1997)]
A
t high latitudes winters are long, days short and temperatures frequently below —20°C for prolonged periods.
Still a few small passerines of the genus Pants are resident the
whole year even far north in the taiga zone. Weighing only 10
to 12 grams these birds have to gain almost 10% of their body
mass in fat each day to have enough energy for a winter night
(Haftorn, 1989).
Species such as the Siberian tit Pants cmctus, the willow tit
P. montanus, and the boreal chickadee P. hudsonicus are winter residents even in the northernmost part of the taiga zone.
To gain enough energy during the short day is a difficult task.
During the autumn when food is abundant, however, these
species store large amounts for consumption during the winter (Brodin, 1994a; Haftorn, 1956). Memory for individual
caching positions does not appear to last from autumn to winter (Brodin et aL, unpublished data; Hitchcock and Sherry,
1990), but hoarders store food in typical positions and will
later rediscover it when foraging in their territories. The
stored food is an important part of their winter diet (Haftorn,
1956;Jansson, 1982).
Parids also use caches in "a much shorter time perspective,
when food is consumed within one to two days after hoarding
(Brodin, 1992, 1993a; Cowie et al.( 1981; Stevens and Krebs,
1986). Since the positions of these caches are remembered
they are readily available and can be viewed as an alternative
to carrying body fat It seems thus that there are two types of
hoarding in parids: short-term hoarding when hoarders retrieve caches that they still remember and long-term hoarding
when they leave caches for the winter. These two types of
hoarding occur in the same population of Scandinavian willow tits (Brodin, 1992, 1993a; Brodin and Ekman, 1994).
To investigate which factors influence long-term hoarding
decisions among parids we used stochastic dynamic programA. Brodin is now at the Department of Zoology, Stockholm Univenity, S-106 91 Stockholm, Sweden.
Received 27 February 1996; fint revision 9 May 1996; Kcond revilion 30 May 1996; accepted 7 June 1996.
1045-2249/97/J5.00 O 1997 International Sodety for Behavioral Ecology
ming (e.g., Houston et aL, 1988; Mangel and Clark, 1988), an
especially weD-suited technique since some effects of decisions
will occur relatively far in the future. The food hoarding behavior in parids has previously been modeled twice using dynamic programming (Lucas and Walter, 1991; McNamara et
aL 1990). Both of these models, however, were fundamentally
different from this one, since they focused on short-term
hoarding. For most parameters McNamara et aL assumed that
retrieval of caches must occur the same day as they were
stored, while Lucas and Walter assumed that most caches
would be pilfered within five days. In this model we instead
consider a long period of food abundance, the autumn, when
hoarding at a high rate is possible. The autumn is followed
by an equally long period of scarcity, the winter, when only a
low rate of food hoarding is possible. Hitchcock and Houston
(1994) modeled the long-term use of caches in the acorn
woodpecker Mdanerpes formiavorus, but they did not treat
the actual build up of the stored supply.
METHODS
The components of our dynamic model of hoarding behavior
are as follows. First, the basic time unit is one 24-hour day,
consisting of two segments called "day" and "night" AD foraging and hoarding activities occur during day hours; nighttime metabolic costs reduce the bird's fat reserves. We use t
= (0,1,.. .7) to denote current date over autumn and winter.
Winter begins on a date Tw when parameters such as night
metabolic costs, energetic gain from foraging, and predation
risk change (see Appendix B for details).
During the day the bird allocates its time among various
activities: (1) forage and eat all food found; (2) search for
and store seeds; (3) retrieve remembered seeds; and (4)
perch in safe environmeat
Each activity (0 incurs certain metabolic costs c, and instantaneous predation risks (L, (see Appendix B for parameter values). The model employs three state variables: X(<) « body
fat reserves (kj); Y{() «* remembered seeds (kj); and Z(Q =
long-term supplies (kj). The dynamics of these sate variables
179
Brodin and Clark • Long-term food hoarding
Overflow of memory ^M|wUy
00
Long-term
stores K(\
lUCXCSSO i n IMMM^"
food supply
Natural food
Remembered
seeds FTA
Retrieval (P)
Loss of r
supplvr
Body fat reserves ATrt
Foraging W
abend seeds
are shown in Figure 1, and precise equations are given in
Appendix A.
Hoarders remember stored seeds only for a few days; specifically, we assume that they forget a constant fraction y, of
remembered seeds each day. Furthermore, there is an upper
limit to how many seeds a hoarder can remember; it forgets
seeds stored in excess of this limit Forgotten seeds, unless
pilfered, accumulate to form a bank of long-term stores. Although the birds will not remember the exact locations of
these seeds, they may still encounter the seeds while foraging
in their territories. Both remembered and long-term stores
are subject to a constant daily loss. We do not adopt a gametheoretical approach to the issue of theft between individuals
but assume that storers will find their caches also after forgetting them (Brodin and F.Wman_ 1994). Since the number
of long-term caches normally is very large, we presume that
long-term stores are not affected by the hoarder's own consumption but disappears at a constant rate.
We use the term "daily foraging strategy" to describe a strategy whereby the day's time allocation to activities A-D is specified in terms of the current time and state variables t, X, Y,
Z, plus an environmental variable, W. The value W[Q represents current weather conditions during the day which may
be either "good" or "bad." Under bad weather conditions
foraging success is reduced. The foraging outcome is a rate
of energy gain that is fixed during the day and stochastic between days. Bad conditions can also occur during the night
but as an unrelated event to the weather during die day. A
bad day may result from precipitation while a bad night is just
unusually cold, with extra energy loss for the sleeping bird.
By an "optimal" foraging strategy we mean a strategy that
maximizes the bird's fitness, or probability of surviving the
non-breeding season (autumn and winter), and only fitness
under the optimal policy is considered. The threats to survival
are starvation (which occurs if body reserves X fall to zero)
and predation. Long-term stores (2) increase energetic gains
from foraging in winter.
The optimal foraging strategy can be computed by dynamic
programming; details appear in Appendix A. Like most dynamic models, we iterate backward, and fitness is die survival
probability until the morning after die last day. For each time
interval, the maximum survival will be die result of the optimal behavior at that time. For aO possible combinations of
variables (body fat, short- and long-term caches, weather) die
Lots of kmg-tenn supplies
Figure 1
Dynamics of Kate variable* in
the model.
optimal behavior each day is saved in a strategy matrix. In die
forward iterations^ve seed 1000 individuals, using this strategy
matrix. The starting condition day 1 is 20% body fat and no
cached food.
A day is split in 20 parts that the forager can allocate to die
various behaviors (A-D). For instance the bird can spend 0,
1/20, 2 / 2 0 , . . . 20/20 on activity A (forage and eat), etc In
a similar way, we discretized die three state variables from 0
to 10. Continuous values of variables are necessary for accurate calculations and we therefore interpolate fitness values in
die backward iteration and strategies in die forward iteration.
We start the account of the results by exploring how survival
depends on die state variables. Then we investigate under
which circumstances retrieval, Le., short-term hoarding, will
occur. After that we "seed" individuals into the model in forward iterations, first under baseline assumptions and then under different parameters. Finally we discuss how the predictions of die model relate to observations of real birds and
what conclusions that we can draw from die model.
RESULTS AND DISCUSSION
Fitness depending on «*»«•>»»«, fat
Early autumn
In die morning of day 10 fitness is around 0.6 and hardly
affected at all by the variables (not shown). Since food is abundant, there is little need of short-term caches and still many
days to build up long-term supplies. Even if die weather is bad
the maximal energy gain from foraging will be 68 kj, many
times more than the night requirements. It does not matter
if fat deposits are almost zero since there is no starvation risk
during die day.
Since the birds cannot forage at night, survival in the evening will essentially be a function of whether die forager has
enough body fat to survive a bad night or only a good night
(not shown).
Early wrnttr
On day 110 survival increases from 0 to 0.7 depending on die
level of long-term supplies that apparently are crucial for winter survival (Figure 2a,b). High levels of short-term supplies
will hardly increase survival at all (Figure 2a,b). Since shortterm supplies eventually will transform into long-term ones,
180
Behavioral Ecology VoL 8 No. 2
a) early winter, 10% fat, good weather
b) early winter, 10% fat, bad weather
c) late winter, 10% fat, good weather
d) late winter, 10% fat, bad weather
Figure 2
Survival for different levels of
cache* and under different
weather conditions. Survival
(from 0.0 to 1.0) for 10% body
fat is ihown at the z-axes and
relative amount of short- and
long-term cupplie* (from 0.0
to 1.0) at the x- and y-axes. (a)
good weather, day 110, (b) bad
weather, day 110, (c) good
weather, day 190, and (d) bad
weather, day 190.
the reason for this is the relatively small number of seeds that
hoarders can remember compared to the amount that accumulates in the seed bank.
Even for individuals with low levels of long-term caches
weather will have little effect unless the level of remembered
caches also is very low (Figure 2a,b). This depends on the feet
that even in winter, under bad weather conditions, a day's
foraging will give 20.0 to 48.4 kj, depending on the level of
long-term supplies. This means that the energy expenditure
also for an unexpectedly cold night, 34.8 kj, will be possible
to gain if the level of long-term supplies is sufficient.
strategies. During autumn there is no retrieval since food is
plentiful anyway. In early winter (day 110) only very lean birds
that also have very low levels of long-term caches will retrieve
if weather is good (Figure 3a). If weather is bad lean birds
will retrieve also at higher levels of long-term caches (Figure
3b). At the end of winter all birds except for very fait birds
retrieved under both good and bad weather conditions (Figure 3c,d) with somewhat more retrieval occurring if weather
conditions were bad.
Late winter
On day 190, 10 days before the end of winter, survival under
good weather conditions is almost 1.0 unless the supplies of
both short- and long-term stores are low (Figure 2c). If weather is bad, also birds with somewhat higher levels of long-term
caches may be affected (Figure 2d).
The forward iteration is a simulation with 1000 birds seeded
into the model, and Figure 4 shows the means of the three
state variables for these birds. Under baseline conditions (Appendix B) the evening fat deposit is kept above 40% of maximum until a few days before winter (Figure 4a). Then there
is an increase in fat three days before winter starts followed
by a permanent winter fattening. The early start of winter
fattening depends on the assumption that the onset of winter
is deterministic Birds then "know" when winter is coming
and gain energy when it is still cheap, ia autusaa. A gradual
onset of winter between day 90 to 110 moved the gain of
winter fat about 10 days earlier (not shown) but did not otherwise affect the curves shown in Figure 4.
During the winter there is a gradual further increase in fat
deposits (Figure 4a). This compensates for the declining food
Short- versus long-term hoarding
We did not design the model to resolve which factors will
promote the development of short- or long-term hoarding
strategies. Short-term caches may be an insurance against stochastidty in the environment but only drawn upon in case of
bad conditions. Still we can see under what conditions retrieval (i.e., short-term hoarding) occurs by checking the optimal
•dheratk
Brodin and CUrk • Long-term food boarding
a) early winter, good weather
c) late winter, good weather
181
b) early winter, bad weather
d) late winter, bad weather
Figure 3
Retrieval as a function of fat
and long-term caches at a level
of 10% ihon-term cache*, (a)
good weather day 110, (b) bad
weather day 110, (c) good
weather day 190, and (d) bad
weather day 190.
supply, depending on the decreasing long-term stores. The
difference between evening and morning fat is higher in winter when the birds need more energy for the night than in
autumn (Figure 4a). Even if the energy demands for extra
cold nights are relatively the same in both seasons, the absolute variation is larger in winter. This would get gradually
more dangerous as winter progresses and the food supply decreases. The extra morning fat deposits, "true winter fattening" (Lehikoinen, 1987), hedges against the larger stochasticity.
Extra overnight energy expenditure under "cold night conditions" is 2.8 kj in autumn and 5.8 kj in winter, corresponding to 4.3 and 8.9% of mavinuim fat deposits. The mean level
of morning fat is around 21% in autumn and between 30%
and 50% in winter. This means that birds carry more fat than
needed just to hedge against one extra cold night Houston
and McNamara (1993) found that for most parameters, small
birds would keep enough energy for several days. If the bird
forages enough to stay on a balanced energy budget, 30% fat
in winter is enough to guard against three consecutive cold
nights, the chance of this being only 1 X 10*3.
The "birds build up short-term caches to mayimmn levels
around day 11-12, after which they continue to store at high
rates (Figure 4b). As winter arrives, conditions are no longer
beneficial for large scale hoarding and short-term supplies
drop (Figure 4b). There is still some hoarding at low inten-
sities in winter, having the effect of maintaining a small, readily available reserve.
Long-term stores reach their highest levels, about 80% of
ma-rimiiin at the onset of winter. If weather is good, a whole
day of foraging would then give an energy gain of 50.4 kj,
dearly more than the 29 to 34.8 kj needed for die night This
means that there is room for other activities besides eating,
such as food storing or perching. Although there may be some
storing in winter it mainly occurs in autumn with the purpose
of building up long-term supplies. The small frequency of
food storing in winter probably depends on the low availability
of storable food.
Since hoarders do not fill long-term supplies to 100%, they
probably aim to keep long-term supplies over a certain minimum level during the winter. The lowest level of long-term
supplies in winter is 36% (Figure 4b), which will result in an
energy gain of 28.1 or 38.1 kj (depending on weather) from
foraging the whole day. Even if weather is bad the birds win
then almost (-0.9 kj) gain the energy required for the night
Depending on foraging success and weather birds may deviate from the mean curves shown in Figure 4. However, most
birds will be dose to the optimal levels and during the autumn those birds wiU spend 24% to 27% (depending on weather
conditions) of the days foraging and the rest hoarding (Figure
5). As winter starts, hoarding decreases dramatically (good
weather) or ceases completely (bad weather). Then 60% to
182
Behavioral Ecology VoL 8 No. 2
a)
a)
50
I.U
b)
b)
ropo rti
c
o
Q.
150
y==
0.8 -Ij
0.6 B •
0.4 JB
*
•
•
•
' •'
==l
0.2 0.0
-\
1
50
1
1
100
Days
1
1
1
150
Figure 5
Optimal strategies for birds with mean levels of state variables (ice
Figure 4a and b). (a) good and (b) bad weather conditions. The
white area is foraging, the vertically striped perching, the
horizontally striped hoarding and the black area retrieval.
c)
Figure 4
Mean levels of state variables as a function of time for 1000 birds in
a forward iteration, (a) evening fat (thick line), morning fat (thin
line), and evening fat adjusted for the changing day length at 60* N
(dotted line), (b) short-term (thick line) and long-term stores (thin
line), (c) evening fat (thin line), ihort-cerm (thick line), and longterm stores (dotted line) under higher environmental variation
than under baseline conditions. In (c) the mean energetic gain is
the same as under baseline assumptions but bad days are worse
(reduction 50% instead of 30%) and more frequent (probability J
instead of .05).
90% of the days are spent foraging with the highest proportion foraging occurring under bad-weather conditions at the
end of winter (Figure 5). When birds are not foraging in winter most time is spent perching, probably since it is a predation-safe strategy. In late winter no further hoarding occurs,
and if weather is bad there is also some retrieval. Birds spend
a larger proportion of the day foraging if weather is bad than
if weather is good (Figure 5).
Forward iterations with conditioiis other
100
die baseline
We here consider the effects of altering three important factors: memory (longevity and capacity), winter rlimat#> (severity
and variability), and predation.
Mtmory dtcay and memory capacity
So far our model predicts that short-term hoarding is less important than long-term hoarding. However, we started with an
assumption that memory decayed quickly and that the birds
could not remember all stored seeds. Still, increasing memory
longevity from 10 to 50 or 100 days did not increase survival.
The same was true if memory capacity increased; if birds
could remember 2300 kj (12300 seeds) instead of 460 kj
(2500 seeds) mortality remained the same. The explanation
may be that we assume that long-term supplies are built up
when storage locations are forgotten. The gains from a better
memory are then counteracted by a slower buildup of longterm supplies. This assumption is supported by the feet that
even a memory limit as small as 46 kj (250 seeds) would not
decrease survival. In fact, a long-lasting memory had to be
combined with a higher memory capacity to increase survival
at alL The implication of this is that there is not much to gain
from the evolution of a long-term memory unless the capadty
to remember many seeds also is high.
Hitchcock and Houston (1994) modeled how acorn woodpeckers should retrieve cached food. They found that a small
remembered stored supply would increase survival since it acts
as an insurance against unpredictability in the environment
In our model the case is different since we assume that the
forgotten caches still remain an essential part of the winter
diet
Winter chmaU
At lower latitudes winters are shorter and have a higher food '
supply and smaller energetic demands. A 20% shorter winter
(80 instead of 100 days) with 20% more accessible food and
20% lower night metabolic cost had the effect of delaying the
onset of high intensity storing until mid-autumn and also de-
Brodin and dark • Long-term food hoarding
creased the number of items stored. Mortality decreased from
38% to 13%.
Not only may the climate differ in means, but also in variability. We increased the reduction in energy gain during bad
weather from 30% to 50% and the probability of bad weather
from 5% to 30%. At the same time we adjusted the energetic
gain from foraging so that the overall mean remained the
same as before. Now the buildup of winter fat starts gradually
already about day 60 (figure 4c) and the birds carry more fat
in winter than under baseline assumptions. More short-term
caches are maintain^ throughout the winter, 15% to 20%.
Also the level of long-term supplies is higher and reaches
100% when winter starts (Figure 4c).
To keep body mass at a level resembling natural conditions
there has to be some form of mass dependent cost, such as
higher energetic loss or predation risk for fatter birds (e.g.,
Houston et aL, in press; McNamara and Houston, 1990). With
no costs for carrying fat our birds kept almost 100% fat deposits throughout the whole period and mortality decreased
about 2%. An increase in the cost of carrying fat (* 1.0 instead
of 03) will have no visible effect in autumn but decreases fat
with an average of 8% during winter (not shown). Obviously,
minimal fu deposits are optimal during the antiimn since
food is abundant During the winter, when conditions are
more critical, fat levels have to be carefully adjusted for optimal survival.
In our baseline assumptions, predation risk per day was
higher in winter (0.005) than in autumn (0.001). If predation
in winter was the same as in autumn hoarding would be more
frequent in winter, gradually decreasing and disappearing
about day 170. While hoarding increased the birds would
perch less and not at all before 135. Perching would then
gradually increase as hoarding decreased. The large amount
of perching in Figure 5, is apparently caused by the high predation risk.
inMnihwiifAiw wits ouselvAlions on real birds
The fitness surfaces in Figure 2c-f show that it is necessary to
build up long-term supplies and then use these during the
winter. Since building up such a long-term seed bank takes
time, a long-term storer has to be resident in the winter territory already in early autumn and, of course, stay there the
whole winter. In fact, willow tits and crested tits, which have
been observed to store large amounts of food in autumn (Brodin, 1994a; Haftorn, 1954,1956; Lens et aL, 1994), are strongly resident on their winter territories (e.g., Ekman, 1979b).
Also, territorial establishment already in early autumn is very
important for survival (Ekman, 1979a). In the beginning of
the ?vtiimn there were no fitness gains from stored supplies.
In nature there is no high-intensity storing until mid-August
(Brodin, 1994a; Haftorn, 1956).
The model supported that there are two components of the
winter weight increase, higher morning body mass and a larger daily amplitude. This has been shown both in hoarding and
non-hoarding birds (e.g., Lehikoinen, 1987; Haftorn, 1989).
Figure 4a shows that winter fat levels should gradually increase as winter progresses. In reality, fat levels decrease as
winter progresses (e.g., Haftorn, 1989; Koivula et aL, 1995),
doubtlessly because the nights gradually shorten and shorter
nights require less energy. The dotted line shows the change
in body fat adjusted for the change in daylengdi (Figure 4a).
Relative to a non-hoarder a hoarder would show a less pronounced decrease in fat deposits depending on smaller fat
deposits in early winter.
The model predicts that even a very low memory capacity
183
(250 seeds or 46 kj) would not decrease survivaL In most retrieval experiments parids show a decay in retrieval performance already after the firrt few seeds (e.g., Shettleworth and
Krebs, 1982) and it is possible that our baseline memory capacity of 460 kj (2500 seeds) may be too high.
CONCLUSIONS
The forward iterations indicate that we have succeeded in creating a model that simulates the conditions for a small passerine hoarding in a boreal climate. There was no effect of
higher levels of stored supplies early in autumn and birds did
not build up long-term reserves to the highest possible leveL
Since there is a daily loss of stores it would be a waste to start
large scale hoarding too early.
From an evolutionary perspective it is plausible that retrieval works the way we have suggested (Le., that short-term retrieval should rely on memory while long-term should not).
The rationale for short-term hoarding is to increase the available energy reserve without carrying excessive fat around. To
be "available" in a way comparable to body fat the hoarder
must be able to return to the cache upon the decision to
retrieve it (Le., know the exact position of the cache). Longterm hoarding on the other hand may have an important
effect on survival just by increasing the general food supply
during the worst part of the winter. To maintain an accurate
memory of tens of thousands of caches for a prolonged period
is probably physiologically expensive compared to remembering a small number of caches for a week or so. The small
additional fitness gain of a long lasting memory predicted by
this model would make the selection for such a memory very
weak.
APPENDIX A
For the dynamic programming computations we introduce
the following "fitness functions" (see Mangel and Clark,
1988):
Fw(x, y, x, t) = iMYimiim probability that the bird survives
from the beginning of day tto the
beginning of day T + 1, given current state
X(r) - x, Y(t> - y, Z(t) = x and W(t) - w
at the beginning of day t
(Al)
F(x, y, z, <) =» same, but evaluated at the end of
day t, Le. before night
(A2)
We then have, directly from these definitions:
F(x, y, z, T) -
fl
\ix>ct{T)
/>, if c,(1) <
ct(T)
(A3)
if * S
where <*(7) and <^(7) denote night-time metabolic cost for
good and bad nights (day 7), and p^ is the probability for a
good night This equation simply says that the bird will always
survive the final night if its energy reserves x are larger than
<»(7), survive with probability />_ if x is larger than cJT) but
smaller than c,(T), and finally, the bird will always die if reserves are smaller than cf(T).
For t < T and * > 0, we have the following dynamic pro-,
gramming equations:
F.(x,y,z,0 = max o(l and
184
Behavioral Ecology Vol. 8 No. 2
- c/.tl.y.x.t + 1) + (1 + 1)]
et aL (unpublished data). The proportion of stored seeds that
are lost per day, -ft " 0.01, and the proportion of long-term
stores that disappear per day, % " 0.01, have been measured
in the field (Brodin, 1993b). In autumn food is abundant and
+ (1 - p<JFt(x + 1)]
small, unstorable food items are often disregarded, so
(A5)
rKTOMj " 1.0. During the winter small invertebrates that are
too small to be stored are an important part of the diet (JansAlso
son, 1982) so r4<»totcr, " 0.5. The energy gain from foraging
F.(0,y,z,<) *> 0
(A6)
in autumn rK(t, g) =» 200 is estimated from food encounter
In Equation A4, a*= (a^, O&, a^, an) where a* is the proportion frequencies in Brodin et aL (1994, 1996) times energy contents in the food (Brodin, 1994a). Minimal energy gain in
of the day spent in activity A, etc (a* + a* + a^ + at) D 1.
winter
rA{t, g) is set to 40 kj/day slightly below the actual
Also, x^m denotes the level of bodily fat leserves Xat the end
mean energy turnover of 45 kj/day measured by Moreno et
of the current day t, assuming X m x at the beginning of the
aL (1991) with doubly labeled water. Under bad-weather conday, that activities are allocated according to a, and that the
ditions foraging success rA(t, b) is reduced to 0.8 * rA(t, g).
weather conditions are w. .Similarly for j ^ w and x^.
The increase in stored supplies (JO from hoarding rB is then
We next specify these values exactly (see Figure 1).
rA (t, w) * -ft. Since territories are large and caches scattered,
Values of the state variables at the start of day (are denoted
the rate of recovering remembered stores rD is probably not
by x, y, x. The environmental state is u First we consider remuch larger than rK so we set it to 1JS • rA (t, tf. The maximembered stores («• caches), y (see Appendix B for parammum possible fat reserves x^ , are estimated to 65.0 kj coneter values):
sidering measurements by Haftorn (1989). With a memory
lasting 10 days and natural hoarding intensities, %**. maxicaches forgotten
- ybmta = -y,j»
mum possible remembered stores will be 460 kj if half of the
caches then remaining
s j , •» y — y^,^
stored seeds are remembered. Based on field studies of hoardretrieved from store (y\) = bymt— " m i n ^ , acr c (0)
ing intensities (Brodin, 1994b; Haftorn, 1956; Pravosudov,
energy obtained by
- A A » ^ . ~ <*ArA(t, w)
1985), loss of caches (Brodin, 1993), and die proportion seeds
foraging
in the stored supplies (Brodin, 1994b; Haftorn, 1956), we asStore after retrieving,
= j , » (jp, - by^tfc*
sume that a maximum of 25,000 juniper seeds, or 4600 kj are
foraging and loss to
+ A ^ wi — y.)
available for the winter. We set p, the maximum effect by longpilfering
term stores on the energy gain from foraging to 0.7. With
excess stores
= y^^. - min(jp, - j ^ , , 0)
100% long-term stores this means that 35% of the energy in
stores at end of day
= y' = j , •
winter will consist of stored food.
Next, for the lost caches s
The coefficient that makes metabolic cost dependent on
additional caches
= Ax •» j f e _ + j
body mass, k, is set to 0.5, which will increase metabolism 30%
lost stores at end of
« i^m = mm((l — it)
for foraging while carrying one extra gram of fax. Moreno et
aL (1991) calculated metabolic costs in winter of activities (C*,
C& and CQ » 12.0 kj/day in winter), and we use a formula in
For fat reserves, x, we have:
Lucas and Walter (1991) to estimate such costs in autumn.
rA(t, w) i f * < T .
Since day length in autumn is twice as long as in winter we
feeding rate
assume that the daily energy cost is equal in both seasons.
Metabolic cost of perching (Q, •=» 10.0 kj/day) is almost as
if 12 T.
high as for active foraging since the heat production from
amount of fat obtained
activity can be used for maintaining the body temperature.
by foraging
+ Og[r, —
We consider the predation risk to be similar for eating and
™ A'l.m.,, =
amount of fat obtained
storing. The risk is higher in winter (0.005 per day) than in
by retrieving
autumn (0.001), since birds then have less time to scan for
metabolic cost
» Q, = aAQ4 + "BCt
predators (Ekman, 1987). Our baseline values will give a morhx
tality of about 38%, which corresponds to a harsh but not
=
+
fat reserves at end of
T *" w
* ^Xtonf
extreme winter.
day
-t-Ax^^
(
- c.
Finally, predation risk |x is given by:
predation risk
= u. • OAJ1^1
APPENDIX B
Sources for and calculation of baseline parameters.
The time horizon (7) is 200 days and winter ( 7 J sorts day
100. CJi.t), metabolic cost a normal night on day (is 14.0 kj
during the autumn and 29.0 during the winter (Reinertsen
and Haftorn, 1986). Metabolic cost for a cold night, Q(l) (1.2
* CJt)), was calculated from C^((), using a formula in Lucas
and Walter (1991). We consider one night of 10 to be colder
than normal (£„ » 0.90) and one day of 20 unsuitable for
foraging (pj. = 0.95). The proportion of stored seeds forgotten per day (-ft = 0.10) is reasonable considering the memory
decay suggested by Hitchcock and Sherry (1990) and Brodin
We wish to thank Alasdair Houston, Chris Hitchcock, and Jan Ekman
for valuable comments on the manuscript. A 3 . was supported by a
postdoctoral fellowship from the Swedish Natural Science Research
Council.
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