AMER. ZOOL., 23:303-313 (1983)
A Class of Patch-Use Strategies'
JAMES N. MCNAIR
Department of Biological Sciences, Purdue L'niversih.
West Lafa\ette. Indiana 47907
SYNOPSIS. A class of models for economic, or optimal, foraging in a patchy habitat is
characterized. All models in the class share a number of ecologically important properties,
but there are important differences, as well. These models are well suited to studying
properties of a variety of simple behavioral rules which foragers might use in deciding
when to leave a patch. Several extensions and limitations of the class are also pointed out.
INTRODUCTION
Three important qualitative results from
the empirical patch-foraging literature are:
(1) Foragers usually spend more of their
time in the better patches of their habitat
than in the worse ones (Hassell, 1966, 1970,
1971, 1978: Smith and Dawkins, 1971;
Hassell and May, 1974: Krebs et al., 1974:
Smith and Sweatman, 1974: Murdoch et
al., 1975: Zach and Falls, 1976: Hubbard
and Cook, 1978;Waage, 1979: but see Tinbergen et al., 1967: Croze, 1970). (2) A
greater yield is usually obtained in the better patches (Tinbergen et al., 1967: Croze,
1970: Hassell, 1970, 1978: Krebs et al.,
1974: Smith and Sweatman, 1974: Zach
and Falls, 1976). (3) Animals usually spend
more time foraging in patches when the
time and energy cost of traveling between
patches is higher (Cowie, 1977: Cook and
Cockrell, 1978: Sih, 1980). When one
thinks of foraging as a problem in economics, then each of these results seems almost
inevitable. That this is so suggests powerfully that such thinking may serve as a valuable instrument in the further study of foraging behavior.
My purpose here is to characterize a class
of models for economic, or optimal, foraging in a patchy habitat and to present
some of the important properties of these
models. As I will show, all optimal patchuse models in the class 1 characterize share
the three properties just listed. My emphasis will fall heavily on the models and their
properties. Useful reviews of the empirical
literature can be found in Pyke et al. (1977),
Krebs (1978, 1979), Cowie and Krebs
(1979), Krebs et al. (1981), and McNair
(1982).
DESCRIPTION OF THE MODELS
We can think of these models as consisting of three basic components: the foraging cycle, a "repeating environment,"
and a "repeating forager." I will give a
rough sketch of these now, and then add
a bit more detail on pages 306-307. The
present portrayal should suffice for those
who do not like the appearance of that
section.
The backbone of these models is the foraging cycle (Fig. 1). The forager's time is
classified into two exhaustive categories,
travel and patch residence. During the former, the forager actively searches, or
receptively waits (for example, if patches
are actually prey items), for a patch. No
energy intake occurs here. Patches are
encountered as point events according to
some process which, for generality, we
assume is stochastic. When a patch encounter occurs, a process of energy intake can
begin. After foraging in the patch for some
time, the animal leaves and travels (or waits)
until the next patch is encountered, and so
on.
Patches are classified into types, each type
being distinct by virtue of properties of the
forager's energy intake process there.
Thus, all patches with the same such properties are the same type. The patches might
be at fixed spatial positions with the forager actively searching for them, or they
1
From the Symposium on Optimization of Behavior
might
move (if they are prey items, say), in
presented at the Annual Meeting of the American
which
case the forager might also move or
Society of Zoologists, 27-30 December 198 1, at Dalmight simply wait for them to arrive. In
las, Texas.
303
304
JAMES N. MCNAIR
patch
residence
But before addressing this question, some
simple yet rather general formulas will be
developed.
The foraging process unfolds as a
sequence of cycles. Let Y(k) denote the
energy yield from the k-th patch visit in
this sequence, and let R(k) denote the associated residence time. Let r<k> denote the
travel time during the k-th cycle, and let
0(k) denote the energy expended during
travel. Let £,, denote the ratio of the cumulative energy intake after n cycles to the
cumulative duration of the n cycles. Thus,
FIG. 1. The foraging cycle.
either case, it is assumed that the relative
frequency of each patch type among the
total available remains constant through
time, in a probabilistic sense, resulting in
a so-called repeating environment.
The energy intake process in a patch
might be a deterministic function of time
spent in the patch (for example, think of
an ambush bug sucking the contents from
a captured fly) or might be some stochastic
process (for example, think of a bird
searching the bark of a tree for insects) (see
Fig. 2). For generality, we allow the energy
intake process to be stochastic, and we generally assume some sort of depletion results
from foraging activity in a patch.
A rule is used by the forager to judge
when to leave a patch. Such a rule is called
a strategy. In the models I consider, once
a strategy is chosen, the same rule is applied
to a given patch type every time it is
encountered, irrespective of the history of
the foraging process prior to entering the
patch; different patch types may, however,
be treated differently. Thus, in addition to
the previously mentioned assumption of a
repeating environment, it is equally important that we have a repeating forager.
The strategy applied to a patch is important in determining the forager's residence
time there, as well as in determining the
patch \ield: i.e.. the net energ\ intake in
the patch. It is natural, then, to ask whether
there is some optimal choice of strategy.
Typically £n is a random variable for each
n. However, under rather general conditions, as n -» cx> we find that
_ E(Y) - E(g)
E(R) + E(T)
with probability one (see pp. 306-307). It
is not necessary that the Y(k)'s be independent, nor the R(k)'s, nor the 0(k)'s, nor the
r(k)'s (see the semi-Markov example below).
Note that we are not concerned with E(£n)
but with £n itself, and we certainly do not
claim that E(£n) = £. (See Templeton and
Lawlor, 1981 and responses by Turelli et
al., 1982 and Gilliam et al., 1982.)
In the models I will discuss, £ is used as
the measure of how good a strategy is; the
larger £ is, the better the strategy. I do not
mean to suggest this is necessarily the best
criterion to use in all cases (see pp. 3 1 1 312).
EXAMPLES OF STRATEGIES
I now provide several examples intended
to illustrate and clarify the sort of strategy
I have in mind. There are many types of
such strategies, of which I will mention
but a few. Notice the implicit biological
difference between them. In each case, I
assume there are m patch types, labeled 1,
2
, m.
Residence tune strategy
Each time a patch is encountered, the
forager begins to keep track of the time
305
PATCH-USE STRATEGIES
since patch entry. If the patch is type i, the
forager remains until its residence time
reaches a fixed value R, and then leaves.
The strategy is specified once the threshold residence times R,, R2, • • • , Rm are
chosen. Note that for any patch type i, the
residence time is a deterministic quantity
while patch yield commonly will be stochastic. Papers which consider residence
time strategies include Krebs et al. (1974),
Charnov (1976), Charnov et al. (1976),
Cowie (1977), Sih (1980), and McNair
(1982).
Gwing-up time strategy
Each time a patch is encountered, the
forager first keeps track of the time since
patch entry. If the patch is type i and no
food encounter occurs before time t,
elapses, the forager leaves; if an encounter
occurs during this time, the forager
remains. It forgets when it entered the
patch and now keeps track of the time since
the first food encounter, leaving if another
encounter does not occur before time t,
elapses; if an encounter occurs during this
time, the forager remains. And so on. The
threshold time t, is called the giving-up time
in patch type i. The strategy is specified
once the threshold giving-up times t,, t2,
. . . , tm are chosen. Note that the givingup time is a deterministic quantity for each
patch type, but the residence time and yield
usually will be stochastic. Papers which
consider giving-up time strategies include
Breck (unpublished), Iwasa et al. (1981),
and McNair (1982).
Yield strategy
Each time a patch of type i is encountered, the forager keeps track of its cumulative energy intake and stays until a yield
of Y, is achieved, at which time it leaves.
The strategy is specified when the threshold yields Y,, Y2,. . . , Ym are picked. These
are deterministic quantities, but the residence time in any given patch commonly
will be stochastic.
Instantaneous rate strategy
Suppose patch yield Y, is a differentiate
deterministic function of residence time R,
for every patch type i. Then each time a
patch of type i is encountered, the forager
X(t)
B
t
FIG. 2. The energy intake process within a patch.
In example A the process [X(t)] is smooth and deterministic; in B it is stochastic, a portion of one realization being shown. In both examples, t denotes time
since patch entry and X(t) denotes net energy intake
up to t. The forager will eventually leave a patch, at
which time t = R and X(t) = X(R) = Y
keeps track of its instantaneous rate of
energy intake (dY,/dR,) and leaves when it
falls to a rate of r,. The strategy is specified
once the threshold rates r,, r2, . . . , rm are
chosen. Note that patch residence time and
yield are deterministic, in addition to the
r,'s.
These examples illustrate that in order
to specify a strategy, we must state the type
of strategy as well as pick the associated
thresholds. In general, a forager might
apply different types of strategies to different patch types. However, every time a
given patch type is encountered, the strategy type and the thresholds must be chosen
in the same way.
If the forager can identify patch types—
306
JAMES X. MCXAIR
and we usually assume identification is
accurate and instantaneous upon patch
encounter, though both assumptions can
be relaxed—then each patch type can be
treated differently; that is, a different
threshold can be chosen for each. However, if patches are not identifiable, then
all patches are treated alike. This means
either the same threshold is used in each,
or possibly that with each patch encounter,
a threshold is picked from one and the same
probability distribution. Note that the case
of unidentifiable patches is essentially a
constrained version of the identifiable case.
In all four of the above examples, the
thresholds involve features of the energy
intake process in a patch which the forager
can experience directly and monitor. This
is an important feature. It means that such
thresholds can serve as simple behavioral
rules for a forager. But note, for example,
that an instantaneous rate strategy could
not be used in a patch where the energy
intake process is a point process, as in Figure 2B. Likewise, a giving-up time strategy
could not be used in a patch where yield
is a differentiable deterministic function of
residence time, as in Figure 2A. In each
case, the problem is that the energy intake
process in a patch lacks the very property
which the forager must monitor. Thus, the
nature of the patches in a habitat may rule
out certain types of strategies.
DF.i AILS
Having given a rough sketch of the class
of models I am concerned with, I now give
a more precise characterization. The
reader who is not interested in such things
can simply skip this section. Some of the
assumptions I will state can probably be
relaxed and still retain the properties presented on pages 309 and 310.
First I make several assumptions concerning interpatch travel. Energy is
expended during travel, but none is taken
in. The duration of travel time may be stochastic. It may be influenced by the type
of the last patch visited ("patch type" is
defined below) but must not be influenced
directly b\ am other patch \isit nor b\ the
outcome of foraging within the last patch
visited or anv other. We assume the mean
travel time and mean energy expended
during travel are finite.
Now for some assumptions concerning
patch residence. As soon as a patch is
encountered, a process of energy intake
and expenditure begins. Let {X,(t): t ^ 0}
denote this process, where t represents time
elapsed since entering the patch and -co <
X,(t) < co represents the cumulative net
energy intake up to t (Fig. 2). We allow this
process to differ between patches, and all
patches with the same process are said to
be the same type. The subscript i denotes
the patch type, and we assume there are
only finitely many types. (This assumption
can be relaxed, though it seems pointless
to do so.) We assume {X,(t)} is independent
of the forager's history before entering
patch type i. Ordinarily the forager will
eventually leave each patch, but it is convenient to define {X,(t)} for all t > 0.
The forager is assumed to monitor only
a small portion of the {X,(t)} process. This
monitored facet constitutes a secondary
process {Z,(t)} which is generated by the
primary process {X,(t)} (Fig. 3). All foragers monitoring the same information (i.e.,
having the same {Z,(t)} process) are said to
use the same type of strategy. We restrict
attention to types of strategies for which
the secondary process is one-dimensional.
As an example, for a residence time strategy, Z,(t) is simply the time since patch
entry; that is, Z,(t) = t. Other examples
appear in Figure 3. A threshold is a fixed
attainable value of the secondary process
<Z,(t)[ having the property that the forager
leaves the patch exactly when -| Z,,(t) ^
achieves this value for the first time (Fig.
3). We require the same threshold to be
applied to a given patch t)pe every time it
is encountered, and the within-patch foraging processes and thresholds are assumed
to be such that the mean patch yield and
mean residence time are finite.
The final assumption I will mention here
involves all facets of the foraging process.
For each patch type i, the process of successive visits to i constitutes a renewal process, and the mean time between successive
visits to i is finite. Note that different patch
types ordinariK will have different associated renewal processes.
307
PATCH-USE STRATEGIES
Let £n and £ be as denned on page 304.
Under the assumptions listed in this section, £n — £ with probability one (Strong
Law of Large Numbers for Renewal
Reward Processes: see Ross, 1970, pp. 5 1 54).
As an example, suppose that foraging
constitutes a semi-Markov process, where
we say the process is in state i if the last
patch entered was type i. The sequence of
patch types encountered is a Markov chain
with transition probabilities {P,,} and stationary distribution {TT,}. Then
2)
2
2
where E(Y,) and E(R,) are the mean yield
and residence time in a patch of type i and
E(0,j) and E(T,J) are the mean energy and
time expenditures during travel from a
patch of type i to one of type j . The ir,'s
are determined by the Py's according to
the usual relation,
ir, = J) 7T.P.,,
Z(t)
Z(t)
i = 1, 2, . . . , m.
OPTIMAL THRESHOLDS
In seeking a strategy which maximizes £,
there are two tasks. One of these is locating
the type of strategy, and the other is choosing the proper thresholds. I will ignore the
first task and concentrate solely on the second. Thus, I look for properties of an optimal giving-up time strategy, or an optimal
residence time strategy, and so on. This
procedure is less ambitious than attempting to locate the best of all types of strategies, but it is better suited for empirical
application. The reason is that, in studying
the foraging behavior of a specific animal,
one ought to find out what behavioral rule
is actually being used. The rule being used
tells one what type of strategy to incorporate in the foraging model whose predictions are to be tested. Since the type of
rule an animal uses probably depends
heavily on the animal's physical and physiological capabilities, which in turn depend
heavily on which species the animal is, the
practical value of asking what type of rule
the animal should use is limited. Given the
c
A
Z(t)
A
t
FIG. 3. The secondary process in a patch. Three
examples of the monitored secondary process (Z(t)}
are shown. In A the forager uses a residence time
strategy and Z(t) = t. In B an instantaneous rate strategy is used and Z(t) = dX(t)/dt. C illustrates a givingup time strategy, where Z(t) is the time since the last
food encounter or since patch entry, whichever is
smaller. For each example, the dashed horizontal line
represents a threshold. As soon as Z crosses it, t = R
and the forager leaves the patch (arrows).
308
JAMES N. MCNAIR
\
\
A
1
2
V
\
->
\
\
independent of x, (i =t j), and 8 and f are
independent of every x r We write
but note that £ depends on other quantities, as well; notably 8 and f.
What choice of x,, x2) . . . , xm maximizes
ij? The most interesting case is that where
each optimal threshold x.-t is positive and
finite, and I will restrict myself to it. For
convenience, suppose f is differentiable, and
define functions Q,(x,) by
^
l(lXj
dE(Y,)/dXi
dE(R,)/dx/
We assume Q, > 0. Q, measures the rate
at which mean patch yield (the benefit)
changes as mean residence time (the cost)
does, both changes occurring as a result of
varying x,. We might call Q; the marginal
value of patch type i, by analogy with Charnov(1976).
Some examples may help. For a residence time strategy, we have x, = R,,
dE(R,)/dx, = l,andQ,(x,) = dE(Y,)/dx,. For
an instantaneous rate strategy, x, = r, and
Q.0O = r,.
We now make the crucial assumption that
each patch type obeys a law of diminishing
FIG. 4. Some properties of optimal strategies. In both
A and B cases appear where mean patch yield and
returns; that is, the rate at which the benresidence time increase as the threshold x increases, efit accrues with repeated increments in
and where patch type 1 is of uniformly better quality
cost decreases. This means that Q, decreases
than type 2. In A the solid horizontal line indicates
{.„. The values of x at which Q, and Q2 cross this as E(R,) increases, but it should be recalled
that the increase in E(R,) is due, in turn, to
line are x, and x2. Note that x, > x2. The arrow shows
the effect on £„,„ of increasing f or B. Note that x,
a change in the threshold, x,. Now for some
and x2 increase. In B the open circles on the horizontal strategies, increasing x, causes E(R,) to
axis represent x, (right) and x2 (left). Note that with
optimal thresholds, E(Y,) > E(Y2) and E(R,) > E(R2) increase, while for other strategies E(R;)
(arrows). Note also that as x, and x2 increase as in A, will decrease. In the former case, we require
so do the mean yields and residence times.
Q, to decrease as x, increases (Fig. 4); in
the latter, Q, must increase as x, increases.
For example, dE(R,)/dx, > 0 for a resitype of rule in use, however, it is still useful dence time strategy, so we require dE(Y,)/
to ask how it should be used. Indeed, most dR, (=Q,) to decrease as x, (=R,) increases.
predictions from the models are based on When Y, is a differentiable deterministic
answers to this question. In our terms, given function of R,, diminishing returns
the type of strategy, we ask how the thresh- obviously means that dY,/dR, decreases as
R, increases. For an instantaneous rate
olds should be chosen.
Now, for some fixed type of strategy, let strategy, then, R, increases as x, (=r,)
the thresholds be x,, x2, . . . , xm where x, decreases, and since Q, = r, here, Q,
is the threshold applied in patch type i. The increases as x, increases.
Under our assumptions, there is a unique
mean \ield FCV ) and mean residence time
E(R,) in patch type i are functions ot x, but set of optimal thresholds x,, . . . , xn, at
PATCH-USE STRATEGIES
which £ achieves its maximum, and x; satisfies
Q,(-\) = *„,„
for every patch type i. Thus, the thresholds
are adjusted so the marginal value of every
patch is the same and equal to the maximum habitat rate £max. This result is a generalization of the marginal value theorem
of Charnov (1976).
UNIFORMLY BETTER QUALITY
309
4). As a result, the mean patch yield E(Y,)
and residence time E(R,) under the optimal
threshold both increase.
Next are some properties which rely on
the definition of uniformly better quality.
Let patch type i be of uniformly better
quality than typej. If the mean yield and
residence time in each type increase
(decrease) as the threshold is increased,
then x, > Xj (x, < £,). Only part (i) of the
definition of uniformly better quality is
needed here. From parts (ii) and (iii), it
follows that E(Y.) > E(Y,) and E(R,) > E(R,)
(Fig. 4).
These shared properties make a great
deal of sense. Basically, they say that a forager should stay longer in patches when it
costs more to leave, and that more effort
should be spent on the better patches of a
habitat than on the poorer ones. These
properties are also important since they
provide ways to test all strategies in this
class simultaneously in an empirical situation.
An important general question in optimal foraging theory is: Should a forager
treat the better quality patches of a habitat
in a different way than the poorer ones?
But what do we mean when we say one
patch is of better quality than another?
Quality cannot be an intrinsic property of
a patch, because the properties of the
energy intake process depend not only on
the patch but on the strategy type and
thresholds, as well.
Rather than define patch quality itself, I
have found it useful to define the relation,
uniformly better quality. Patch type i is of Differences
uniformly better quality than typej if and only The strategies I consider differ in
if all three of the following conditions hold numerous ways, not the least of which is
for all choices of x, and Xj such that x, = Xj: the behavioral rules used. The differences
(i) Q, > Q,, (ii) E(Y.) > E(YJ), and (iii) are important since they may result in difE(R,) > E(Rj). Thus, if each pair of func- ferent values of £max. They also permit
tions is graphed against the common empirical tests to distinguish between types
threshold x, the curves for type i never fall of strategies. I now illustrate these differbelow those for j , and Q, must actually lie ences with some examples.
above Q, (Fig. 4).
First is an example where two different
This relation is a fairly restrictive one, types of strategies turn out to be matheand not all parts of its definition are nec- matically equivalent but biologically very
essary for every one of the properties which different. Let the patch types be 1, 2, . . . ,
I mention in the next section. However, in m and let each patch yield Y, be a differorder for all the properties to hold for all entiable deterministic function of resistrategies of the class considered here, all dence time R,. Further, let dY,/dR, be posthree parts are needed.
itive at R, = 0 and steadily decrease as R,
increases, approaching zero asymptotiPROPERTIES OF OPTIMAL STRATEGIES
cally. For a residence time strategy, Q, =
Similarities
dY,/dR,. Let patch type i be of uniformly
Optimal strategies of the sort I am con- better quality than type i + 1 for i = 1,2,
sidering share several important proper- . . . , m — 1. Then we know that R, > R2 >
ties. In mentioning some of these, I will . . . > Rm. Now consider the same habitat,
but apply an instantaneous rate strategy.
assume patches are identifiable.
First, for each patch type i, the optimal Then the threshold x, is r, (=dY i /dR [ ) and
threshold x, increases (decreases) when f Qi = r,. This means that for every patch
or 6 is increased, provided that dE(Y,)/dx, type, the graph of Q, against x, is a straight
and dE(R,)/dx, are positive (negative) (Fig. line of slope one which passes through the
310
JAMES N. MCXAIR
T A B L E 1.
Two comparisons of optimal gn'tng-up tune and
residence time strategies.
Optimal residence
time strateg)
Optimal giwng-iip
time slrateg\
Ex- .
ample
{„.
1
2
1.302
1.019
E(V)
F.(R)
3.204 2.061
3.422 2.958
U.
F(Y)
F.(R)
1.199
1.040
2.315
2.092
1.614
1.612
origin. Therefore f, = f2 = . . . = fm. But
note that in this example, R, under a residence time strategy must be the same as
the residence time when r, = f, under an
instantaneous rate strategy, and | max must
be the same in both cases. From a mathematical point of view, then, it makes no
difference which of the two types of strategies is used. There is, however, a biological difference, and it goes well beyond the
implicit difference in types of behavioral
rules used. For a residence time strategy,
patches must be identified and a different
threshold applied to each type. If patches
cannot be identified, the same threshold is
used in every patch, and a lower £max results.
In contrast, for an instantaneous rate strategy, the same threshold is applied to each
patch whether or not it can be identified,
and the same value of £,„,„ is obtained.
Ordinarily there will be neither mathematical nor biological equivalence between
types of strategies, and I now give two
examples where this is so. In each example
I assume there is only one patch type.
Within a patch, food items are encountered as a renewal process. The only difference between the two examples is in the
properties of this process. Details can be
found in McNair (1982). Three properties
[£max> F-(Y), and F.(R)] of the optimal givingup time strategy and optimal residence time
strategy are compared in Table 1. Note
that all quantities compared differ between
strategies. In particular, note that £mav is
larger for a giving-up time strategy in the
first example but larger for a residence time
strategy in the second.
destroying the simple character of the
models. For example, we might incorporate mandatory patch identification time
or allow for errors in identification
(Hughes, 1979), or we might allow the
environment to fluctuate temporally in
certain ways. I now give an example to
illustrate how more general notions of a
patch visit can be incorporated in our
models.
When patches are prey items, it is often
reasonable to allow additional patches to
be encountered while one is being fed upon:
that is, to allow overlapping encounters. As
a simple model of this situation, let there
be but one patch type, and let the yield be
a deterministic function of residence time.
Suppose a forager encounters patches as a
Poisson process with intensity X when it is
not feeding on one: when it is, let the
encounter process have a reduced intensity
X' < X. Suppose energy is spent at a constant rate c during travel time. For simplicity, let the same yield be extracted from
each patch, and let the same residence time
be spent. Thus, if there are n overlapping
encounters, the total yield is nY and the
total residence time is nR. This means that
for any specified R the total yield and residence time in the cluster of overlapping
encounters (which I will call an extended
encounter) are determined by the random
total number of overlapping encounters,
which is equivalent to the total number of
arrivals in the busy period of a Takacs process (Takacs, 1955). The mean number
E(N) of overlapping encounters is 1/(1 —
X'R), where we require X'R < 1 to ensure
that E(\) < co. The mean total yield F(Y*)
and residence time E(R*) are YE(\) and
RE(N) respectively.
The random total yield and residence
time in an extended encounter now play
the role of the single yield and residence
time in the previous models. With probability one,
In
F.x i FVSIONS
The class I ha\e described can be
extended in a \ariet\ of wa\s without
E(Y*) - cf Y/(l - X'R) - c/X
E(R*) + f ~ R/(l - X'R) + 1/X
where r = 1 /X. Let the forager use a residence time strategy in each individual
311
PATCH-USE STRATEGIES
patch. Applying the theory developed earlier, we find that
dE(Y*)/dR
Q(R) = dE(R*)/dR
The optimal threshold R satisfies Q(R) =
£max. Rearranging,
Thus, when encounters can overlap (A' >
0), the forager remains in a patch beyond
the point at which its instantaneous rate of
intake equals the maximum habitat rate:
when encounters cannot overlap (X' = 0),
the forager leaves when the two rates are
equal. But when encounters can overlap,
the individual patch does not play the role
of the individual patch in previous sections.
Instead, this role is played by the cluster
of patches in an extended encounter, and
the theory handles this generalized patch
in the same way it treated the individual
patch previously.
LIMITATIONS
I will end on a cautionary note. There
are many alternative ways to view optimal
foraging problems, and each has limitations. The present case is no exception, and
I will mention several of these limitations
now.
We have looked at an asymptotic rate of
energy intake, though we realize that a forager will visit only a finite number of
patches. Our hope is that the asymptotic
rate provides a good approximation to the
actual rate for relatively small numbers of
patch visits. A question of interest, then,
is: How small can the number of visits be
while still allowing the asymptotic rate to
provide a good summary of the process?
The answer depends on the amount of
variability present in each component of
the process (patch yield, residence time,
and travel time), as well as on what we mean
by "good."
As an example, consider the case with
only one patch type, where patch yield is
stochastic under a residence time strategy
Suppose travel time is a deterministic
quantity, always being exactly T. The only
source of variability, then, is patch yield.
We now ask how small the number n of
patch visits can be while still ensuring that
the ratio £„/£ will be close to one with high
probability. As a conservative approach, we
first find that
I"1
J
by Chebychev's inequality, where o/\i is
the coefficient of variation of patch yield
Y. Next we require the bound on the right
to be no larger than a (0 < a < 1). We
find that n must obey
n >
Thus, if we ask that £ n / | be within 0.1 (=8)
of one with probability at least 0.8 (=1 —
a), then a number of patch visits of at least
500 X (a/n)2 suffices. If patch yield has a
coefficient of variation of about 0.3 (based
on data of Cook and Cockrell, 1978), then
50 patch visits should suffice: if 0.1, then
only 5. In fact, since the bound obtained
here is rather loose, a smaller number
might suffice. But what if additional sources
of variation were allowed? We then would
probably require more patch visits.
Even if we judge the utility of a strategy
by an asymptotic rate of energy intake, we
need not restrict attention to the class of
strategies I have described. For example,
a forager might use several different
thresholds in a single patch. Oaten (1977)
and Green (unpublished) have examined
strategies of this more general type.
In some cases it might be better to use
a criterion other than an asymptotic rate.
For example, maybe one should try to maximize the probability of obtaining at least
some critical amount of energy in a small
number of patch visits. The solution to this
problem usually will differ from that discussed in previous sections.
Finally, there is a long list of biological
factors we have ignored, and these could
be important in particular applications. For
312
JAMES N. MCNAIR
example, we have not considered how a
forager acquires knowledge of the state of
its environment. As numerous authors have
suggested, the forager might be wise to
allot some of its time to sampling its habitat
(Gibb, 1962; Royama, 1970«, b: Tullock,
1971;KrebsetaL, 1978; Cowie and Krebs,
1979). And what about forager risks, such
as starvation or predation? What about the
need to maintain a balanced diet? What
about the need to schedule competing nonforaging activities?
Because of the limitations of the
approach examined here, and of each other
alternative, one must be careful in generalizing from results obtained by any one
approach.
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