AMER. ZOOL., 27:249-258 (1987)
The Effect of Variability on the Optimal
Size of a Feeding Territory 1
JAMES N. MCNAIR
Department of Biological Sciences, Purdue University,
West Lafayette, Indiana 47907
SYNOPSIS. Previous empirical and theoretical work has focused on how feeding territory
size is governed by average levels of food availability and intrusion pressure; the potentially
important effects of variability have not yet been studied in detail. Here I incorporate
variation in food availability and intrusion pressure in some simple optimality models of
territory size. The results show that the possible effects of variability are diverse, including
both increase and decrease in territory size. And in some cases, variation in food availability
produces qualitatively different effects than variation in intrusion pressure.
INTRODUCTION
Factors governing the size of feeding
territories have been of interest to behavioral ecologists for some time. Based on
numerous field studies, the two principal
determinants appear to be food availability
(Stimson, 1973; Simon, 1975; Carpenter
and MacMillen, 1976; Kodric-Brown and
Brown, 1978; Mares et al, 1982) and intrusion pressure (Myers et al, 1979a, b; Norton et al., 1982). The relative importance
of these factors, however, seems to vary
from one species to another (Myers et al,
1979a, b; Mares et al, 1982).
Studies of territorial behavior have thus
far dealt almost exclusively with the dependence of territory size on prevailing average
levels of food availability and intrusion
pressure. But as Myerses al. (1979a, i)have
stressed, there is often pronounced temporal variation about these average levels.
To date, we know virtually nothing about
the potentially important effects of this
variability on territory size.
Several authors have suggested that (ceteris paribus) territories should be larger in
variable environments than in constant
ones. For example, Myers et al. (1979a, b)
contend that territory size should continually be readjusted in a varying environment. But while decreasing the size of a
territory is easy, increasing its size may be
considerably more difficult, due to resis-
1
From the Symposium on Territoriality: Conceptual
Advances in Field and Theoretical Studies presented at
the Annual Meeting of the American Society of Zoologists, 27-30 December 1984, at Denver, Colorado.
tance from excluded individuals. Myers et
al. therefore argue that residents ought to
maintain territories larger than currently
needed. In this way, territory size will
already be adequate, should food abundance plummet. Mares et al. (1982) observe
that some species maintain nearly constant
territory size, despite pronounced variation in food availability. These authors note
that, in such cases, the territory size should
be large enough to suffice during the worst
periods. As a result, most of the time the
territory will be larger than current conditions require.
The basic idea behind these arguments
is the same. Variation should increase territory size since, by defending a larger territory size than is currently required to
break even energetically, a territory resident protects itself against future energy
shortfalls resulting from decreases in food
availability or increases in energy expenditures. This simple idea is quite reasonable, provided one assumes that a territory
has the smallest size which permits the resident to avoid starvation. In a variable environment, the worst conditions will be worse
than the average, so the minimum safe territory size will be larger than in a constant
environment with the same average conditions.
There is, however, no direct evidence
that animals choose the smallest territory
size which prevents starvation. And several
theoretical studies (e.g., Hixon, 1980;
Schoener,1983; see also the present results)
have shown that, when criteria other than
merely avoiding starvation are taken into
account (factors such as maximizing daily
249
250
JAMES N. MCNAIR
energy intake or minimizing daily foraging
time), the optimal territory size in a constant environment is typically greater than
the smallest required to avoid starvation.
This result is important. It means that even
if the environment is constant, the optimal
territory size will still be large enough to
provide a hedge against starvation in a variable environment with the same average
conditions. It is therefore no longer intuitively clear whether variability would shift
the optimal territory size upward or downward.
What effect, then, does variability have
on the optimal size of a feeding territory?
My purpose here is to take a first step
toward answering this surprisingly thorny
question by developing some simple models
of territory energetics which explicitly
incorporate variation in food availability
and intrusion pressure. These models will
be used to determine, among other things,
whether the optimal territory size is
increased, decreased, or left unaffected by
variation.
dence of these parameters on the physiological condition of the resident, (3) the
risk involved when the resident's physiological condition deteriorates due to an
energy shortfall, (4) the impact of successful intrusions on food availability within
the territory, and (5) time and processing
constraints (sensu Schoener, 1983). In each
case, one has considerable latitude in
choosing assumptions, since there is very
little pertinent evidence and rather different assumptions often seem equally reasonable.
With so many options and so little evidence, it is wise to begin with simple models.
Once a theoretical framework has been
established, it is a relatively straightforward task to flesh it out by modifying specific assumptions. This modus operandi is
especially important in the present case,
since the incorporation of variability results
in a substantial increase in complexity over
the constant-environment models of territoriality.
Here I will take several steps to reduce
complexity. First, I will assume territory
defense is completely effective, so intruTHEORETICAL FRAMEWORK
sions do not reduce food availability. (While
Sources of variability
not the norm in nature, total exclusion of
As I mentioned in the Introduction, intruders has been observed; e.g., see Carempirical evidence suggests there are two penter and MacMillen, 1976.) Assumpprincipal determinants of territory size. I tions regarding the relationships between
will focus on the same two determinants as territory size and the models' rate paramsources of variability. These are: (I) food eters are stated near the end of this section.
availability, where variation occurs in the Other simplifying assumptions will be inditime between food discoveries and in the cated as the models are described.
I now describe three models, each corenergy values of food items, and (2) defense
cost, where variation occurs in the time responding to a different kind of environbetween territory intrusions and in the ment. The first kind lacks variability
energy cost of expelling an intruder. In entirely. In the second, there is random
each case, I will restrict attention to ran- variation in food availability, but all other
dom variation about constant average facets of the environment are constant. In
the third kind, the sole source of variation
levels.
is defense cost. These models will be used
in the next section to determine how variDescription of the models
ability affects the optimal territory size.
In developing optimality models of terConstant environment. Here food encounritoriality, there are several potent sources ter and territory defense are assumed to
of complexity. These include (1) the forms occur at constant rates. Energy from food
of the relationships between territory size enters the resident in a steady stream, while
and various rate parameters (such as the energy spent on defense and other activiaverage food encounter rate and the aver- ties leaves in a steady stream.
age intrusion rate), (2) the forms of depenLet X(t) represent the level of stored
OPTIMAL FEEDING TERRITORY SIZE
energy reserves of the territory resident at
time t. Then
X(t) = X(0) + [XF - (m + s + 7Q]t, (1)
where
X = the rate of encounter with food items,
F = the energy value of a single food item,
m = the rate of energy expenditure on
activities not related to searching for
food or defending the territory,
s = the rate of energy expenditure on
searching for food,
7 = the intrusion rate,
C = the energy cost of expelling a single
intruder.
All these quantities are constant through
time. Several, however, are functions of
territory size (see below).
From (1), the resident's average rate of
net energy intake between times 0 and t is
given by
t"'[X(t) - X(0)] = XF
- (m + s + C)
(2)
This formula will be useful later.
Note that (1) includes the energy benefits and costs of feeding and defense, but
not their time costs. Ignoring time costs is
simply a way to reduce the complexity
which results when variability is added to
the model.
Variable food availability. Here I incorporate variability in the time between food
discoveries and in the energy value of food.
All other processes occur at constant rates,
as above.
Suppose food items are encountered as
a Poisson process with rate X. Let the energy
value of the i-th food item encountered be
F: > 0, having probability density f(») with
mean F. Let N(t) denote the number of
food items encountered between times 0
and t. As is well known, N(t) has a Poisson
distribution with mean Xt (e.g., Karlin,
1975). X(t) is given by
N(t)
X(t) = X(0)
- (m + s +
(3)
with Fo = 0 and other symbols as in (1).
251
The resident's average net rate of energy
intake between times 0 and t obeys a strong
law of large numbers. Specifically, as t -»
oo.
t-[X(t) - X(0)] - XF
- (m + s +
(4)
with probability one (e.g., Ross, 1970). For
sufficiently large t, then, XF — (m + s +
YC) summarizes virtually everything that
is important about the resident's average
net rate of energy intake.
Another result which will be useful later
is the following. Let p(x) denote the probability that the level of stored reserves
eventually reaches 0, given that the level
starts out at x; that is, p(x) = P[X(t) = 0
for some t|X(0) = x]. Then p(x) is determined by the pair of relations,
p(x) = e-<1-p>
p = f[(l -
(5a)
(5b)
(Takacs, 1955), where f(z) is the Laplace
transform of f with transform parameter
z, \p is given by
(6)
m+ s+
and p is the smallest root of (5b). If XF >
m + s + YC, then 0 < p < 1 and 0 <
p(x) < 1; otherwise p = p(x) = 1. This means
there is a chance the reserve level will avoid
ever reaching 0 if, on average, energy
intake exceeds expenditure. Otherwise it
is certain that the reserve level will eventually reach zero—and this includes the
case where intake exactly balances expenditure, on average.
Variable defense cost. I now incorporate,
variability in the time between territory
intrusions and in the cost of expelling an
intruder. All other processes, including
food intake, occur at constant rates.
Suppose that territory intrusions occur
as a Poisson process with rate y, but all
other processes are deterministic. Let the
energetic cost of the i-th defense be C, >
0 with mean C. Let M(t) denote the number of territory intrusions (or defenses)
between times 0 and t. Then M(t) has a
Poisson distribution with mean Yt- X(t) is
given by
252
JAMES N. MCNAIR
be defended as reasonable. To limit the
present labor somewhat, I impose the following plausible restrictions, which hold
for all territory sizes:
A
X(t)
xT
t
t
t
FIG. 1. Behavior of the reserve level X(t) in a constant environment (A) and in two types of variable
environment. With variable food availability (B),
upward jumps correspond to food discoveries. With
variable defense cost (C), downward jumps correspond to intruder expulsions. With either type of variability, both jump size and time between jumps are
random variables.
K, 7a, sa > 0; Xaa < 0;
Taa, saa > 0; ma = Fa = Ca = 0.
The subscripts denote differentiation with
respect to territory area. These restrictions
imply that the food encounter rate increases
at a decreasing rate as territory area
increases. The rates of intrusion and energy
expenditure on search also increase, but at
rates which either remain constant or
increase. The rate of energy expenditure
on activities unrelated to searching for food
X(t) = X(0) + [XF - (m + s)]t
or defending the territory is assumed not
M(t)
to vary with territory size, and the same is
Z
(7) true for the average value of a food item
and the average cost of expelling an
with Co = 0 and other symbols as in (1).
intruder.
The average rate of net energy intake
obeys the strong law of large numbers indi- Objectives of the territory resident
cated in (4). With p(x) defined as above, we
The theoretical framework is now nearly
also have, approximately,
complete. Essentially, I am viewing a terp(x) =
(8) ritory resident as an energy-procuring
machine whose operation can be adjusted
if 4> > C, and p(x) = 1 otherwise {e.g., Bart-only by changing the territory size. An optilett, 1978), where
mal territory size is one which causes this
4> = 7~'(XF — m — s).
(9) machine to run at peak efficiency.
But efficiency can only be denned in
Equation (8) presumes that C has an expo- terms of some objective which the resident
nential distribution, meaning that while is trying to achieve. And in studying how
most defenses are cheap, a few are quite variability affects the optimal territory size,
expensive. (The theory is intractable with I soon found that the results hinge on which
a general cost distribution.) Note that 0 < objective is assumed. Here I will focus on
p(x) < 1 when (8) applies. Thus, there is a three. Under energy rate maximization
chance that the reserve level will avoid ever (ERM), the resident's goal in choosing its
reaching 0 if \f/ > C. Using (9) and rear- territory size is to maximize its average net
ranging, this condition is seen to be equiv- rate of energy intake. Under starvation risk
alent to the analogous condition where food minimization (SRM), the resident's objecavailability varies, and it has the same bio- tive is to minimize the probability that its
stored energy reserves will fall to some
logical interpretation.
Sketches of the behavior of X(t) in each dangerously low level. Cautious energy rate
of the above situations appear in Figure 1. maximization (CERM) is a hybrid between
the previous two objectives. Here the resident chooses its territory size to make its
Dependence of rate parameters on
rate of energy intake large while simultaterritory size
As indicated above, the quantities X, F, neously limiting its risk of experiencing
7, . . . may depend on territory size, and dangerously low reserve levels. (Details are
many different forms of dependence can presented in the next section.)
OPTIMAL FEEDING TERRITORY SIZE
253
10
FIG. 2. Dependence of a* on food availability and
defense cost. Increasing F (A) increases a*; this is
equivalent to increasing X by a constant proportion.
Increasing C (B) decreases a*; this is equivalent to
increasing 7 by a constant proportion.
RESULTS
I now consider the effects of variability
on the optimal territory size for each objective mentioned above. The primary questions of interest are twofold. First, does
variability increase, decrease, or have no
impact on the optimal territory size? And
second, if there is an impact, is it similar
or qualitatively different for the two sources
of variability? To avoid trivialities, I assume
that a territory size exists for which AF >
m + s + 7C; that is, for which energy intake outweighs expenditures, on average.
FIG. 3. The af interval in a constant environment.
Curve 1 represents AF, the average rate of energy
intake. Curve 2 represents m + s + -yC, the average
expenditure rate. The starvation probability is unity
where expenditure exceeds intake and is zero elsewhere. This yields the indicated optimal interval, which
includes both endpoints. The vertical arrow marks
af under the ad hoc minimum principle. This graph
can also be used to locate af (approximately) in a constant environment: it is the area at which the vertical
distance between the intake and expenditure curves
is greatest. Note that this value lies well above the
smallest area permitting the resident to avoid starvation (vertical arrow).
time frames short enough to prevent this
limiting rate from adequately summarizing
the resident's rate of energy intake.
Starvation risk minimization
I will model starvation by equating it to
Energy rate maximization
X(t) reaching zero, and I will restrict attenI will restrict attention to time frames tion to time frames long enough for p(x)
long enough for the limiting rate (4) to to provide a good approximation to the
closely approximate the territory resi- starvation probability. The resident's
dent's rate of energy intake. The resident's objective is to choose its territory area to
objective is thus to maximize XF — (m + minimize p(x).
s + 7C) by suitable choice of territory area.
Constant environment. Here p(x) = 1 for
The optimal area af satisfies
every area a such that XF < m + s + 7C,
and p(x) = 0 for every a such that XF >
= 0.
(10) m + s + 7C. Any area satisfying the latter
KF ~ (s. +
Implicit differentiation in (10) shows a* is inequality therefore qualifies as an optimal
increased by increasing f, decreasing C, territory size af. Typically this yields an
increasing X by a constant proportion (i.e., interval of values (Fig. 3). The endpoints
multiplying X by a constant larger than of this interval converge as X or F is
unity), or decreasing 7 by a constant pro- decreased, or as 7 or C is increased.
portion (see also Fig. 2).
One might argue that factors extraneous
Since (10) applies with or without vari- to the present analysis (e.g., predation risk,
ation, I conclude that variability has no laziness) will favor the smallest area in the
effect on the optimal territory size. Of optimal interval. [This view seems implicit
course, this conclusion merely reflects the in the ideas of Myers et al. (1979a, b) and
convergence indicated in (4), which guar- Mares et al. (1982) discussed in the Introantees that only average levels matter. duction.] Invoking this ad hoc minimum
Other factors may become important over principle, a.* becomes the left endpoint of
254
JAMES N. MCNAIR
M
(8) and is minimized by maximizing <f>, which
is defined by (9). Here a* satisfies
food
A
B
7(XaF - sa) 02
n
10
.2
-
100 10
^
100
p
100
defense
c
D
02
0LS=
10
-
100 10
FIG. 4. Dependence of aj on food availability and
defense cost in a variable environment. A and B: variable food availability. C and D: variable defense cost.
For convenience, the source of variation is indicated
on the figure. Dashed curves show a? in a constant
environment under the ad hoc minimum principle.
Corresponding solid curves apply to a variable environment with the same average rates. Notice that
variability increases a*. Increasing F or C is equivalent
to increasing X or 7 (respectively) by a constant proportion. Note: parameter values in A and B differ
from those in C and D.
7a(XF
- m - s) = 0. (12)
Note that a? is independent of C and of
constant-proportion changes in 7 (though
it does depend on the shape of 7 as a function of a). It is increased by decreasing F
or by decreasing X by a constant proportion, as implicit differentiation of (12) shows
(see also Fig. 4). Here, then, territory size
is governed primarily by food availability
rather than defense cost, opposite the previous case.
Once again, XF > m + s + 7C at af, so
that af is an internal point of the optimal
interval in a constant environment. Under
the ad hoc minimum principle, I again conclude that variability increases the optimal
territory size.
Cautious energy rate maximization
If X(t) = 0 corresponds to starvation,
then the previous objective of ERM is foolhardy in ignoring this risk. On the other
hand, SRM seems overly conservative in
ignoring the potential rate of energy intake.
To amalgamate these two objectives, I
introduce a new reserve level process, X*(t).
the optimal interval. It therefore increases For any realization of the X process where
as X or F decreases, or as 7 or C increases. X(t) > 0 for all t, I define X*(t) = X(t). But
Variablefood availability. Here p(x) is given for any realization where X(t,) = 0 for some
by (5). It is minimized by maximizing \J/, t, (and t, is the first such time), I define
defined by (6). The optimal area a* satisfies X*(t) = X(t) for t < t, and X*(t) = 0 for
t > t,. X(t) is now a hypothetical reserve
Xa(m + s + 7 C) - X(sa + 7aC) = 0. (11) level process which continues even after
Note that a* is independent of F. It is also starvation. It shows how the reserve level
unaffected if X is changed by a constant would behave if starvation did not result
proportion (though it does depend on the when X(t) reached 0. X*(t) represents the
shape of X as a function of a). But implicit resident's actual reserve level, and it is
differentiation of (11) shows af is increased absorbed at zero if starvation occurs.
Under CERM, the resident's objective is
by decreasing C or by decreasing 7 by a
constant proportion (see also Fig. 4). Ter- to maximize the expected limit of
ritory size is therefore primarily governed t-'[X*(t) - X*(0)]. If starvation occurs
by defense cost rather than food availabil- (probability = p(x)), the limit is clearly zero.
If starvation does not occur (probability =
itySince XF > m + s + 7C at af, the optimal 1 - p(x)), the limit is XF - (m + s + 7C)
territory size here lies within the range of with conditional probability one. (This
optimal values in a constant environment. result follows from the strong convergence
Invoking the ad hoc minimum principle, I property of the X process, indicated by (4).)
conclude that af is increased by variability. Thus, the quantity to be maximized by
Variable defense cost. Now p(x) is given by appropriate choice of territory area is
255
OPTIMAL FEEDING TERRITORY SIZE
food
B
A
10
=
100 10
-
100
defense
c
100
D
y
10
-
10010
200
300
X(0)
-
100
i
FIG. 5. Dependence of af on food availability and
defense cost in a variable environment. A and B: variable food availability. C and D: variable defense cost.
For convenience, the source of variation is indicated
on the figure. Dashed curves show af in a constant
environment. Corresponding solid curves apply to a
variable environment with the same average rates.
Notice that variability decreases af. Parameter values
in A and B agree with those in C and D.
FIG. 6. Dependence of af on initial reserve level X(0).
Curve 1: variable food availability. Curve 2: variable
defense cost with the same average rates as in curve
1. The arrow marks af in a constant environment.
Note that at low reserve levels, af is smaller with
variable food availability than with variable defense
cost, while the opposite relationship holds at high
reserve levels. This result reflects a shift in the relative
importance of starvation risk for the two sources of
variability.
able food availability. Referring to the right
side of (13), the first term is maximal at
af, the second at af. The product must be
maximal at a value af obeying a* < af <
af. Since af = af in a constant environment, I conclude that variability decreases
the optimal territory size (see also Fig. 5).
E{lim t_-'[X* (t) - X*(0)]|X*(0) = x}
= [XF - (m + s + 7 Q][1 - p(x)]. (13) But the amount of decrease depends on
the resident's physiological condition, as
reflected
in X(0) (see also Fig. 6). When
Constant environment. Here (13) is identically zero unless the territory area is cho- X(0) is large enough for the risk of starsen within the optimal interval under the vation to be negligible, af is virtually unafstarvation risk minimization objective. For fected by variability. (In fact, af — af as
any area in this interval, p(x) = 0 and (13) X(0) -> oo.) When X(0) is small enough so
reduces to (2). This means that the optimal the risk of starvation is significant, variterritory size af is identical to af in a con- ability results in a substantial reduction in
af. (But af is not the limit of af as X(0) —
stant environment.
Variable food availability. Setting the 0.)
derivative of (13) to zero, implicit differVariable defense cost. Here af has the same
entiation of the resulting equation shows qualitative properties as with variable food
that the optimal territory size a? is increased availability, including the dependence on
by the same factors which increase af (see X(0) (see Figs. 5, 6). The effect of varialso Fig. 5). But an additional factor now ability is once again to decrease af by an
plays a role; namely, the initial reserve level amount which varies with X(0), being negX(0). Decreasing X(0) causes af to decrease ligible for X(0) large enough, but protoward the value which minimizes the risk nounced for X(0) sufficiently small.
of starvation. Increasing X(0) increases
DISCUSSION
af toward the value which maximizes the
potential rate of energy intake.
Although my primary concern has been
How does variability affect af ? It follows with direct effects on territory size, some
from (10) and (11) that af < af with vari- of the most interesting results deal with
256
JAMES N. MCNAIR
TABLE 1. Summary of results.
Effect
on a* of
X(0) variability
Effect on 21* of increasing
Source of variability
X
F
7
C
Energy rate maximization
None
1 T 1 i
Food availability
T T 1 1
Defense cost
I T 1 I
Starvation risk minimization*
None
1 1 T T
0 0 I I
Food availability
Defense cost
I 1 0 0
Cautious energy rate: maximization
None
T j 1 i
Food availability
T T i 1
Defense cost
T T 1 i
0
0
0
0
0
0
0
0
T
T
0
T
T
1
I
* Invoking the ad hoc minimum principle.
other consequences of variability (see summary in Table 1). A particularly surprising
one occurs under SRM. Here variability
can substantially alter the roles which food
intake and defense cost play in determining territory size. Increasing average food
availability (by increasing X or F) decreases
territory size in a constant environment
or with variation in defense cost; it has
no effect with variation in food availability.
Increasing average defense cost (by increasing y or C) increases territory size in
a constant environment, decreases it with
variation in food availability, and has no
effect with variation in defense cost.
A consequence is that territory size is
governed mainly by defense cost when food
availability varies but mainly by food availability when defense cost varies, showing
that the two sources of variability can have
qualitatively different effects. This difference is interesting in connection with the
debate over whether food availability
(Mares et al, 1982) or defense cost (Myers
etal., 1979a, b) is the most important determinant of territory size. To minimize its
risk of starvation, a territory holder evidently should worry more about the predictable features of its environment than
the unpredictable ones. Either food availability or defense cost might therefore be
more important, depending on which is
more predictable.
Another interesting result occurs under
CERM. In a constant environment, the res-
ident's initial level X(0) of stored energy is
not involved in determining territory size.
But with either source of variability, X(0)
becomes a factor: decreasing X(0) decreases
the optimal territory size toward the value
which minimizes the risk of starvation.
CERM is the only objective under which
X(0) was found to have an impact on territory size.
Both of these effects are indirect, in the
sense that they concern ways in which variability modifies the roles of other factors
in determining territory size. But the results
(Table 1) also demonstrate a direct effect,
which can be either to increase or decrease
the optimal territory size. The possibility
of a decrease contrasts sharply with the
simple, intuitive notion that variability
should increase territory size. As I indicated in the Introduction, however, this
idea rests on an implicit assumption that
animals choose the smallest territory size
permitting them to avoid starvation. There
is no good evidence that this assumption
is correct. The notion that variability
increases territory size must therefore be
regarded with suspicion, despite its intuitive appeal.
I do not, of course, mean to suggest that
the other possible consequences of variability are not intuitive. They are. Consider, for example, the decrease in territory size caused by variability under CERM.
Let us accept that the territory size which
minimizes the risk of starvation is less than
the size which maximizes the potential rate
of energy intake; that is, af < a?. (Briefly,
this holds because a small territory reduces
variability in X(t), which is important to do
under SRM but not under ERM.) In a constant environment, the optimal territory
size under CERM (af) is identical to a?,
since there is no risk of starvation (as long
as the net rate of energy intake is nonnegative). If we now add variability, we likewise create a risk of starvation. In response,
the optimal territory size decreases somewhat toward the value (af) which minimizes
this risk, just as one would expect.
Returning to Table 1, it is apparent that,
as in the constant-environment models of
Hixon (1980) and Schoener (1983), the
assumed objective of the territory resident
OPTIMAL FEEDING TERRITORY SIZE
plays an important role in determining
properties of the optimal territory size. This
applies not only to the dependence of territory size on average rate parameters such
as X, F, 7, and C, as in Hixon's and Schoener's work, but also to the effect of variability, which can be either to increase
(SRM), decrease (CERM), or have no
impact on (ERM) the optimal territory size.
The importance of the resident's objective, while interesting, is perhaps a bit
disheartening. We presently have no
acceptable way to directly determine the
appropriate objective. One might resort to
further modeling, dealing directly with fitness rather than energy. In this way, it
might be possible to predict the proper
objective, or to eliminate the need for these.
Yet there seems to be a legitimate question
as to whether such a procedure would
create more problems than it would solve,
given the vast array of poorly understood
factors that would have to be incorporated
in a plausible fitness-based model.
An alternative is to compare the properties of each objective with available
empirical evidence. In this way, the objectives can be evaluated indirectly. Applying
this procedure, SRM appears best supported. Several studies provide evidence
for either a decrease or no change in territory size following increased food availability (Stimson, 1973; Simon, 1975;
Kodric-Brown and Brown, 1978; Myers
et al., 1979a, b; Mares et al, 1982). Both
results are consistent with the properties
of SRM but are contrary to those of the
other two objectives. Several studies also
demonstrate a decrease in territory size
following increased intrusion pressure
(Myers et al., 1979a, b; Norton^ al, 1982).
Since variability was present in each case,
this result, too, is in rough agreement with
the properties of SRM.
As I indicated at the outset, the present
study is merely a first step and can be
extended in many ways. One is to incorporate the time costs of feeding and territory defense. Two questions will be particularly interesting to pursue, once this is
done. First, how do the new properties of
CERM compare with empirical evidence?
Intuitively, I find this objective the most
257
reasonable of the three considered, and it
was a surprise to me that its properties did
not compare with available evidence nearly
as favorably as did the properties of SRM.
The second question is: do the roles played
by food availability and defense cost in
determining the optimal territory size
under SRM continue to hinge on the source
of variability?
It may also be fruitful to consider additional territory objectives, such as those
studied by Schoener (1983) in the absence
of variability. For the models I have dealt
with, one can show that ERM is equivalent
to Schoener's energy maximization and
time minimization (provided we restrict
attention to expected values of daily energy
intake and time to achieve the daily energy
quota). This equivalence disappears when
the time costs of feeding and defense are
incorporated in the present models.
Extending the theory in these and other
ways is likely to further multiply the possible effects of variability. But as the present results suggest, the predicted effects
should be quite testable, particularly in an
experimental context. And nothing would
be more helpful in advancing the theory
in this area than further empirical work of
an experimental nature.
ACKNOWLEDGMENTS
Thanks to Mark Hixon, Jim Quinn, John
Gillespie, and especially Lynn Carpenter
for many helpful comments.
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JAMES N. MCNAIR
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