AMER. ZOOL., 27:259-291 (1987)
Time Budgets and Territory Size: Some Simultaneous
Optimization Models for Energy Maximizers1
THOMAS W. SCHOENER
Department of Zoology, University of California,
Davis, California 95616
SYNOPSIS. A set of optimization models in two variables of choice, territory size and time
spent patrolling for intruders, is presented for energy maximizers. Models vary in the
curvilinearity of the relationship between territory circumference and both intrusion rate
and cost of expelling a single intruder. Models are analyzed both with and without constraints; constraints are on processing rate and on the time spent patrolling, feeding and
actively defending. The models all include the concept of "intruder equilibrium," an
equilibrial density of intruders in a territory resulting from a balance between intrusion
rate and expulsion by the defender. This equilibrial density can be considered a measure
of territorial exclusiveness.
The two-variable models predict effects on territory size and patrol time of variation
in food density, intrusion rate, costs of expelling a single intruder in energy and time,
food-consumption rate of an intruder, area of detection while patrolling, total time available for territorial and feeding activities, time to eat a unit of food energy, energy cost
of patrol per time, and processing-rate capacity. With increasing intruder rate, optimal
territory size usually decreases, whereas optimal patrol time behaves much more irregularly. With increasing food density, optimal patrol time usually decreases, whereas optimal
territory size behaves irregularly. In particular, when intrusion rate and expulsion costs
accelerate sufficiently with increasing territory size and no constraints exist, the higher
the food density the smaller the optimal territory size. When food density is large enough
for a constraint to be effective, the opposite relation can hold and will always hold for a
processing constraint.
When a particular parameter changes, optimal territory size and optimal patrol time
may covary or one may increase while the other decreases, depending on the parameter
and model.
A new set of one-variable models is suggested by the two-variable models; models
optimizing patrol time while holding territory size constant could correspond to a tightly
packed system of territories initially determined by settlement patterns. A unified onevariable analysis suggests that how food density affects territory size when patrol time is
constant depends upon whether a constraint is operating: Provided that invasion rate does
not vary with density of intruders on the territory, time minimizers and constrained energy
maximizers decrease territory size with increasing food density; unconstrained energy
maximizers do the opposite.
The addition of a second optimization variable to a one-variable model can change
qualitative predictions about variation in particular parameters (e.g., food density) and can
increase the number of parameters predicted to affect optimal territory size and patrol
time.
INTRODUCTION
The past few years have seen the development of a large number of "simple"
models of optimal territory size (Davies,
1978; Kodric-Brown and Brown, 1978;
Dill, 1978; MacLean and Seastedt, 1979;
Tullock, 1979; Ebersole, 1980; Hixon,
1980; Schoener and Schoener, 1980; Myers
et al., 1981; Wittenberger, 1981; Schoe-
ner, 1983). Because they allow only one
decision variable, territory size, such
models have necessarily made rather
restricted assumptions about time budgets
of territorial defenders. The various times
devoted to particular activities have been
considered constant, or considered strictly
a
by-product of the machinery optimizing
territory size, or not considered at all.
However, a one-variable model that
directly optimizes time expended in some
territorial activity, e.g., patrolling for
• From the Symposium or, Territoriality: Conceptual
Advances in Field and Theoretical Studies presented at
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the Annual Meeting of the American Society of Zool- that optimizes territory Size. In particular,
ogists, 27-30 December 1984, at Denver, Colorado, as shown below, the assumption that all
259
260
THOMAS W. SCHOENER
intruders are immediately expelled, which
is made in most published one-variable
models of territory size, need not be made
in the former type of model.
Moreover, a several-variable model
optimizing territory size simultaneously
with optimizing time allocation may provide considerably more biological realism
than either one-variable model by itself.
Such models provide a richness of predictions not possible with the one-variable
models. For example, predictions about
how many intruders are on the territory
in a steady-state situation are perforce
impossible when all intruders are assumed
immediately expelled. Or, if intruders are
always expelled, so that defense time has
priority over feeding time, predictions
about the effect of simultaneous defense
and feeding are again not possible.
Somewhat more negatively, results from
the two-variable models do not always
accord with those from one-variable
models. For example, when time devoted
to monitoring the territory for intruders
is a variable in addition to territory area,
even qualitative predictions from the onevariable models can be overturned. Empiricists may bemoan such inconsistencies
between models and may even view the
present more complicated modelling effort
as sadistic. However, a theoretician's task
includes examining the sensitivity of models
to variation in assumptions toward which
variation an empiricist is indifferent, i.e.,
lacks data on. If such variation produces
highly variable predictions, that is not the
fault of the theoretician, who merely reveals
what logic implies. Of course, testing theory will then be more laborious than hoped
from initial modelling efforts; more precisely specified data (i.e., numbers) and
more kinds of data will typically be necessary. How to test predictions from complex models of territory size is discussed
further below and by Hixon (1987).
This paper takes some initial steps toward
incorporating time allocation into models
of territorial defense. It briefly investigates
one-variable models in which time
expended in some territorial activity, rather
than territory area, is optimized. The bulk
of the paper, however, is devoted to simul-
taneous optimization of time and area variables.
The approach distinguishes three types
of activities: patrolling, feeding and
defending. Defenders are assumed to patrol
the territory for intruders; only when
intruders are encountered during patrol
time can they be expelled. Patrol time can
be simultaneous or not with other activities, but it is always assumed to incur a fixed
energy cost per-unit-time. (A second group
of two-variable models in which intruders
can be detected at any time [for example
because of the smallness and openness of
the territory] is also briefly discussed.)
Feeding includes both the search for and
consumption of food; energy costs of feeding are assumed constant per-unit-foodenergy gained. Defending is restricted to
the expulsion of intruders once they are
encountered.
Of the three activity times, patrol time
is considered a variable to be optimized
directly, i.e., a decision variable. Feeding
and defense time are not directly optimized but are considered by-product variables; they are calculated from formulae
containing certain parameters (which are
constant, e.g., energy cost per patrol time)
and the optimal values of the decision variables (patrol time and territory area).
The territorial defender is assumed to
be an energy maximizer (Schoener, 1969,
1971), an animal whose goal it is to maximize the energy obtained on the territory
over some period of time. Energy maximizers may be quite common in nature;
Belovsky (1986) has shown that many
mammalian species are apparently in this
category. The energy available to the
defender is assumed equal to the potential
food energy in the territory minus that
energy consumed by intruders. The
defender's ability to gain energy, however,
is constrained in the models by how much
energy can be processed and by the time
available for the three activities: patrolling,
feeding and defending.
All constraints on the defender can be
assumed acting simultaneously if appropriate. Explicit representation of time and
energy constraints is absent from many
previous models of optimal territory size
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
and greatly adds to the present models'
biological realism. An innovative feature
of the present treatment is the distinction
of activities that are simultaneous and those
that are not, an issue of increasing interest
(e.g., Ydenberg and Krebs, 1987). Whether
or not activities are simultaneous has key
implications for optimal territory size and
optimal patrol time, as will be shown.
The tradeoffs involved in optimizing territory size and patrol time can be visualized
intuitively as follows. The larger the territory, the more food available, but the
more intruders that will invade per unit
time, and generally, the more costly an
individual act of expulsion. The larger the
time patrolling for intruders, the greater
the number of intruders expelled, but the
more energy lost by the defender because
of the energetic cost of patrol, and the less
time available for other activities (feeding
and active defense). The models to follow
show how an optimal balance between these
considerations can be obtained.
The following questions will be addressed:
1. What is the effect on optimal territory size and optimal patrol time of changes
in the following parameters: food density,
intrusion rate, time cost of expelling an
intruder (defense time per intrusion),
energy cost of expelling an intruder, energy
cost of patrolling per unit time, area surrounding the defender within which
intruders can be detected while patrolling,
total time available, time to eat a unit of
food energy, food-processing rate of the
defender, and amount of food eaten by an
intruder?
2. What is the effect, on optimal territory size and optimal patrol time, of the
manner in which intrusion rate and expulsion cost vary with territory area?
3. Is there some simple relationship
between optimal territory size and optimal
patrol time as parameters change?
4. How does the assumption, that only
intruders encountered during patrolling
can be expelled, affect predictions from
one-variable models of optimal territory
size?
5. Do one- and two-variable models differ fundamentally in their predictions about
261
the effects of various parameters, such as
food density, on optimal territory size?
INTRUDER EQUILIBRIUM
"Intruder equilibrium" refers to that
steady-state number or density of intruders
that will eventually exist in a territory for
a fixed territory area, c2, and patrol time,
t. (All symbols are summarized in Appendix 7.) The concept (Schoener, 1971) is
essential for modelling systems in which all
intruders cannot be immediately expelled.
Such systems involve territories that are so
large a n d / o r densely vegetated that
intruders cannot be immediately detected.
For example, territories of most mammals
(e.g., chipmunks, Tamias striatus [Getty,
1981; Mares and Lacher, 1987]), many
birds, especially in forested habitats (e.g.,
Davies and Houston, 1981; Ydenberg and
Krebs, 1987), and many lizards (Stamps,
1977; Schoener and Schoener, 1983) fall
into this category, as do cases whose detection is difficult for sensory reasons, e.g., owl
limpets Lottia gigantea (Stimson, 1970).
Organisms that do not fall into this category are hummingbirds and certain fishes
(e.g., Ebersole, 1980; Hixon 1980; Hixon
etal., 1983; Carpenter, 1987).
The simplest form of intruder equilibrium, used in the models below, is given in
Figure 1. Two curves, intrusion rate and
expulsion rate, are plotted against density
of intruders in a territory.
Intrusion rate is defined as the number
of intruders invading a territory during
some large time period, T (see below). The
simplest assumption on intrusion rate is that
it is unrelated to intruder density. Hence,
the relationship of intrusion rate to
intruder density is linear and parallel to
the abscissa; it is given by qc", where q is
an "intrusion intensity" factor to be clarified in the next section. Realistic values of
x are 1 and 2, as when intrusion is proportional to territory circumference (invasion occurs laterally; see Hixon, 1980) or
to territory area (invasion occurs from
above), respectively.
Expulsion rate is defined as the number
of intruders expelled from a territory during some large time period. It is most simply assumed increasing in direct propor-
262
THOMAS W. SCHOENER
JtN/c 2
Density of Intruders
(N/c 2 )
Equilibrium
N*/c2
FIG. 1. Intruder equilibrium. In this simple version,
the intrusion rate is unrelated to density of intruders
on the territory, whereas the expulsion rate is proportional to intruder density. Where the two rates
are equal gives the equilibrial intruder density.
tion to intruder density on the territory,
N/c 2 ; this corresponds to random encounter. The proportionality factor for this
relationship must also include J, the area
within which an intruder can be detected
per unit patrol time, and t, the patrol time.
Intuitively, the number of intruders
encountered per-unit-time is proportional
to the intruder density, the speed at which
the defender moves through the territory,
and the defender's perceptual abilities. In
this main model, all intruders encountered
are assumed expelled once found; any individual intruder, of course, can reinvade.
(The assumption that all encountered
intruders are expelled is relaxed in a subsequent section.) As a technical note, the
total time during which intruders can
invade is incorporated into q; this time
might be equal to the entire activity period
of the defender, and it would ordinarily be
greater than t. But this time could conceivably be smaller than the defender's
activity period and even smaller than t.
Further, q may incorporate per-unit-area
differences in intrusion rate as a function
of territory shape (e.g., Stamps etal., 1987).
T h e intrusion and expulsion curves
intersect at some value of intruder density,
called the equilibrial intruder density, N * /
c2. For the model of Figure 1,
N*/c 2 = qcVJt.
(1)
N* is then the number of intruders on the
territory at equilibrium. It is considered a
long-term temporal average. Even though
all encountered intruders are expelled,
N*/c 2 can be very high if encounter rate
with intruders is very low or if intrusion
rate is very high. N*/c 2 may also equal
some fraction of an intruder, i.e., territories may be intruder-free most of the time
and still have positive N*/c 2 . Notice that
N*/c 2 is a determinant of the fraction of
food appropriated by intruders (see below).
Moreover, the higher the intruder density,
the greater the proportion of individuals
on the territory that are intruders. In this
sense, N*/c 2 is a measure of the exclusivity
of a territory from the owner's perspective.
Given a choice of territory size and patrol
time, exclusivity is automatically specified,
so that predictions about the former two
variables are also predictions about exclusivity.
The forms assumed for intrusion and
expulsion curves (Fig. 1) are the simplest
ones possible; even so, model output is very
complicated, as will be shown. These simple forms imply fairly strong assumptions,
especially for intrusion rate, as I now discuss.
The models assume that intrusion rate
is unaffected by density of intruders already
in the territory. The most reasonable general alternative, as used in my original
"complex" model of optimal territory size
(Schoener, 1971), is that intrusion rate
declines with increasing intruder density.
Possibly this would happen if the pool of
intruders were sufficiently finite; a model
of this situation would have to be more
global than any dealt with below. A very
unlikely mechanism for a declining intruder
rate is that potential intruders monitor the
number of intruders already on the territory: We have postulated that detection of
intruders by the defender is often far from
instantaneous, and detection would certainly be even more difficult for an intruder
rarely on the territory, much less familiar
with it. If intrusion declined because of
decisions by potential intruders not to trespass, the most likely mechanism is via monitoring of food density on the territory. In
fact, an increasing intruder rate with
increasing food density has been shown for
sanderlings (Calidris alba; Myers et al., 1979)
and many hummingbirds (Hixon et al., 1983
and references; but see Mares et al. [1982]
for chipmunks, Tamis striatus). If intrusion
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
rate is proportional to density of food on
the territory, it will show a linear decline
with intruder density under the assumptions made in the next section; this is the
form assumed previously (Schoener, 1971).
Consequences of assuming a declining
intrusion rate can be quite different from
those assuming a constant intrusion rate.
The obvious difference is that equilibrial
intrusion rate can vary down to zero in the
former case but is constant in the latter.
Another difference is that energy gained
by defender would typically be less sensitive to a given decrease in patrol time when
intrusion rate declines; an upper limit on
intruder density exists no matter how little
patrolling is done. This results in feasible
(and optimal) values of c and t being more
restricted when constant intrusion is
assumed than when intrusion declines.
Despite these differences, the models with
constant intrusion and declining intrusion
must approach one another when the
expulsion rate is sufficiently large. Hence
behavior of the latter should include at least
as much complexity as behavior of the former, which as mentioned is intricate
enough. How a declining intrusion rate
affects predictions will be treated in a subsequent paper.
A linearly increasing expulsion rate that
is proportional to patrol time, t, also implies
some assumptions that do not always conform to reality. First, when intruder density is very high, the defender may become
swamped and expel only a few intruders
or even give up defense altogether. This
behavior is documented in the sanderling
(Myers et ai, 1979). The result will be a
flattening or even declining expulsion
curve. Second, the act of patrolling may
itself deter intruders, so that active expulsion is unnecessary in some cases. This is
particularly likely when defenders vocalize
and/or perform visual displays. Consequences in terms of the models here are
that 1) the intrusion intensity, q, is a
decreasing function of the patrol time, t;
or 2) that intruders are deterred without
entering the territory, so that active defense
costs (Cd below) are zero. Under condition
1, if q were inversely proportional to t and
expulsion after encounter on the territory
263
were entirely absent, equation (1) would
exactly result, so the algebra would be
identical. Under condition 2, the definition
of the territory could be enlarged to include
that area within which intruders are
deterred, rather than just chased, and the
model would again be much the same. Were
both active and passive exclusion occurring, things would of course be more complicated.
THE MAIN MODEL
Before giving the optimization equation,
it is useful to discuss its ingredients from a
biological point of view. The following are
the parameters of the model:
Food density, D, is the density of food,
measured in units of net energy that can
be gained by the defender per-unitdefended-area. Net energy is the difference between gross energy intake and the
energetic costs of searching for, pursuing,
handling, swallowing, and processing food.
It is an average taken over all food item
types actually selected by the defender.
Variation in selected food types as a function of number of intruders on the territory is not considered here; such variation
may be of considerable importance {e.g.,
the compression hypothesis, MacArthur
and Wilson, 1967; Schoener, 1971, 1974)
and will be discussed in a subsequent paper.
Food density is assumed to renew to a
fixed level at the end of some large time
period, T. That amount of the renewal
period during which the defender can be
active is defined as the defender's activity
period, P. This activity period could be part
of a day, as in many aquatic insects (Hart,
1985, 1987) but it might be considerably
longer when resources do not renew rapidly and defenders take substantial time to
traverse their territories.
Area of detection of intruders per unit patrol
time, J, is the actual area surrounding the
defender within which an intruder can be
detected, integrated over some unit of
patrol time. The concept is analogous to
those employed in population-dynamical
search models {e.g., Holling, 1966); one
imagines a moving disc that sweeps out a
path shaped like a rectangle with a semicircle at either end. J will increase with the
264
THOMAS W. SCHOENER
patrol speed of the defender and will
increase with its perceptual acuity. J will
ordinarily decrease with the density of
foliage or other structures in the territory.
J might be smaller for terrestrial than volant
organisms, except where the former can
use elevated perches. The sequence mammals -• lizards -• birds -» fish might represent an increasing J, at least for visual
cues.
Intrusion intensity, q, is the number of
intruders invading a territory per c* during
some large time period; it is proportional
to the number of intrusions per-unit-intrusion time and to the amount of time during
which intrusion is possible. It is multiplied
by c* to obtain the intrusion rate, the quantity corresponding most closely to "intruder
rate" in the single-variable models.
Energy eaten by a single intruder, M, is the
average food eaten from a territory by an
intruder during T, the same large time
period as used above. When intruders are
acting as time minimizers, M would correspond to a fixed metabolic cost. Then,
the smaller the intruder, the smaller would
be M, and ectothermic organisms would be
expected to have smaller Ms than endothermic ones (although they should also
have lower food-processing rates; see
below). Implied in the existence of this term
is that all intruders are assumed similar in
their ability to appropriate the defender's
potential food, and that intruders that do
not appropriate food are in effect not considered intruders (Hixon, 1980; Hart,
1985). The latter should not be actually
expelled but rather should leave of their
own accord, as has been observed for pied
wagtails (Motacilla alba; Davies and Houston, 1981).
Energy cost per-unit-patrol-time, C s , is the
energy expended per-unit-patrol-time.
Ordinarily, this would be some multiple of
basal metabolic rate {e.g., Bartholomew,
1982).
Energy of defense, or energy cost of expelling
dent of territorial circumference, as when
a vocal or visual display or a short chase is
sufficient to dispel an intruder, or when
intruders are turned back right at the territorial boundary. When y equals 1, expulsion costs are proportional to territory
radius (Hixon, 1980). This condition would
exist when an intruder is observed initially
from a central perch, or when (more
roughly) intruders are chased from some
average point in the territory to the boundary, or escorted across several territorial
boundaries as in certain harriers {Circus
cyaneus; Temeles, 1986), or well beyond
the boundary as in certain hummingbirds
{Calypte anna; Ewald and Carpenter, 1978)
and pied wagtails {Motacilla alba; Davies and
Houston, 1981). (Note that this average
point, assuming a random distribution of
intruders in the territory, is more likely to
be close to the boundary than deep inside
the territory, because the annuli corresponding to different distances from the
territory's center are larger in area, the
greater that distance.) When y equals 2,
the intruder might be imagined as being
chased back and forth within the boundaries of the territory before being expelled,
as when the intruder treats the territory as
a unit; a tendency for intruders to stay
within the boundaries of a particular territory for a certain time, then treat subsequently occupied territories in the same
manner, has been reported several times
in the literature (review in Buechner, in
preparation). The condition y = 2 might
also correspond to the case (as in various
mammals) in which olfactory "messages"
are distributed throughout the territory to
repel intruders, provided that the number
of such messages is proportional to the
density of intruders. Otherwise, the cost of
such scent marking might better be treated
as a fixed, intruder-independent cost proportional to c2 and renewed with period T;
it can then simply be incorporated into the
first term of equation (2) below, with
appropriate redefinition of the proportionality factor.
an intruder, Cd, is the energy involved in
ousting a single intruder per c^. This energy
is multiplied by c to the power y to obtain
The energy (and time) cost of expelling
the expulsion cost per intrusion. Realistic an intruder appears sometimes to vary as
values for y are 0, 1 and 2. When y equals part of a territorial strategy. For example,
0, expulsion energy (and time) is indepen- Anna hummingbirds {Calypte anna) can use
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
long chases, short chases, or visual displays
and vocalizations (Ewald and Carpenter,
1978). Presumably the former are most
effective in deterring future intrusion; otherwise they would never be used (unless
the less costly defenses did not in fact expel
the intruder). Under these conditions the
intrusion intensity, q, would decline with
Cd, a relationship not considered in this
paper.
Energy, E T , gained by defender from the ter-
ritory during T can then be written as
Net food
Net food
Energy _ energy available _ energy
gained
to defender
eaten by
and intruders
intruders
Energy
Energy
— expended — expended in
in patrol
expulsion
depletion term (2) and the expulsion term
(4). Increasing patrol time t increases ET
via the intruder-depletion term (2) but
decreases it via the patrolling cost term (3).
Assuming no constraints, the optimization
objective is to maximize ET with suitable
choice of c and t, given the values of the
six parameters, q, J, D, Cs, Cd, and M. However, constraints in both food-processing
rate and time available for eating, patrolling and defense may exist, as I now discuss.
CONSTRAINT EQUATIONS
Processing constraints
A limit may exist on the amount of food
energy the defender is able to process per
unit time. This limit, Ep, is the amount of
food energy potentially consumable by the
defender, i.e., the first two terms of equation (2):
EP < Dc2 - qMC+VJt.
or
2
+2
265
(3)
Note that EP is measured over the same
large period of time T (the renewal time)
as the terms in equation (2); this time
The first term is just the food density times includes one complete activity period P of
the area. The second term is MN*, the the defender.
energy eaten by each intruder from the
The processing constraint is thus a functerritory per T (M) times the number of tion of c and t and generates a curve in the
intruders in the territory at equilibrium c-t plane. The shape of this curve is most
(N*). N* is obtained by multiplying N*/c 2 easily seen by rewriting equation (3) so that
(equation 1) by c2. The third term is the t is a function of c:
product of patrol time and the energy cost
per time patrolling. The fourth term is the
qMc"
t <
(4)
product of the number of intrusions in a
J(Dc 2 - EP)
large unit of time and the expulsion cost
per intrusion. The former is obtained by Equation (4) says that t has to be smaller
solving the expulsion equation at equili- than the function of c on its right-hand side
brial density, i.e., Jt(N*/c 2 ) = qc\
in order that the processing constraint be
Equation (1) actually gives a family of satisfied. Figure 2 shows the permissible
models, since x and y can take on a variety values of t for a particular set of parameof values. In this paper, I will be concerned ters. In this figure, the function oft is plotwith four combinations: x = l , y = 0 ; x = ted as separate curves for x = 1 (Models 1,
1, y = 1; x = 2, y = 1; x = 2, y = 2. These 2) and x = 2 (Models 3, 4). For a given c,
will be referred to as Models 1, 2, 3 and 4, values of t below the appropriate curve are
respectively. Another reasonable (inter- permissible.
mediate) combination, x = 1, y = 2, is not
considered, but as we shall show, qualita- Time constraints
tive results are generally independent of
We may consider a variety of different
which of Models 1-4 is considered.
time constraints that can act singly or
Notice from equation (2) that increasing jointly. Such constraints need to be evalterritory size c increases ET via the density uated in two ways: first, the type of activity
term (1) but decreases it via the intruder- (or activities) involved, and second, the
E T = Dc - q M c * / J t
- C,t - qCdcx+y
(2)
266
THOMAS W. SCHOENER
r
Defense
20
~ 15 Ul
\
2
Patrolling
io
o
i
1/2
(TERRITORY AREA)
(c)
FIG. 2. Processing constraints for x = 1 (Models 1,
2) and x = 2 (Models 3,4) in the c-t plane. Parameters
are D = 100, q = 100, J/M = 1, Ep = 100, x = 1 or
2. The horizontally hatched region corresponds to
permissible combinations of c and t for x = 1. The
horizontally and vertically hatched region corresponds to permissible combinations for x = 2. The
curves are calculated from equation (4); in this function, the value c = (E p /D) 05 forms an asymptote along
which the positive portion of the function originates;
as c increases, t first decreases, then increases. The
left-hand portion of this function increases to oo as c
decreases because so little food is available on the
territory that there is no processing constraint. Eventually, c approaches a curve in which t is an increasing
power function of c.
simultaneity or lack thereof in the performance of these activities. The three types
of activities are patrolling, feeding and
defense (expelling intruders). When two
activities are incompatible, i.e., they cannot
be performed simultaneously, they must be
written into the same constraint equation.
For example, if defense and feeding are
incompatible, and P is the total time available, the inequality constraint is written
defense time + feeding time < P.
Partly for ease of analysis, I consider a
hierarchy of constraints; those at the lower
end are perhaps less likely to occur in nature
than the others. First, three one-activity
constraints exist, one each for patrolling,
feeding and defense. These are written:
patrolling
t < P
feeding
2
- qMc* +2 /Jt) < P
tf (Dc
defense
qt d c* +y —: P
(5a)
(5b)
(5c)
where tf is the eating time per-unit-food
i
i
20
15
5
10
1/2
(TERRITORY AREA)
(c)
FIG. 3. One-activity time constraints in the c-t plane.
Parameters for the feeding constraint are like those
of Figure 2 (Models 1, 2) where P/t f = Ep. Other
parameters are P = 10, td = 0.0077 (Model 1). When
the appropriate constraint operates, permissible values of c and t lie to the left of the defense line, below
the patrolling line, or to the left and below the feeding
curve (as in Fig. 2).
energy and t d c y is the time to expel a single
intruder (thus td is the time analog of the
energy term C d in eq. 2). In these constraints, none of feeding time, patrolling
time or defense time can separately exceed
P, the total time available.
Just as for the processing constraint (eq.
4), these constraints can be described by
functions in the c-t plane. Figure 3 gives
an example for x = 1. T h e patrol and
defense constraints are, respectively, given
by single values of t and c, so they are perpendicular to those axes. Values of c and
t below the patrol constraint or to the left
of the defense constraint are permissible
when the respective constraint is in operation. T h e feeding constraint has exactly
the form of the processing constraint (eq.
4 and Fig. 3). Indeed, notice that if the
identification is made that E P = P / t f , they
are identical. (This identification says that
the energy limit equals the total time available to feed divided by the energy that can
be eaten per time.) As before, for c fixed,
values of t below the feeding constraint are
permissible.
Exactly three two-constraint equations
and one full or three-constraint equation
are possible. Of the former, that which
assumes feeding and defense incompatible
in time is a priori the most likely; the second-most-likely, perhaps, is that which
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
considers patrolling and defense incompatible. Both will be discussed below.
Where both exist, defense is incompatible
with any activity, and this seems quite likely
most of the time. In addition, I shall analyze in detail the full time constraint, in
which no activity can be performed simultaneously with any other.
The several-variable time constraints are
written as combinations of the one-variable
constraints. For example, that for patrol
and defense is:
t + qtdcx+y < P.
The various several-variable time constraints have different representations in
the c-t plane. That for feeding and defense
is roughly similar to that for processing.
That for defense and patrolling is always
represented by an increasing power function (with an intercept) between c and t;
for Model 1, the power equals one and the
constraint is linear. The function for the
full constraint is most complex. It begins
at t = P, curves downward, then either
increases so as to merge with a power function between c and t (Models 1-3) or
asymptotes to a constant (Model 4). Figure
4 plots examples for the full constraint.
Not all time constraints can exist in a
given situation. If the full constraint is
operating, that is the only one possible. In
contrast, all three of the one-activity constraints can be operating if all activities can
be performed simultaneously. Or some
combination of one- and two-activity constraints may exist. For example, if the
feeding-is-incompatible-with-defense constraint exists, we might have in addition 1)
the one-activity constraint for patrolling
(this means that patrolling is compatible
with the other two activities), or 2) the
patrolling - is - incompatible - with - feeding
constraint, or 3) the patrolling-is-incompatible-with-defense constraint. Determining which constraints are operating in a
field situation may be no simple task. Indeed
one paper in this symposium (Ydenberg
and Krebs, 1987) describes the extensive
set of behavioral experiments required to
show that patrolling and feeding are separate (nonsimultaneous) activities for the
great tit {Parus major).
267
(TERRITORY AREA)1"1 (c)
FIG. 4. Full time constraints for Models 1-4 in the
c-t plane. Parameters are D = 100, q = 100.J/M =
1, tf = 0.1, td = 0.2, P = 10. Permissible values of c
and t are below the curve for the appropriate model.
The constraints appropriate for a particular situation form a set of intersecting
(and perhaps tangled) curves in the c-t
plane. Given certain constraints, the maximum value of energy gain, ET, is the largest such that it is within (closer to the origin) all constraint lines or on a constraint
line. Unless there is a local maximum, i.e.,
a "peak" in the surface of ET plotted as a
function of c and t, the second possibility
must occur.
RESULTS FROM THE MAIN MODEL
One-variable versions without
constraints
Before analyzing the two-variable model,
it is illuminating to analyze equation (2) as
two one-variable models. That is, if patrol
time, t, is assumed constant, what are the
properties of territory size (c) optimization
and vice versa? As some of these one-variable models for optimizing with respect to
territory size do not correspond to the
assumptions of any model for territory size
now in the literature (see also below), this
analysis is of particular interest. Moreover,
holding territory size constant and optimizing patrol time is relevant to nature: it
could correspond, for example, to cases
where territory size is constrained by tightly
packed neighbors. In such cases, varying
the time devoted to territorial activities may
be the only behavioral option open to the
defender.
For constant patrol time, t, we find (by
268
THOMAS W. SCHOENER
TABLE 1. Effect of increase in parameter values on optimal territory size (models 1-4).1[
Time
Consump- Energy/time
Time cost lo eat
Protion of an
cost of Activity
of an
a unit cessing
period expulsion of food capacity
intruder
patrol
(M)
(p)
(t,)
(CO
OJ
Food
density
I ntruder
rate
Energy
cost of an
expulsion
Area of
detection
(D)
(q)
(CJ
a)
1
I
I
T
1
0
—
—
—
—
T
1
1
T
1
I
—
—
—
—
1
0
1
1
T
T
—
—
—
T
T
1
I
T
1
0
1
—
—
—
0
I
0
0
0
0
T
1
—
—
I
0
1
1
T
T
T
—
I
—
T
1
1
T
i
T
T
I
—
—
1
JorT
IorT
1
T
lorl
1
lorl
I
—
I
—
1. One-variable
models for c"
2. Local maxima
(Models 3, 4)
3. Processingenergy constraint
4. Patrol-time
constraint
5. Defense-time
constraint
6. Feeding-time
constraint
7. Patrol-defensetime constraint*
8. Feeding-defensetime constraint*
9. Full time
constraint*
IorT
IorT
IorT
IorT
IorT
lorl
1
1
I
11 = territory size increases; 1 = territory size decreases; 0 = no change; — = not in model; underline
means £85% of simulated cases—see text.
* Based on computer simulation.
1
Only one example of 1 (for Model 3) found.
" copl also increases with patrol time, t (fixed).
differentiating eq. 2 with respect to c and
setting the result equal to zero) that each
of Models 1-4 (recall that these are defined
by different values of x and y) produces a
single optimum territory size, copt. The
equations of copt are, respectively:
Model 1
D + I(D» - 3q 2 C d M/Jt)° 5
^-opt
3qM/Jt
(6a)
Model 2
copt = 2Jt(D - qC d )/3qM
(6b)
Model 3
- 3 C d + (9Cd2 + 32DM/Jtq) 05
8M/Jt
(6c)
Model 4
copt = DJt/(2qM + QJt).
(6d)
Equations (6a)-(6d) give explicit formulae for how various parameters affect
optimal territory size when no constraints
are operating. No qualitative difference
exists between models in how parameters
affect copt. Note that optimal territory size
increases with increasing food density (D)
and decreases with increasing intruder rate
(q). The larger the (fixed) patrol time (t),
the larger the optimal territory size. Other
results are displayed in Table 1 (Row 1).
They show in particular that although the
"beneficial" terms D and J have the same
effect on optimal territory size, the "harmful" terms Cd, M, q, Cs and t do not. This
kind of result will become quite general as
the analysis proceeds to more complicated
cases, although its particular manifestation
will vary.
For constant territory size, the optimal
patrol time has the general representation
Topt = (qMc*+VJCs)05
(7)
Models 1 and 3 produce in addition a local where it is to be recalled that x = 1 (Models
1, 2) or 2 (Models 3, 4). Optimal patrol
minimum in E T at positive values of c.
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
269
time in this situation varies only with
intruder rate (q), area of search (J), energy
consumed by an intruder (M), search cost
(Cs) and territory size (c). It increases somewhat faster than territory size (either as c 15
or c2). Table 2 (Row 1) gives the qualitative
effect of parameter variation on topt; note
that the "harmful" parameters q and Cs
have opposite effects.
This brief treatment of one-variable
models provides some of the machinery
necessary to analyze two-variable models.
A fuller treatment of one-variable models,
including the effects of constraints, is given
in a later section.
The two-variable model with no
constraints
In contrast to each one-variable version,
the two-variable version does not give a
local maximum in Et for the simplest two
models (1 and 2). Rather, for those models
the two-variable surface (for c and t both
positive) is an infinite ridge (Fig. 5); for
each value of c or t there is always a maximum (follow any line perpendicular to an
axis), but the surface climbs forever
upwards for certain combinations of c and
t. To see this algebraically, note that if the
value of t obtained by setting dE T /dt = 0
(eq. 7) is substituted into equation (2), ET
will eventually increase without bound with
increasing c, provided ET is ever positive.
This is because the highest power of c in
the ET expression is 2, and this occurs in
the only positive term. When an infinite
ridge exists, the implication is that the optimum territory size and patrol time are
always found on a constraint curve (see
below for analysis of such curves).
Models 3 and 4 both produce a single
local maximum in ET. When the maximum
lies within all constraint curves, the optimum territory size and patrol time will be
located here. Examples of the maximum
falling within and outside constraint curves
are given in Figures 6 and 7, respectively.
The equations for optima when on the local
maximum are:
Model 3
copt = (2JZD - 4qM)/3qC ( JZ
(8a)
FIG. 5. Three views of an infinite ridge generated
by Model 1. The constraints are shown as heavy lines
(dashed line is the full time constraint, solid line is
the processing constraint). In this situation, the feasible maximum is on the full time constraint. Parameters are D = 100, q = 100, J/M = 1, C, = 0.1, Cd =
0.05, td = 0.1, t, = 0.1, E,, = 1,000, P = 100,x = 1,
y = 0.
270
THOMAS W. SCHOENER
FIG. 6. Two views of a local maximum for Model 3
(point X). The local maximum is within (satisfies) the
full time constraint (dashed line) but not the processing constraint (solid line), so the feasible maximum is the highest value of ET on the processing
constraint. Parameters are D = 20, q = 100, J/M =
1, C, = 0.1, Cd = 0.05, td = 0.1, tf = 0.1, EP = 40, P =
200, x = 2, y = 1.
Model 4
(8b)
copt = [(JZD - 2qM)/2C d qJZf 5
05
where Z = (qM/JC s ) . The equation for
optimal patrol time for both Models 3 and
4 is:
(9)
'•opt
Equations (8) and (9) allow us to determine how the various parameters affect
optimal territory size and patrol time when
those are on a local maximum (Tables 1
and 2, Row 2). When comparisons are possible, results mostly parallel those for the
one-variable version without constraints,
FIG. 7. The same ET-surface as in Figure 6, but now
the local maximum is within the processing constraint
but not the time constraint. Here the feasible maximum is on the time constraint. New parameter values
are EP = 100, P = 100. Note by combining this and
the previous figure that the local maximum can also
be 1) within both constraints, in which case it would
be the feasible maximum or 2) within neither constraint, in which case the highest point on a constraint
line and satisfying the other constraint is the feasible
maximum.
e.g., optimal territory size increases with
food density. The one exception is for
intruder rate, q; now, the greater the
intruder rate the smaller the optimal patrol
time. Also, unlike the one-variable models,
each of patrol time and territory size is
affected by each parameter. Thus we find
that increasing food density increases optimal patrol time (as opposed to having no
effect in the one-variable model); increasing defense cost decreases optimal patrol
time. By and large, when constraints are
271
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
TABLE 2.
Effect of increase in parameter values on optimal patrol time (models 7-4). t
Food
density
(D)
Intruder
rate
(q)
Energy
cost of an
expulsion
(Ca)
Area of
detection
0)
Consumption of an
intruder
(M)
Energy/
time
cost of
patrol
(Q)
1. One-variable
models for tc
0
1
0
1
1
2. Local maxima
(Models 3, 4)
1
[
I
I or T l o r 1
3. Processingenergy constraint"
1
I
T
i
I
I
4. Patrol-time
constraint
0
0
0
0
0
0
5. Defense-time
1
constraint
0
lorO'O
|
J
6. Feeding-time
constraint
i
T
1
1
I
I
7. Patrol-defensei
time constraint*
1
lorj
j
|
f
8. Feeding-defenseor |
time constraint*
| or }
J or J
i or J
[ or ]
1 or |
9. Full time
constraint*
1 or f
I or J
| or f
| or J
J or |
1 or | b
11 = territory size increases; [ = territory size decreases; 0 = no change;
means >85% of simulated cases—see text.
* Based on computer simulation.
' Model 4 only; Models 1-3 show J only.
" Based on computer simulation, except for C,, Cd, Models 1 and 4.
b
Model 4 shows J only.
c
topt also increases with territory size, c (fixed).
not operating, increasing "beneficial"
parameters increases territory size and
patrol time, whereas increasing "harmful"
parameters decreases both. The one
exception is for the area of detection, J; in
Model 4, for example, an increase in J
decreases optimal patrol time when J° 5D >
(16qMCs)05, i.e., when detection area, J,
and/or food density, D, are sufficiently
high. This is by far the most likely outcome, because for most parameter combinations when the inequality is false the
maximum is not feasible (eq. 8b gives a
negative number inside the parentheses).
The situation is similar but less extreme
for Model 3. It is to be emphasized that
these results apply only when the maximum is within all constraint curves.
The mathematical analysis supporting
the foregoing results is given in Appendix
1. That appendix also shows that Models
1 and 2 have a critical point for c and t
both positive (a point at which both <9ET/
dC and 3E T /dt equal zero). Rather than a
Activity
period
(?)
-
Time cost
of an
expulsion
(tj
ProTime to cessing
eat a unit
caof food pacity
(E,)
(W)
-
—
T
T
—
—
—
?
I
—
—
—
I
—
T
lorj
or f
—
—
I or J
[ or f
J or f
—
| or |
[ or J | or f —
— = not in model; underline
maximum, however, as in Models 3 and 4,
the critical point in Models 1 and 2 is a
saddle. Exactly at the saddle, ET is always
negative, so the critical point itself is outside the biological domain of interest.
Nonetheless, the saddle influences the ET
topography, as Figure 8 illustrates.
A model not analyzed in detail, in which
x = 1 and y = 2, is intermediate to the
foregoing models. It has both a saddle and
a maximum. For practical purposes, the
saddle can be ignored, so that the behavior
of this model is, except for details, similar
to that of Models 3 and 4.
The two-variable model with a
processing constraint
As detailed above, one possible constraint on ET is the amount of food that
can be processed in some large period of
time, T. Equation (3) gives this constraint.
In principle, when the optimum values of
a pair of variables are on a constraint line,
those optima can be obtained by substitut-
272
THOMAS W. SCHOENER
Explicit expressions such as those for copt
cannot be obtained for topt, nor does implicit
differentiation help. One can show analytically for parameters not in the constraint
equation but only in the ET equation that
if copt changes in a certain way topt will
change in a certain other way. The key to
obtaining these results is to show that the
maximum attainable ET always lies on the
left arm of the constraint equation, i.e., that
arm closer to the asymptote (Fig. 2).
Appendix 2 details the proof, only possible
for Models 1 and 4. The upshot is that for
the parameters Cs and Cd, in Models 1 and
4, analytical predictions about their effects
on patrol time can be obtained, but for the
FIG. 8. A saddle for Model 1. This view is blown up other parameters and/or models, simulaclose to the origin. Parameters are D = 10,000, q = tion is required. Table 3 gives parameter
100, J/M = 1, C, = 0.1, Cd = 0.05, td = 0.1, tf = 0.1,
Ep = 1,000, P = 100, x = 1, y = 0. Full time constraint values used in simulations; the system has
six free parameters (J and M always cois dashed line and processing constraint is solid line.
occur as J/M, so can be combined for simulation purposes), and I used three values
ing the constraint equation into the equa- for each, giving a total of 3 6 = 729 simution to be optimized (e.g., substitute eq. 3 lations per model. All simulations gave the
into eq. 2) so as to obtain an equation in same effect for a given change in a particone variable. Then normal differentiation ular parameter.
is performed to obtain the maximum. As
Results for optimal territory size and
we shall see below, rarely does this prooptimal
patrol time, given that the procedure give simple explicit expressions for
cessing
constraint
is acting, are displayed
the optima. For the processing constraint,
however, Models 1 and 4 both have qua- in Row 3 of Tables 1 and 2, respectively.
dratic solutions for copt that are relatively The most striking is the absence of an effect
of q, the intruder intensity, on optimal tersimple:
ritory size (see, e.g., eqs. 10a, b). More generally,
optima lying on the processing conModel 1
straint
usually show a different qualitative
E P [2CJD/M + 3CS
2 05
relation
to a particular change in a param+ (8CJQD/M + 9CS ) ]
(10a) eter than those lying on maxima (Tables 1
c
opt 2D(CJD/M + O
and 2, compare Row 3 with Rows 1 and 2).
An increase in food density now causes a
Model 4
decrease in optimal territory size and opti- C , + 4E P CJ/M
mal patrol time. This means that when the
+ (C52 + 8CSEPCJ/M)°-5 . (10b) processing constraint is in operation, the
relation of territory size to food density is
4JC d D/M
different from that when the constraint is
Similarly readable equations cannot be not operating: In general, the larger the
obtained for Models 2 and 3, which require food-processing rate relative to food denthat a quartic be solved. However, by sity, the less likely the constraint is to be
implicitly differentiating the expression limiting, so that continuous variation in
that results when equation (3) is substituted food-processing rate can cause a reversal
in how food density affects optimal terriinto equation (2), qualitative effects of all tory size. Put another way, as food density
parameters can be obtained for c and t both increases, its effect on optimal territory size
positive. As Appendix 2 shows, these effects can first be increasing, then decreasing
are exactly like those for Models 1 and 4.
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
(once the processing-rate capacity is
reached). Other differences between this
and the previous optima include that search
cost, Cs, and defense cost, Cd, now have
opposite effects: increasing search cost
increases copt and decreases topl; increasing
defense cost does the reverse. In general,
when a single parameter is varied, copt and
topt change in the same direction for about
half the parameters: copt and topt covary for
a change in food density, area of detection,
food requirements of an intruder, and processing-rate capacity, but vary in opposite
directions for a change in intruder rate,
search cost and defense cost.
The two-variable model with
time constraints
As detailed above, possible time constraints include three for when a single
activity is competitive with all others, three
for when two activities are incompatible in
time, and one for when all three activities
are incompatible.
One-activity constraints. We have already
produced all the machinery necessary for
a complete solution of the three one-activity constraints. The constraint on feeding
(eq. 5a) is exactly of the form of the processing constraint (eq. 3), so all results for
the latter apply to the former. Recall that
EP can be identified with P/t f , so that the
effect of P (total time available) is qualitatively exactly the same as the effect of EP
(processing capacity), whereas the effect of
tf (time to eat a unit of food energy) is
exactly the reverse. The constraint on
patrolling (eq. 5b) behaves exactly as does
the one-variable version above (eqs. 6a-6d)
for t constant; in this case t equals P. The
constraint on defense (eq. 5c) behaves
exactly as does the one-variable model for
c constant; in this case c = (P/qtd)1/x+>r. To
determine how the various parameters
affect patrol time, this expression for c
would be substituted into the single-variable expression for optimal patrol time (eq.
V).
Results for the one-activity constraints
are collected in Tables 1 and 2 (Rows 4 6) and show that those constraints can vary
in how changes in parameters affect territory size and patrol time. Increasing food
273
density, for example, increases optimal territory size for the processing constraint and
decreases it for the feeding constraint; it
has no effect when the defense constraint
is in operation. In general, the three constraints affect optima quite differently.
Multiple constraints. Unfortunately, no
simple readable expressions exist for how
the more complicated time constraints
affect optimal territory size and patrol time:
the simplest expression is a cubic, and a
number of expressions are of fifth degree
or higher, which in principle are impossible to solve explicitly (Uspensky, 1948).
Moreover, implicit differentiation does not
provide expressions of unchanging sign
with changes in copt and topt, as was the case
for the processing constraint. Consequently, we must resort to computer simulation to obtain predictions.
Simulations were performed as follows.
First, the same parameter values (3 of each)
as were used to evaluate the processing
constraint's effect on topt, plus three values
for each new parameter, were used for the
complex time constraints (Appendix 3).
Parameter values were taken in all combinations to yield a total of 37 = 2,187 or
36 = 729 simulations for each combination
of Models 1-4 and constraint equation. The
constraint equations used were those
assuming 1) incompatibility of patrol and
defense, 2) incompatibility of feeding and
defense, and 3) incompatibility of all activities. For each simulation, that combination of c and t on a constraint line giving
the maximum ET was located. Simulations
for the two latter constraints had to be done
on the Burroughs 7800 version of doubledouble precision (22 significant digits; the
highest available); double precision (16 significant digits) was sufficient for the first
constraint. Such precision was necessary,
among other reasons, because optimization surfaces could be very flat. (Of course,
under such circumstances an optimally
performing defender is relatively indifferent to broad variation in territory size.)
One should realize that even extensive simulations may miss certain combinations of
parameter values giving results different
from those tried, as a number of studies in
population biology have illustrated {e.g.,
274
THOMAS W. SCHOENER
Cramer and May, 1972). Finally, an additional series of simulations was performed
for certain parameters when informal continuity arguments (see below) and other
considerations (values too small to affect
ET much) suggested them desirable. These
are listed at the end of Appendix 3.
Results of the simulations are presented
in Tables 1 and 2 (Rows 7-9). The following conventions are adopted. When an
increase in a parameter causes both an
increase and decrease in the optimization
variable, both directional arrows are
included. However, when in all sets of simulations a particular direction occurred in
85% or more cases, the corresponding
arrow is underlined. A simulation set is
defined as being restricted to a particular
model and particular combination of
parameters such as those in Section A,
Appendix 3. Because of precision problems, cases where no change occurred in
the optimization variables are not listed, as
one cannot be certain of their reality.
No several-variable constraint showed
exactly uniform behavior with respect to
all parameters. Indeed, the patrol-defense
constraint was the only one that even came
close. All territory-size changes were unidirectional using that constraint (Table 1,
Row 7), but three patrol-time changes were
not (Table 2, Row 7)—those resulting from
variation in intruder rate (q), total activity
time (P), and time to expel an intruder (td).
For the other two several-variable constraints, it is easier to list changes that are
unidirectional. For the feeding-defense
constraint (Table 1, Row 8), territory size
decreased with increasing food density (D),
decreased with increasing area of detection
(J) or decreasing food consumption per
intruder (M), increased with increasing total
activity time (P) and decreased with
increasing total time to eat a unit of food
energy (tf). No parameter affected patrol
time unidirectionally (Table 2, Row 8). For
the full constraint, territory size decreased
with increasing energy cost (Cd) and probably time cost (td; see Table 1, Row 9) of
expelling an intruder, increased with
increasing total activity time (P), and
decreased with time to eat a unit of food
energy (tf). Again, no parameter affected
patrol time unidirectionally (Table 2, Row
9).
The fact that the several-constraint
equations are linear combinations of the
one-constraint equations allows partial
understanding of these results. The full
time constraint, for example, by sufficiently lowering tf and td can be brought
arbitrarily close to the one-activity constraint for patrol time. Hence under suitable parameter choice we would expect the
full time constraint to show the same relation to food density, for example, as shown
by the simple patrol-time constraint—territory size increases with food density. We
might also suspect the opposite relation to
food density would occur were td and tf
sufficiently large, because then defense and
feeding portions are dominant. However,
this part of the argument is quite informal,
because t is an optimization variable, and
we need simulation to be sure. Indeed, as
inspection of Tables 1 and 2 show, arguments of the latter kind appear not always
to hold.
Overall, the hierarchial rule exists that
any several-activity constraint will show at
least those directions of change shown by
its component activities when they occur
as one-activity constraints. However, rarely
several-variable constraints may show
additional directions of change that no
component activity shows by itself. For
example, the patrol-defense constraint
(Table 2, Row 7) gives a decreasing optimal
patrol time with increasing food density,
whereas neither component (Table 2, Rows
4, 5) shows any effect of food-density
change.
The rule between one-activity and several-activity constraints does not generalize to comparisons among several-activity
constraints. Thus the feeding-defense constraint shows both an increasing and
decreasing effect of energetic defense cost
(Cd) on optimal territory size, whereas only
the latter is shown by the full constraint.
In general, the feeding-defense constraint
and the full constraint behave irregularly
to about the same degree.
Finally, except for the effect of intruder
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
rate (q) on optimal territory size for the
full constraint and two possible additional
cases (Table 1, Row 9), Models 1-4 are
identical with respect to the directional
effects of parameters—each effect shown
in Tables 1 and 2 can be obtained with each
model, although distributions of directions
and magnitudes of change may vary substantially between models.
In summary, noting the caveat on simulation expressed above, the following
generalizations are possible about all time
constraints. Optimal territory size usually
or always decreases with increasing energy
cost of expulsion (Cd), increases with
increasing energy cost of patrol per unit
time (C5), increases with increasing total
activity time (P) and decreases with time to
eat a unit of food energy (tf). Optimal patrol
time usually or always decreases with
increasing food density (D), decreases with
increasing energy cost of patrol per time
(Cs) and increases with total activity time
(P). Other results are too irregular for generalization, and for all irregular predictions, caution should be used in their evaluation. In particular, the effect of food
density on optimal territory size is irregular, and claims that changes in this parameter alone support theoretical predictions
must be made very carefully; the particular
model giving those predictions must be
specified and justified. For several-activity
constraints, one can argue from the dominance of one or another component activity that predictions should run as if the
latter were the only activities constrained.
Where such dominance is not extreme,
however, quantitative evaluation is necessary.
By-product variables
The preceding discussion is limited to
effects of parameter change on the two
optimization variables territory size, c, and
patrol time, t. In addition to these "pure"
variables, the models make predictions
about a number of "by-product" variables.
Four of these are of special interest:
total time spent defending
275
total time spent feeding
= tf[Dc2 - (qMC +2 /Jt)]
equilibrium density of intruders
= qcVJt
fraction of available food consumed
by intruders
= qc x M/DJt
where totals are measured over some large
time period T. Notice that these four quantities are composed of both parameters and
optimization variables. To determine their
values in any given situation, copt and topt
are substituted along with values of the
required parameters.
The by-product variables are important,
because predictions about how they are
affected by parameter changes can be used
along with similar predictions for the optimization variables to provide a more robust
validation of a particular model. The temporal variables are especially useful in this
regard, as they are relatively easy to measure in the field. Moreover, predictions
about by-product variables are interesting
in their own right; the equilibrium density
of intruders, for example, is, as discussed
above, a measure of territorial exclusiveness.
Effects of parameter changes on byproduct variables are given in Appendices
4-6. In a number of cases, effects could be
determined analytically. In others, simulation was required; for them, results from
the same simulations (Appendix 3) as were
used to construct Tables 1 and 2 are given
in the main body of the appendix tables.
For models not simulated in Tables 1 and
2 but requiring simulation here, the "main
set" (Appendix 3, A) was used. In certain
cases where it was suspected that a broader
range of qualitative outcomes could be
obtained with new simulations, these were
performed and are so noted in the appendices.
Predictions about the by-product variables show the following general characteristics. First, predictions are at least as
irregular as those for c and t. In fact,
because simulations were designed to
investigate variation in c and t, predictions
276
THOMAS W. SCHOENER
for by-product variables are probably even
more irregular than represented here.
Greater irregularity for by-product variables is unsurprising, because such variables (except total defense time) incorporate both optimization variables, which
themselves often do not covary (see above).
Second, as with c and t, several-variable
time constraints give more irregular predictions than one-variable constraints.
Third, which of Models 1-4 is used makes
more of a difference for by-product variables then for c or t. Despite the substantial
complexity, a number of trends do emerge
for each by-product variable.
Total defense time (Appendix 4). This variable generally increases with increasing
patrol cost (Cs), total activity time (P) and
time to expel a single intruder (td); it always
increases with increasing processing-rate
capacity (EP). Total defense time generally
decreases with increasing cost to expel a
single intruder (Cd) and time to feed on a
unit of food (tf). Total defense time shows
a highly irregular relationship to changes
in food density (D), intruder intensity (q),
area of detection (J) and food consumption
of an intruder (M).
Totalfeeding time (Appendix 5). This variable generally increases with increasing
food density (D), area of detection (J), total
activity period (P) and time to feed on a
unit of food (tf); it always increases with
processing-rate capacity (EP). Total feeding time generally decreases with increasing intruder intensity (q), food consumption of an intruder (M) and time to expel
a single intruder (td). Total feeding time
shows a highly irregular relationship to
changes in cost of expelling a single
intruder (Cd) and patrol cost (Cs).
Equilibrium density of intruders (Appendix
6). This variable generally increases with
increasing food density (D) and patrol cost
(Cs). Equilibrium density of intruders generally decreases with increasing cost to
expel a single intruder (Cd), area of detection (J) and food consumption of an
intruder (M). Equilibrium density of
intruders shows a highly irregular relationship to changes in intruder intensity
(q), total activity period (P), time to expel
a single intruder (td), time to feed on a unit
of food (tf) and processing-rate capacity (EP).
Fraction of food consumed by intruders
(Appendix 6). This variable is very similar
to the last, differing in the addition of two
terms, M and D. Therefore all predictions
are identical to those for equilibrium density of intruders except that 1) fraction of
food consumed by intruders generally
decreases with increasing food density (D),
and 2) fraction of food consumed by
intruders generally increases with increasing food consumed by a single intruder (M).
MODELS THAT VARY THE FRACTION OF
DETECTED INTRUDERS EXPELLED
In my original territorial equilibrium
model (Schoener, 1971), I included a third
variable of choice, f, the fraction of detected
intruders expelled. In that model, it is easy
to show that f must equal one if t is positive,
as follows. First, wherever f occurs, patrol
time, t (called T s in Schoener, 1971) also
occurs: in order to expel an intruder, it
must first be detected. Next, suppose that
we calculated the value of the currency
being optimized for a vector of variables
c, t and f such that f were less than one.
Then so long as t is sufficiently greater than
zero, the same number of intruders
expelled could be obtained by decreasing
t and increasing f to one. In so doing, however, less energy would be required to
patrol (Cst would decline), and a net benefit
would be obtained. Hence, f will always be
set at one in the optimum. More generally,
for any model in which f and t are related
as in that of Schoener (1971), which may
include all sensible models, the same argument holds. Only where t is zero or
excluded from the model (perhaps because
it is always simultaneous with another
activity, e.g., feeding) would the argument
fail and the possibility exist that optimal f
would be less than one.
This latter possibility may in fact hold in
cases where intruders are always automatically detected when they invade. Such a
situation could occur, for example, for very
small a n d / o r open territories, or for
defenders that always have a good vantage
point no matter what activity is being per-
277
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
formed. Possibly certain hummingbird territories would be an example of a system
in which patrolling is always simultaneous
with another behavior and which therefore costs no extra time or energy (Hixon
et al., 1983; Carpenter, 1987). Then the
most suitable model would be one with two
variables, territory size and the fraction of
intruders chased. Indeed, Ewald and Carpenter (1978) and Carpenter (1987) demonstrated a variable f. Where f varies and
2
4
6
8
12
14
10
other parameters stay roughly constant, the
(TERRITORY AREA) 1/2 (c)
equilibrium density of intruders would also
vary and thereby degree of exclusivity FIG. 9. The full time constraints for models with t
would vary as well. In the Carpenter study, fixed but the fraction of intruders expelled, f, varying.
optimum is always where the left arm of the time
however, a low f was associated with low The
constraint crosses the line f = 1. Parameters are D =
intrusion, q, because during such times food 100, q = 100, J/M = 1, td = 0.05, tf = 0.1, P = 10,
was superabundant; this may result in about x = 1 or 2, y = 0, 1 or 2.
the same equilibrium density of intruders
as a higher f and q.
For the territorial equilibrium model ing c. Thus the smaller the c the better,
presented here (eq. 2), if f were a variable and c will decrease until f equals one, at
of choice and t were fixed, expulsion rate which point the optimum will occur. Figwould include f as a factor and equal ure 9 illustrates the situation. When the
JfPN*/c 2 ; note that we assume intruders defense constraint alone occurs, the optican be detected during the entire activity mum will be at the intersection of that conperiod P. As a consequence, everywhere J straint and the line at which f equals one.
appears in subsequent equations, f will
In conclusion, no form of the models
appear also. Equation (2) in particular given here, when modified to include the
would be written
fraction of intruders expelled, allows that
fraction
to be anything other than one.
E T = Dc 2 - qc x + 2 M/JfP . (11) However, all this is based on the assumpNote that equation (11) does not provide tion that territorial equilibrium is detera maximum in f; the larger the f, the higher mined as in Figure 1. On the other hand,
the ET. Hence, given no constraints, f = 1 were intruder rate, q, to fall with (nonequiis always optimal. (It may seem paradoxical librial) number of intruders on the territhat the number of intruders expelled in tory (as in Schoener, 1971), f would affect
some large time period does not depend the number of intruders expelled, and
on f. But note that this number is propor- equation (2) would have separate terms in
tional to N*/c 2 , and that term in turn is which f had an increasing and decreasing
inversely related to f. So halving the frac- effect. Hence the non-results on f are not
tion of intruders expelled, for example, necessarily expected to generalize to
doubles the equilibrium density of intrud- models with more complex invasion paters and results in the same absolute num- terns; such models will be discussed in a
subsequent paper.
ber of intruders expelled.)
The optimal value of f should equal one
ONE-VARIABLE MODELS REVISITED
when various constraints apply as well.
When either the analog of the feedingAs outlined above, the two-variable
defense-time constraint, or the processing energy-maximizer model discussed in this
constraint (eq. 3), or the feeding constraint paper has one-variable versions, one of
(eq. 5a) is substituted into equation (11), f which optimizes territory size. How does
disappears and ET decreases with increas- this model compare with published one-
X...
278
THOMAS W. SCHOENER
2000 r
ill
1500 -
8
2
4
6
TERRITORY AREA (C 2 )
10
FIG. 10. Two ways of representing cost-benefit curves
for one-variable versions of the energy-maximizer
model of equation 2. Models 3 and 4 only. Top. Benefit
includes only the first term of equation 2. Bottom. Benefit includes the first and second terms of equation 2.
The optimal territory size is where the difference
between the benefit and appropriate cost curve is
maximum. Parameters are D = 100, q = 100, J/M =
0.5, C, = 0.1, Cd = 0.05, t = 10, x = 2, y = 1 or 2.
variable models of territory size? As
reviewed in Schoener (1983, table 1), these
models can be divided into four groups on
the basis of environmental inputs: food
density alone changes, intruder rate alone
changes, increasing food density increases
intruder rate, and increasing intruder rate
decreases food density. The present model
would seem to be of the fourth kind,
because the second term in equation (2)
represents the decline in food available to
the defender resulting from intruder consumption. However, models in the present
paper, even in their one-variable form, do
not make the assumption that all intruders
can be expelled from the territory, but
rather allow a "standing crop" of intruders
in part determined by how intruder rate
varies with territory size. It is this fact that
makes the one-variable version of equation
(2) unlike any of the one-variable models
previously published.
To illustrate, we can convert the onevariable version into the cost-benefit format used by Schoener and Schoener (1980)
to model an energy maximizer. To do this,
we must decide which terms are included
in each of the cost and benefit curves. Two
possibilities exist, depending upon whether
the second term in equation (2)—the diminution of food caused by intruders—is part
of the cost or benefit curve. If it is a cost,
then the benefit rises proportionately to c2
(area) and the cost rises as a function of c
to various powers (Fig. 10, top). Depending
upon whether we have Models 1, 2, 3 or
4, the highest power of area in the cost
curve equals 3/2 or 2. Hence the cost curve
is accelerating, and we have the prediction
(Schoener, 1983, table 1) that increasing
food density raises the benefit curve and
therefore causes an increase in optimal territory size. This is essentially the simplest
model given by Ebersole (1980; his fig. 1).
(It is interesting that the foregoing set of
assumptions provides a rationale for an
accelerating cost curve not given in previous papers.) The second way to incorporate the second term of equation (2) is
into the benefit curve (Fig. 10, bottom).
Under those circumstances, the benefit
curve (Terms 1 and 2 of eq. 2) will have
both increasing and decreasing portions,
as its negative term is proportional to a
higher power of area (3/2 or 2) than its
positive term (1).
To understand how constraints work in
the present one-variable model, it is more
pedagogical to use the second format (Fig.
10, bottom). First consider the processing
constraint (eq. 3). This constraint says that
EP, the processing capacity, is less than or
equal to the potential food benefit, all
expressed in units of energy over some
large time period. Hence it is plotted as a
line parallel to the territory-size axis. For
this kind of graph, optimal territory size
can occur at values of food consumption
smaller than the processing constraint
(when the cost curve sufficiently accelerates) or at values equal to the processing
constraint. In the latter case, the intersection of the processing-constraint line with
279
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
the benefit curve determines the optimum.
Note from Figure 11 (top) that if the EPline intersects the benefit curve at all, it
does so twice. Also note that when the cost
curve rises monotonically, the optimum
territory size, if it is determined by the processing constraint, is represented by the
first intersection. (This can also be seen
algebraically by substituting equation (3)
into equation (2) and noting that the smaller
the territory size c, the higher the gain in
energy ET.)
Time constraints can also be pictured
and analyzed in a similar fashion. For the
full, three-activity constraint, we have that
Food obtained by the defender
< Food obtainable by the defender
during the total activity period,
200
r
150 100
Processing
Constraint, E p
z
HI
50 -
200
r
LU
150 -
O
a.
100
LLJ
2
LU
or
Dc2 - (qc"+2M/Jt)
< (P - t - q C ^
(12)
The terms on either side of the equation
are in units of energy, and the right-hand
portion, which represents the time constraint, can be plotted on the same graph
as the processing constraint (Fig. 11, bottom). The time-constraint curves decline
with increasing territory size (as eq. 12
shows), and again there are two intersections. Because the cost and benefit curves
converge as territory size increases, the
optimum territory size, if it is determined
by the time constraint, is at the first of the
two intersections, just as for the processing
constraint.
The simpler time constraints can also be
represented graphically. The one-activity
constraint on patrol time (eq. 5b) is automatically included in this model, as patrol
time is set to a constant. The one-activity
constraint on defense time (eq. 5c) uniquely
determines a value of c, so no further optimization is needed. The one-activity constraint on feeding time (eq. 5a) acts here
exactly as does the processing constraint,
just as in the two-variable models. The twoactivity constraint on feeding and defense
is basically similar to the three-activity constraint, as the latter only contains (for the
one-variable model) an additional constant
term t. The two-activity constraint on
TERRITORY AREA
(c 2 )
FIG. 11. Top. Same cost and benefit curves as in Figure 10, bottom, with a processing constraint. The
feasible maximum difference between benefit and cost
is at the left intersection of the processing constraint
and the benefit curve. Models 3 and 4 only (x = 2).
Bottom. Same cost and benefit curves as top, with fulltime-constraint curves. The feasible maximum difference between benefit and cost is again at the left
intersection. Additional parameters are tf = 1, td =
0.02, E P = 100, P = 110.
patrolling and defense again sets a unique
value for c.
Analysis of how parameter changes affect
territory size is substantially easier for these
one-variable models than for the two-variable models, because t is not a function of
c but is a constant. For example, increasing
food density increases the benefit curve for
all values of territory size. As can be seen
graphically (Fig. 12), when the processing
or time constraint determines the optimal
territory size, increasing food density causes
the optimal territory size to decrease.
Notice that as food density declines, the
optimum will eventually not be determined
by constraints but rather by the maximum
difference between the cost and benefit
curves. At this point, the reverse relation
of optimal territory size to food density will
take over. The effects of other parameters
280
THOMAS W. SCHOENER
High
200
HIGH
dArea
LOW,'
/•
*
/ J
<
UJ
oc
<
dBenefit
dArea
L H
HIGH
LOW
/
/
CO
CO
O
a.
o
o
O
O
'0
HL 2
4
6
8
10
Optima
2
TERRITORY AREA (c )
FIG. 12. Two values of food density (high and low)
with a processing constraint and a full time constraint
(Model 3). H and L are optimal territory sizes for
high and low food densities, respectively. For either
constraint, the feasible optimal territory size is higher,
the lower the density. D = 100 and D = 125. Other
parameters as in Figure 11.
can be assessed similarly. Note that the
intruder rate, q, affects both the benefit
curve and the full time constraint curve
(see eq. 12); moreover, the effects go in the
opposite direction, so that graphically at
least, the net result is indeterminate.
The preceding analysis points the way
toward a more unified theory of one-variable territory-size models than was previously available (Schoener, 1983). The key
distinction between model situations, so far
as the effect of certain parameters is concerned, is whether or not the defender is
actually up against a limit exerted by a particular constraint (see also Sih, 1984). For
example, when the optimum is determined
by a time or processing constraint, an
increase in food density decreases territory
size; an increase in intruder rate can have
either effect. When the optimum is such
as to fall short of all constraints, the fooddensity relation and sometimes the
intruder-rate relation changes. The same
behavior is exhibited by all the simple
models summarized in Schoener (1983). In
the processing-constraint model (Schoener
and Schoener, 1980), for example, the
effect of increasing food density is reversed
when going from a constrained to unconstrained situation (Fig. 13); an intermediate situation gives one or the other result.
For increasing intruder intensity, optimum
o
•o
cc
UJ
w
z
cc
<
/
L H
HIGH
y'
LOW
y'
UJ
CD
y'
y'
UJ
HL
m
HIGH
LOW
.
.
'
•
'
TERRITORY AREA
FIG. 13. Food-density variation for cost-benefit
curves for the Schoener and Schoener (1980) processing-constraint model of optimal territory size (see
also Schoener, 1983, fig. 3). H and L are optimal
territory sizes for high and low food density, respectively. As one descends the figure, the processing constraint increases in importance as the cost curve shifts
progressively downward. Where the optima are not
constrained, the greater the food density, the greater
the territory size. This relation shifts to the opposite
relation as optima are forced against the constraint.
territory size decreases when all optima are
at values below those in which the processing constraint is limiting; optimal territory size is constant when the opposite
holds (because the benefit curve never
declines). Similarly, in Hixon's (1980) linear time-constraint model, the defender is
always time limited, and territory size
always decreases with increasing food density. When cost curves are not assumed linear, the optimum is not necessarily constrained by available time, and the opposite
result can hold for food density. In this
case, it is optimal for the defender not to
use all of its available time in feeding and
territorial activities, other constraints notwithstanding.Moreover, in Hixon's model
an increase in intruder rate always
decreases optimal territory size; for the
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
model in Figure 12, it is as if only the timeconstraint curve moves down and benefit
is unaltered by intruder rate. If benefit is
also changed, the result may not hold, as
pointed out above and in a different way
in Schoener (1983, fig. 10). Finally, time
minimizers can be viewed as animals that,
by definition, are always limited by a constraint, i.e., the fixed minimal requirements they must obtain. In their case, an
increase in food density results in a decrease
in territory size unless increased food density causes intruder rate to rise in an accelerating fashion (Schoener, 1983, fig. 6).
New models by Hixon (1987), in which
feeding efficiency increases with food density, do not fall into this pattern when the
latter relationship is exponential.
Hence, except sometimes when intruder
rate is affected by food density or when
feeding efficiency changes with food density, whether a constraint is operating or
not determines the relation of food-density
changes to optimal territory size; in the
latter case, the higher the density the larger
the territory size (Fig. 10); in the former
case, the higher the density the smaller the
territory size (Fig. 12). This rule holds only
for one-variable models, as the above analysis of two-variable models has shown. In
particular, for a patrolling-time constraint,
the opposite relation obtains, and the same
may be true for the more complex time
constraints (Table 1). Nonetheless, in
Models 3 and 4, where a local maximum
exists within all constraint lines, the same
reversal of the effect of food-density increase on territory size can occur as for the
one-variable case just discussed.
281
ing cases where there was "no effect," certain trends are apparent. Optimal territory
size always increases with increasing total
activity time, increasing processing-rate
capacity and decreasing time to feed on a
unit of food; it usually (with the reservations stated above) increases with decreasing intruder rate, decreasing energy cost
of defense, increasing energy cost of patrol
and decreasing time cost of defense. Optimal patrol time always increases with
increasing processing-rate capacity; it usually increases with decreasing food density,
increasing energy cost of defense, decreasing area of detection, increasing consumption rate of an intruder, decreasing energy
of patrol, increasing total activity time and
decreasing time to feed on a unit of food.
Effects of other parameters are quite varied; these unfortunately include the effect
of food density on territory size, a relationship of particular interest. Specifically,
food density has different effects, depending upon whether a processing constraint
is limiting or not; this means (for a sufficiently accelerating dependence of defense
time and invasion rate on territory size;
Models 3 and 4) that as food density
increases, territory size will typically
increase (while on the local maximum), then
decrease once the processing constraint is
reached. The time constraints also act in a
very varied manner with respect to an effect
of food density on territory size; where eating and defense time are dominant, time
constraints will behave as do processing
constraints, but if patrol time is dominant,
the opposite will occur.
Despite such a diversity of theoretical
predictions,
in nature territory size usually
CONCLUSION
decreases with increasing food density
Answers to some questions
(review by Hixon, 1980, 1987; Hart, 1987;
We now return to the questions asked in Mares and Lacher, 1987; but see Ebersole,
the introduction and provide at least par- 1980). This suggests that certain constraints are generally in operation if the
tial answers. They are, in order:
1. Tables 1 and 2 summarize the effects defender is an energy maximizer, or that
of the various parameters. They depend the defender is a time minimizer having no
heavily upon which constraint is active, and accelerating invasion rate with increasing
for models with a local maximum, whether food density (Hixon, 1980; Schoener,
the optimum can be attained there. No 1983), or that the defender is an area maxparameter has exactly the same effect on imizer (Hixon, 1987). The natural trend
optimal territory size or patrol time for all also implicates a particular class of stomodels and constraints. However, ignor- chastic models (McNair, 1987).
282
THOMAS W. SCHOENER
5. Qualitative differences can exist
2. An increasingly curvilinear dependence of invasion rate on territory circum- between one- and two-variable models in
ference (proportional to c or to c2) and terms of particular parameters. In the simdefense cost on territory circumference plest two-variable models, no local maxi(not proportional to area, proportional to mum exists even though for each variable
c or to c2) is analyzed. The most important taken separately there is a local maximum.
result is that local maxima are possible for As a result, in the two-variable model, the
the more accelerating dependencies (pro- optimal territory size must be on a conportional to c2 and c, or to c2 and c2, respec- straint curve, whereas it may or may not
tively). These local maxima imply different be in the one-variable analog. This can
relationships of trie optima to parameters result in opposite predictions from the onesuch as food density than those on con- and two-variable models about the effects
straint curves. This result is analogous to of parameter change, e.g., in food density.
the one for accelerating cost curves in the Moreover, even though both one- and twoone-variable models already in the litera- variable models contain the same paramture (Schoener, 1983).
eter set, in the latter all parameters may
3. All relationships between optimal affect optimal patrol time and optimal terterritory size and optimal patrol time are ritory size, whereas in the former a number
possible, given a monotonic change in one of parameters have no effect (compare
of the parameters. This can be seen by flip- Rows 1 and 2, Tables 1 and 2).
ping back and forth between Tables 1 and
2. Almost no generalizations are possible. How to test an optimal-territory-size model
Mathematically, the reason the situation is
Theory in this area of behavioral ecolso complex is that many of the constraint ogy is rapidly outstripping data. Empiriequations are highly nonlinear in the c-t cists may be bewildered and frustrated,
plane, and that typically most parameters perhaps even irritated, by what appear to
appear in both the function to be maxi- be yet more complications. How can a set
mized (ET, eq. 2) and the constraint equa- of models with so many variable predictions.
tions be tested? Clearly the best procedure
4. The assumption that the number of is to determine, a priori, which of the cases
intruders expelled and intruder standing detailed in Tables 1 and 2 best fits a pardensity in a territory varies with patrol time ticular system. Testing can then be done
has profound effects on one-variable in an entirely non-circular manner. A posmodels of territory size. A new rationale is teriori interpretation of results are necesprovided for accelerating cost curves, and sarily less desirable, because of the flexithe benefit curve, under one interpreta- bility of the models, unless 1) strong
tion, declines past a certain territory size. arguments can be made for a natural sitThe new one-variable models suggest a less uation fitting one theoretical case as
cluttered way to group predictions on opti- opposed to another, and/or 2) detailed
mal territory size; in particular, if optimal information on many of the model predicterritory sizes are such that the defender tions is available, so that a multiple check,
is at a processing or time constraint, e.g., across a row of Tables 1 or 2, or using
increasing food density decreases optimal the by-product variables (Appendices 4-6)
territory size provided intrusion rate is can be made (in the manner of Hixon et
al., 1983). In many cases, quantitative
unaffected by food density.
The one-variable model for patrol time rather than qualitative information will be
provides predictions for a territory holder desirable, i.e., when a parameter change
whose territory size is constrained by tightly gives variable qualitative output. Even in a
packed established neighbors. Optimal carefully planned a priori study, varying
patrol time increases with intruder rate, several parameters instead of one will proindividual intruder consumption and ter- vide stronger verification of a particular
ritory size; it decreases with area of detec- model. This technique is especially useful
if several a priori likely models are to be
tion and energy-per-time cost of patrol.
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
discriminated. For example, determining
how a decreasing area of intruder detection (say by decreasing visibility) affects
patrol time will not by itself discriminate
much between models, whereas determining the effect of food-density variation in
addition allows many more models to be
rejected (Table 2). Especially difficult for
any application, however, will be cases
where a number of constraints will be operating; if they give different predictions, the
only resort is to determine their features
quantitatively and plot the resulting curves
in the c-t plane to see how they interlace.
Relation to models on the existence of
territoriality
Carpenter and MacMillen's (1976) early
model of whether or not to be territorial
is still the only such pure model; it compares net benefits of being territorial with
those of being a floater. The models herein,
however, provide another way of looking
at the existence of territoriality. We might
define degree of territoriality according to
position along several axes, such as the time
devoted to territorial activities, the fraction of encountered intruders expelled, or
the long-term standing crop of intruders
in the territory. These continua more-orless correspond to the early criteria for a
territory (review in Schoener, 1968) as a
defended area (Noble, 1939) or an exclusive area (Pitelka, 1959). The farther to
the left an animal is on any of these continua, the less territorial, in some sense,
that animal is. The models presented here
provide a framework for making precise
predictions about locations on such continua, and thereby about operationally
defined degrees of territoriality.
283
items or places preferred, there would be
a diminishing return per feeding time for
the defender as more and more intruders
depleted a given territorial area, because
the defender would find it optimal to
include poorer food types in its diet or
include poorer patch types in its itinerary.
How this phenomenon can be modelled is
suggested elsewhere (Schoener, 1971 and
in preparation). Second, intrusion rate may
diminish with the standing crop of intruders, as also suggested elsewhere (Schoener,
1971, 1977, 1983). The simplest assumption for this situation (Schoener, in preparation) is that intrusion rate declines in
proportion to the food on the territory (in
units of metabolic requirements), minus the
total energy requirements of the intruders
and the defender. The situation is discussed more fully above. Third, the
defender may be a time minimizer. Models
for this type are all rather simple unless the
first assumption, that food energy obtained
is not proportional to food energy on the
territory, is violated; the situation will be
dealt with in a future paper. Fourth, the
deterministic models such as those presented above might be stochasticized in the
manner of McNair (1987) and Lima (1984),
doubtless with unanticipated results.
ACKNOWLEDGMENTS
I thank G. Belovsky, L. Carpenter, C.
Clark, M. Hixon, S. Minta, J. Stamps and
E. Temeles for many useful comments on
a previous draft of this paper. Lisa Palermo
wrote the software linkages for the computer graphics and the optimization simulations. Supported by NSF Grant DEB 81 18970.
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The next theoretical steps
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OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
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main text. The positive solutions for Models 1 and 2
are:
Model 1
Model 2
APPENDIX 1
This appendix solves for the critical points of Models
1-4 and determines their nature.
Critical points of a function of two variables, here
ET(c, t), are obtained by determining those values of
c and t that simultaneously satisfy
dET
and
• 0
(Al)
= 0.
3t
dc
Call these values of c and t c* and t*. The nature of
these critical points depends on the sign of the eigenvalues of the matrix of the second partial derivatives:
/3q + V9q 2 + 8Z 2 JDC d q/M\ 2
4ZJD/M
j
(A6a)
c* = (3qM/(2JZ(D - qCd)))°5.
(A6b)
To determine the nature of these critical points,
we must evaluate A, B, and C in the matrix (A2).
General representations of the derivatives are:
= 2D
=
_ (x + 2)(x + l)qMcJt
- qCd(x + y - l)(x + y)C+>-2
B=
aTat=
(qM(x + 2 c X +
)
')/J t 2
C = ^ 7 = -(2qMc» +2 )/Jt 3 .
2
(A7a)
( A7b >
(A7c)
3
For Model 1, B - AC = qc M(4D - (3qcM/Jt))Jt3.
From
eq. A3a, we find that - 3 q c M / J t = - 2 D + qC d /
3 ET
c, and when this is substituted into the previous
2
2
dc
acat
(A2) expression, B — AC > 0. Hence, the critical point is
a saddle.
For Model 2, B2 - AC = 3q 2 M 2 /J 2 Z < c > 0 (Z =
dcdt
(qM/JC s ) 05 ). Again, the critical point is a saddle.
For Model 3, B2 - AC = 4qc 4 M((-2qc 2 M/Jt) +
A sufficient condition for a local minimum is that the D - 3qcC )/Jt 3 . From eq. A3, -3qcC = (4qc 2 M/
d
d
eigenvalues of eq. A2 are positive; a sufficient con- Jt) - 2D, so
that the sign of D2 — AC is negative if
dition for a local maximum is that the eigenvalues are D > 2qc 2 M/Jt, which is always true for E positive.
negative; a sufficient condition for a saddle is that one Finally, A + C < 0, so that the critical point Tfor Model
eigenvalue is positive and the other negative (Bryson 3 is a maximum.
and Ho, 1969). In terms of the elements of the matrix
For Model 4, B2 - AC equals 8qMc4((2qc2M/Jt) (A2), a local minimum occurs when B2 — AC < 0 and
s
2
A + C < 0; a local maximum occurs when B2 — AC < D)/Jt . 2Again from eq. A3, (2qc M/Jt) - D = -3qcC,,/
2,
so
B
—
AC
is
negative.
Again,
A + C < 0, so the
2
0 and A + C > 0; a saddle occurs when B — AC >
critical point for Model 4 is a maximum.
0 (Kaplan, 1959).
Finally, we show that for both saddles (Models 1
Evaluating eq. Al for eq. 2, we find
and 2), ET is never positive at the critical point. For
Model 1, substitute t* = Zc 15 into eq. 2 to get
(x + 2)qMc2
0 =
ac
•• 2 D c
-
-qC d (x + y)<
0 =
at
ET = c[Dc - c» 5 (qM/JZ + ZC,) - qC d ]. (A8)
Jt
_ qMc"
Jt2"
(A3a)
• - c,.
(A3b)
From eq. A3b,
t* = ±(Mq/C,J)°sc<1'+2><'2 = ±Zc" +2 > /2 .
(A4)
Substituting the positive value of eq. A4 into eq. A3,
0 = 2Dc -
Eq. A8 is negative if the term in square brackets is
negative. Substituting the critical value of c (eq. A6a)
into that term, gives after substantial manipulation
the value -q(9q 2 + 8Z 2 JDC d q/M)° 5 - 3q2 - 4C d q 2 D/
C,2, which is negative.
For Model 2, t* is identical to that for Model 1, so
eq. A8 is again the relevant one. The term in square
brackets reduces to (qM/2JZ) — C,Z, which equals
-1.
(x + 2)qMc*/2
- qCd(x
JZ
(A5)
For Models 1-4, eq. A5 is, respectively, an equation
of second degree in c° 5 (x = 1, y = 0), an equation of
first degree in c 05 (x = 1, y = 1), an equation of first
degree in c (x = 2, y = 1), and an equation of first
degree in c2 (x = 2, y = 2). The positive solutions for
Models 3 and 4 are given as eqs. 8a and 8b in the
APPENDIX 2
This appendix deals with optima on the processing
constraint. It shows 1) by implicit differentiation how
parameter variation affects c,^,; 2) that when the point
(Cop,, top,) is on the processing constraint, it is always
on the left-hand portion; and 3) that if the critical
286
THOMAS W. SCHOENER
value of c along the processing constraint has a critical
t that is positive, the pair of points constitutes a maximum.
Part 1. When the processing constraint (eq. 3) is
substituted into eq. 2, we get
(A9)
Differentiating eq. A9 with respect to c, setting it
equal to zero, and shifting q gives
dET
(x + 2)EPC,c-'-' - xC,Dc + 3
dc ~
j(Dc 2 - EP)VM
+
1
- (x + y)C >~ Cd = 0.
(AlO)
(All)
The equation for T 2 is rather horrendous:
T, = C,(3xDc2 - EP)
+ C"2(y - l)(x
dt
qxDc-+3 - qEP(x + 2)C+1
dc ~
J(Dc2 - EP)2
(A 13)
The roots of eq. A13 are
Eq. AlO does not always yield an easily solved expression for c, but it can be treated as an implicit expression. To determine how variation in some parameter
P affects c, we treat P and c as functions of a third
variable (say a) and differentiate eq. AlO with respect
to a. This results in an equation having two sets of
terms: those with dP/da and those with dc/da. The
equation is dP/da(T,) + dc/da(Tj) = 0, where T, and
T 2 are the two sets corresponding to factors of dP/
da and dc/da, respectively. Then
dc/dP = - T , / T 2 .
on the constraint line. Note that the argument just
given does not depend on whether the critical value
of c actually represents a maximum or not, provided
c is positive. However, Part 3 (below) shows that critical c is always a maximum.
Part 2. The processing constraint curve can be analyzed for critical points by taking the derivative (of
eq. 4) with respect to c:
4cDJCd(x + y)(Dc2 - EP)/M
y)(Dc2 - EP)2
(A 12)
Eq. A12 is always positive when on the constraint line,
since there Dc2 — EP > 0 (see eq. 3), (y — 1) is always
0 or 1, and (for x = 1 or 2) 3xDc2 - EP > 0. Hence
the sign of dc/dP depends entirely on T,. T, for
various parameters is obtained simply by differentiating eq. AlO with respect to the parameter. For
example, T, equals -xC,c» - 2JCd(x + y)C+'(Dc2 EP)/M < 0. So from eq. A1 1 dc/dD < 0, or the larger
the food density, the smaller the optimal territory size
c = ±[EP(x + 2)/xD]° 5 .
(A 14)
It is easy to show that the positive root represents a
local minimum (see Fig. 2). This minimum can be
shown greater than the optimal c for Models 1 and
4 (eqs. 10a, 10b) by setting the proper inequality and
rearranging terms.
Part 3. If we differentiate eq. AlO with respect to
c we obtain
1 d*ZT
f(c)
J(Dc2 - EP)>
(A 15)
q dc2
- (x + y)(x + y - l)c*+*-2.
For Dc2 > EP (c on the positive part of the processing
constraint) and x = 1 or 2, eq. A15 is negative if f(c)
is negative. For x = 1, f(c) = -2Dc'E P C, - 6EP2C,c <
0. For x = 2,
f(c) = -2c2C,(D2c4 - 3Dc2EP + 6EP2). (A 16)
Eq. A16 is negative if the term in parentheses is positive. Because 0 < (D2c4 - V6EP)2 = D2c4 2\/6E P Dc 2 + 6EP2 < D2c4 - 3Dc2EP + 6EP2, that term
is positive. Hence the second derivative is negative
for all critical values of c on the processing constraint
curve, implying that they represent local maxima. This
also implies, for Models 1 and 4, that the larger of
the two roots of eq. AlO (eqs. 10a, 10b) represent
local maxima, as c increases monotonically to infinity
along the constraint line past c = (E P /D) 05 .
287
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
APPENDIX 3
PARAMETER VALUES USED IN COMPUTER
SIMULATIONS
A. Main Sel
D
20,000.0
J(l/M)
10.0
101.0
q
1.0
c,
0.50
1.03
1,001.0
1.0
1,000.0
1.0
10.002
0.10
0.050
0.101
101.0
0.100
100.0
0.100
1.020
0.010
0.005
0.102
10.020
0.100
B. Other Sets
50.001
0.010
200.03
0.00502
p
1.
2.
3.
1.
2.
3.
4.
5.
10,000.2
10.02
10,000.1
10.002
0.005
200.01
0.0102
0.0005
10.001
0.00502
100.02
0.00103
50,000.3
5.003
5,000.3
5.001
0.0005
50.003
0.0051
0.000101
1.0
0.00103
50.001
0.00003
1,001.0
1.003
1,001.00
1.03
0.000101
10.002
0.00011
0.00002
C,
1.
2.
3.
4.
5.
100.002
1,000.1
10,000.2
1,000.2
10,000.3
50.003
500.003
5,000.3
500.1
5,000.1
10.001
100.002
1,000.1
100.003
1,000.2
APPENDIX 3.
CONTINUED.
A. Maill Set
c,
1.
2.
3.
?
1.
10.001
50.002
0.005
1,000.2
0.5001
10.001
0.0012
101.0
0.0503
0.5001
0.00053
10.002
In addition to all possible combinations from the main
set, the following simulations were performed—
parameters used from Main Set unless noted otherwise.
Patrol-defense-time constraint: Vary P (Models 3, 4)
P = Set 2, Set 3.
Defense-eating-time constraint: Vary C, (Models 1-4) C, =
Set 1; Vary C, (Models 2-4) C, = Set 2; Vary C,
(Model 3) C, = Set 3.
Full time constraint: Vary Q (Model 1) Q = Set 1, td =
Set 5, tf = Set 4; Vary Q (Model 1) td = Set 3, tf =
Set 3; Vary q (Model 1) td = Set 2, tf = Set 2; Vary
q (Models 1 -4) td = Set 1, tr = Set 1; Vary Cd (Models
1, 2) td = Set 3, tf = Set 3; Vary Cd (Models 1-4)
Cd = Set 1, td = Set 1, tf = Set 1, P = Set 1; Vary
Cd (Models 1-4) Cd = Set 2; Vary Cd (Models 1-4)
Cd = Set 2, td = Set 1, tf = Set 1; Vary C. (Models
3, 4) C, = Set 4, td = Set 1, tf = Set 1; Vary C,
(Models 1-4) C, = Set 1; Vary C, (Model 4) C, =
Set 5; Vary Cs (Models 3, 4) C, = Set 4. Vary C,
(Model 4) C, = Set 5, td = Set 1, tr = Set 1; Vary C,
(Model 4) C, = Set 5, td = Set 1, tf = Set 1, Cd =
Set 3; Vary td (Model 1) td = Set 4; Vary td (Models
1-4) td = Set 1; Vary td (Models 1-4) td = Set 1, tf
= Set 1.
* EP = P/t f .
288
THOMAS W. SCHOENER
APPENDIX 4
EFFECT OF INCREASE IN PARAMETER VALUES ON TOTAL DEFENSE TIME (MODELS l-4)f
1A. One-variable
models for c*
IB. One-variable
models for tb
2. Local maxima
(Models 3, 4)
3. Processingenergy constraint
4. Patrol-time
constraint
5. Defense-time
constraint
6. Feeding-time
constraint
7. Patrol-defensetime constraint*
8. Feeding-defensetime constraint*
9. Full time
constraint*
Energy/
time
cost of
patrol
Energy
cost of an
expulsion
(Ca)
Area of
detection
U)
Consumption of an
intruder
(M)
T
1
I
I
1
0
—
T
—
—
0
|
0
0
0
0
—
T
—
—
T
i
i
I
i
i
—
T
—
—
I
T
I
i
1
T
—
I
—
T
T
I
I
I
I
0
T
T
—
—
0
0
0
0
0
0
—
—
I
T
i
I
T
O f
—
I
—
T
IorT
I
I
1
T
I I
T
—
—
[
]
IorT
I
T
lor]
1#
T
i
—
i or T
I or 1
i or Tc
I or T
I or T
I or 1
T
ior 1
i o r Td
—
(CJ
Time cost
of an
Activity
perioa expulsion
(p)
(O
ProTime to cessing
eat a unit
capacity
of food
(W)
(E,)
Intruder
rate
(q)
Food
density
(D)
T
0
o r
t T = territory size increases; I = territory size decreases; 0 = no change; — = not in model; underline
means a: 85% of simulated cases—see text.
* Based on computer simulation.
# Can also go in opposite direction (for parameter values not in Appendix 3).
• Also increases with patrol time, t (fixed).
b
Also increases with territory size, c (fixed).
c
Except Model 1.
d
Only one example (Model 3) found.
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
289
APPENDIX 5
EFFECT OF INCREASE IN PARAMETER VALUES ON TOTAL FEEDING TIME (MODELS l-4)f
1 A. One-variable
models for c"*
IB. One-variable
models for tb
2. Local maxima
(Models 3, 4)
3. Processingenergy constraint*
4. Patrol-time
constraint*
5. Defense-time
constraint*
6. Feeding-time
constraint
7. Patrol-defensetime constraint*
8. Feeding-defensetime constraint*
9. Full time
constraint*
ProTime to cessing
eat a unit
caof food pacity
(tr)
<E P )
Energy
cost of an
expulsion
Area of
detection
(CJ
0)
1
lor T
T
1
0
—
—
T
—
T
1
0
T
1
1
—
—
T
—
T
1
I
T
1
1
—
—
T
—
0
0
0
0
0
0
—
—
T
T
T
1
l° r T
T
1
0
T
—
T
—
T
1
0
T
1
1
T
1
T
—
0
0
0
0
0
0
T
—
0
—
T
1
|or|
T
1
IorT
T
1
T
—
T
1
iorj
T
1
|orT
T#
1
T
—
Food
density
(D)
Intruder
rate
(q)
T
Consump- Energy/tinne
Time cost
tion of an
cost of
Activity
of an
intruder
patrol
perioa expulsion
(M)
(C.)
(p>
(U
|orT d
|orT
IorT
iorl
IorT
T IorT IorT —
t T = territory size increases; | = territory size decreases; 0 = no change; — = not in model; underline
means >: 85% of simulated cases—see text.
* Based on computer simulation.
# Can also go in opposite direction (for parameter values not in Appendix 3).
• Also increases with patrol time, t (fixed).
b
Also increases with territory size, c (fixed).
c
Except Model 4.
d
Only Model 4.
J'orJ
1(1)
0(1)
0(1)
1 ° r Tc
(q)
(D)
1
(CJ
1, 0 , o r T"
0)
Area of
detection
1*
(M)
Consumption
of an
intruder
0
(CJ
Energy/
time
cost of
patrol
(p>
Activity
period
(W)
Time cost
of an
expulsion
w
Time to eat Processsing
a unit of food
capacity
(E,)
—
—
—
—
0
—
1
i
1
T
—
—
—
2. Local maxima (Models 3, 4)
0
—
1
1
1
1
—
3. Processing-energy constraint
—
T* (] o r T)
0
—
1
—
T
1
i*
—
] or T*
4.
0
1, 0 , o r Td
—
Patrol-time constraint
1
1, 0, or Td
T(T)
1*
]orT"
—
f
Tor 0
]or 0'
0
5. Defense-time constraint
0(1)
1
1
1
1
—
—
6. Feeding-time constraint
0
]orl*
T* (1 o r T)
iorT
1
1
1
1*
ior T
7. Patrol-defense-time constraint*
lorl
—
TUorT)
lorl
1
iorl
1
T
lor T
lor I
8. Feeding-defense-time constraint* 1 or T (1 or T) l o r l
1 or T
lorl
iorl
1
lorl
lor T
lorT(iorT)
IorT
ior J
1 or T
IorT
9. Full time constraint*
lorl
iorl
l°rTg
=
t T territory size increases; j = territory size decreases; 0 = no change; — = not in model; underline means > 85% of simulated cases—see text.
* Based on computer simulation.
* Effect of patrol time, t (fixed), like effect of J.
b
Effect of territory size, c (fixed): Models 1, 2,1; Models 3, 4, no change.
' Model 4, 1 or T; Models 1-3, 1 only.
d
Model 1, T only; Model 2, no change; Model 3, 1 only; Model 4, 1 or Tc
Models 1,2,]; Models 3, 4, no change.
' Models 1, 2, T; Models 3, 4, no change.
» Only Models 2, 3, T-
1A. One-variable models for c*
IB. One-variable models for tb
Intruder
rate
Food density
Energy cost
of an
expulsion
parentheses; results for M exactly the opposite of those for J).
APPENDIX 6
EFFECT OF INCREASE IN PARAMETER VALUES ON EQUILIBRIAL INTRUDER DENSITY AND FRACTION OF F O O D CONSUMED BY INTRUDERS
(MODELS l - 4 ) f (Effects are the same except for D and M; for D effect on fraction of food consumed by intruders is given in
Z
o
2
&
X
CO
O
OPTIMIZATION MODELS FOR ENERGY MAXIMIZERS
APPENDIX 7
TABLE OF SYMBOLS
c2
c^
Cd
C,
D
Ep
ET
f
J
M
N
N*
P
q
t
td
tf
topt
T
x
y
Z
Territory area; c is proportional to territory
circumference and diameter
Optimal territory size
Energetic cost of defense, i.e., energy of expelling a single intruder per c*
Energetic cost per unit patrol time
Food density, in units of net food energy per
area
Maximum amount of food energy that can be
processed during T
Amount of energy gained by the defender during T
Fraction of encountered intruders that are
expelled
Area of detection of intruders per unit patrol
time
Food energy eaten by a single intruder during T
Number of intruders on a territory
N at intruder equilibrium
Activity period, measured as amount of time
available for activity during T
Intruder intensity; number of intruders invading a territory per C during T
Patrol time, measured as total patrol time during T
Defense time, i.e., time to expel a single intruder
Feeding time, i.e., total time spent in feeding
activities during T
Optimal patrol time
A large time period during which resources
renew on the territory
Power of c such that c" is proportional to intrusion rate
Power of c such that cy is proportional to energetic cost of expelling a single intruder
(qM/JC,)°5
291