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Brown,J.S.Patchuseasanindicatorofhabitat
preference,predationrisk,andcompetition.
Behav.Ecol.Sociobiol.
ARTICLEinBEHAVIORALECOLOGYANDSOCIOBIOLOGY·DECEMBER1987
ImpactFactor:2.35·DOI:10.1007/BF00395696
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Behavioral Ecology
and Sociobiology
Behav Ecol Sociobiol (1988) 22:37 47
9 Springer-Verlag 1988
Patch use as an indicator of habitat preference,
predation risk, and competition
Joel S. Brown*
Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, AZ 85721, USA
Received January 28, 1987 / Accepted September 25, 1987
Summary. A technique for using patch giving up
densities to investigate habitat preferences, predation risk, and interspecific competitive relationships is theoretically analyzed and empirically investigated. Giving up densities, the density of resources within a patch at which an individual
ceases foraging, provide considerably more information than simply the amount of resources harvested. The giving up density of a forager, which
is behaving optimally, should correspond to a harvest rate that just balances the metabolic costs of
foraging, the predation cost of foraging, and the
missed opportunity cost of not engaging in alternative activities. In addition, changes in giving up
densities in response to climatic factors, predation
risk, and missed opportunities can be used to test
the model and to examine the consistency of the
foragers' behavior. The technique was applied to
a community of four Arizonan granivorous rodents (Perognathus amplus, Dipodomys merriami,
Ammospermophilus harrisii, and Spermophilus tereticaudus). Aluminum trays filled with 3 grams of
millet seeds mixed into 3 liters of sifted soil provided resource patches. The seeds remaining following a night or day of foraging were used to
determine the giving up density, and footprints in
the sifted sand indicated the identity of the forager.
Giving up densities consistently differed in response to forager species, microhabitat (bush versus open), date, and station. The data also provide
useful information regarding the relative foraging
efficiencies and microhabitat preferences of the
coexisting rodent species.
can be used to investigate the properties of communities. A recent example of this includes Rosenzweig's (1979 et seq.) theory of habitat selection
where the behavior of individuals behaving optimally is used to predict the outcome of intra- and
interspecific interactions (Rosenzweig 1981, 1985;
Pimm and Rosenzweig 1981; Pimm et al. 1985).
Other examples include short-term apparent competition (Holt and Kotler 1987), moose foraging
dynamics (Belovsky 1978 et seq.), and resource
theory (Tilman 1982, 1985).
Testing models of population interactions
based upon the foraging behavior of individuals
often requires measuring habitat or patch use, habitat preferences, and the rejection or acceptance
of patches. In this paper, I present a measuring
technique to accomplish these goals. The technique
uses artificial or manipulated resource patches to
measure a forager's giving up density. The technique is applicable to communities of active foragers seeking comparatively immobile prey.
I begin by discussing the theory which provides
an interpretation of patch giving up densities. The
theory is an extension of Charnov's (1976) marginal value theorem and it uses what, in economics,
is called the "marginal rate of substitution" (see
Russell and Wilkinson 1979) to include the effects
of predation risk and alternative activities on patch
use. I then give some preliminary results from applying this method to a community of four desert
granivorous rodent species: Arizona pocket mouse
(Perognathus amplus), Merriam's kangaroo rat
(Dipodomys merriami), round-tailed ground squirrel (Spermophilus tereticaudus), and Harris's antelope ground squirrel (Ammospermophilus harrisii).
Introduction
As first suggested by MacArthur and Pianka
(~ 966) and Emlen (1966), optimal foraging theory
Department of Biological Sciences, University of Illinois at Chicago, Chicago, IL 60680, USA
* Current address:
Optimal patch use
Theoretical treatments of optimal habitat use are
applicable to systems where foragers are able to
identify and direct their foraging efforts to subsets
38
of the environment (i.e. patches) that on average
yield higher harvest rates or benefits than the environment at large. Once a habitat patch is located,
a forager must decide whether to accept the opportunity to harvest the patch (Rosenzweig 1974), and
it must decide how much time or effort to devote
to those patches it accepts for harvesting (Charnov
1976). By varying the foraging time allotted to each
patch, a forager can vary the total, average, marginal or net reward from each patch.
In the simplest case of habitat utilization, an
individual can only forage and has no alternative
activities. Assume that resources are distributed in
discrete patches and that foragers deplete the resources as they harvest a patch. However, assume
also, that the distribution of resources among
patches remains fixed. In other words, although
the resources of a given patch are depleted by a
forager while it is in the patch, the resources of
the entire environment are not. Under these assumptions, the marginal value theorem (Charnov
1976) states that the rate of energy gain to a forager
is maximized, and by assumption its fitness is also
maximized, when the forager's quitting harvest
rate in a patch equals its average harvest.
The above assumptions, which lead to the marginal value theorem, are restrictive. More realistically, foraging in patches will affect not only energy gain, but also other aspects of fitness such as
predation risk. Empirical work suggests that foragers balance the benefits of energetic reward and
the cost of predation when making foraging decisions (Milinski and Heller 1978; Sih 1980; Grubb
and Greenwald 1982; Werner etal. 1983; Lima
et al. 1985). Furthermore, most organisms can engage in other fitness-determining activities besides
foraging such as territorial defense, mate finding,
resting, dormancy, grooming, and nest maintenance. Finally, foraging activity of individuals may
not only depress the resources of a given patch
but also, at least for a time, depress the resources
of the entire environment. For example, following
sunrise, hummingbirds and nectarivorous insects
often deplete their nectar resources (Feinsinger
1976; Schaffer et al. 1979; Brown et al. 1981).
In what follows, I relax these restrictive assumptions (1) by assuming that foraging activity
affects both net energy gain and predation risk;
(2) by permitting the foragers to engage in alternative activities; and (3) by permitting the depletion
of resources over the entire environment.
Let fitness, denoted by G, be a function of net
energy gain from foraging, e, the probability of
surviving predation while foraging, p, and a vector
of inputs into fitness from engaging in any and
all alternative activites, a. I assume that the realization of fitness is discrete and occurs following a
fixed amount of time T. If the individual is preyed
upon during this interval its fitness is zero. The
goal of the forager is to divide its time between
foraging and alternative activities so as to maximize fitness.
Net energy gain, e, and the probability of surviving predation, p, are functions of time spent
foraging. Assume that time spent foraging is divided over m resource patches. Resource patches
may vary with regard to initial resource density
and predation risk. Harvest rate is assumed to be
an increasing function of patch resource density.
As a forager devotes harvest time to the patch,
its resources are depleted and the harvest rate declines. Let t I = ( 6 , " , tin) denote the vector of times
allotted to each of the m resource patches. Assume
that there are s alternative activities that contribute
to the other inputs into fitness, a. Let t " = ( t m + b ,
tm+s) be the vector of times devoted to each alternative activity. Note that alternative activities may
also expend energy and incur predation risk. For
accounting purposes, in what follows, I will include
these in the vector of inputs into fitness from alternative activities. Let
G [e (ts), p (t~), a (ta)] = p (t y) .f[e (ts), a (ta)]
denote the expected per capita rate of growth (fitness) subject to the constraint that the sum of the
tj's and t~'s equals T where j = 1,..., m and i = m +
1,..., m + s. Futhermore, 0 < p (t s) _<1 is the probability of surviving to realize fitness as a function
of times exposed to predators while foraging, and
F[e(ti), a(t")]>0 is the expected per capita rate
of growth given the individual survives predation
while foraging.
I assume that energy enhances fitness (i.e. ~ G~
~e> 0) and that spending time foraging decreases
the probability of surviving predation to realize
fitness (i.e. ~p/~tj<O where j = 1,..., m). If patch
j has a lower risk of predation than patch k then
~p/~ t; > ~p/~tk.
The optimal values of t~ and ta can be found
through the technique of Lagrange multipliers
(Chiang 1974). I will consider those patches and
alternative activities such that the optimal values
for tj and ti are greater than zero for j = 1,..., m,
and i = m + 1,...m+s. Setting the necessary condition for the optimal tj equal to the necessary condition for the optimal ti yields for all j = 1,.--,m and
i = m + l,...,m+s:
[~ G (')/~ el [~e (.)/8 tj] + [~ G (-)/~p] [~p (")/~tj]
= [~ G (')/~ a] [ea (.)/eh]
39
where (.) is shorthand for the functions' variables.
Rearranging yields:
~e(.)/~tj
_ IS a (.)/ 8 a] [8 a (.)/8 ti]
P
[~ a (')/8 e]
[~ a(')/~p] [~p(.)/~ tj]
[8 a ( - ) / 8 e]
(1)
Note that the derivatives of G and a with respect
to a and ti respectively are vector valued and the
inner product of these two vectors is a scalar.
The left side of equation (1) is the quitting net
harvest rate of an individual in patch j (quitting
harvest rate minus the energetic cost of foraging).
The quitting harvest rate which satisfies (1) is general. It applies to foragers which can engage in
alternative behaviors and incur a risk of predation
while foraging. The first term on the right side
of (1) is the change in the amount of alternative
activities per unit time weighted (multiplied as the
inner product of two vectors) by the ratio of the
marginal fitness of alternative activites, ~ G (-)/8 a,
and the marginal fitness of energy, ~ G(.)/8 e. The
second term is the change in predation risk incurred per unit time weighted by the ratio of the
marginal fitness of surviving predation, ~ G (.)/~p,
and the marginal fitness of energy.
The cost of predation and missed opportunity costs
When comparing such disparate inputs as energy,
predation risk, nest maintenance, and mate selection, what is the appropriate common currency?
Since energy, predation risk, and alternative activities jointly contribute to this fitness, each can be
considered as a surrogate for fitness (see Cheverton
et al. 1985). To see how one input can be converted
into the currency of the other, consider net energy
gain and predation risk. We ask the question, how
much additional energetic gain would it take to
get the individual forager to accept a higher risk
of predation? This exchange rate is called the marginal rate of substitution, MRS, of energy for predation risk (see Caraco 1979 for the use of MRS
in ecology). Figure I is an example of fitness isopleths in the state space of net energy gain and
probability of surviving predation while foraging.
Each isopleth gives the combinations of energy and
probability of surviving predation such that fitness
is held constant. An isopleth can be expressed as
a function, p (e), which relates the net energy gain
to the value of surviving predation which will hold
fitness constant. The points of the isopleth, given
e
Fig. 1. The relationship between net energy and predation risk,
and fitness. The x-axis is net energy gained from foraging (e);
and the y-axis is the probability of surviving predation while
foraging (p). Each isopleth gives the combinations of energy
and predation risk that yield the same fitness. The fitness of
R2 is greater than that of R1. The isopleths are concave to
the origin because of diminishing returns to fitness from increasing energy and because individuals which have acquired
abundant reserves of energy have more potential fitness to lose
from predation than individuals which have not. The negative
and reciprocal of the slope of any point along an isopleth is
the marginal rate of substitution of energy for predation risk,
MRSev (Caraco 1979)
by p(e), when substituted into the fitness function
must satisfy:
G [e, p (e), a ] = R
(2)
where R is some constant. The negative of the slope
of this isopleth gives the rate at which an individual
should be willing to exchange energy into safety
from predation risk (i.e. the MRS of predation
risk for energy). The MRS is the negative of the
slope of the isopleth. To find the slope of the isopleth take the derivative of expression (2) with respect to net energy gain, e, and rearrange to give:
8p (e)/8 e = - [8 G (-)/8 e]/[~ G (.)/Sp]
(3)
A similar argument can be made to show that
the slope of fitness isopleths in the state space of
net energy gain and alternative activities is given
by:
8 a (e)/~ e = - [8 G (')/~e]/[~ G (-)/8 a]
(4)
where a (e) defines the isopleth that gives the combinations of net energetic gain and alternative activities such that fitness is held constant.
The negatives of expressions (3) and (4) are
the MRS's of predation risk and alternative activities for energy respectively. The negatives and reciprocals of (3) and (4) are the MRS's of energy
for predation risk and alternative activities (Caraco 1979). These exchange rates go from units of
predation risk and alternative activities into units
of energy, MRSep and MRSea respectively:
40
MRSep
[8 G ( ")/ap]/[~ G (')/8 e] = F/p (8 F/~ e)
MRSea = [8 G ( ) / 8 a]/[8 a (.)/8 e]
=
(5)
(6)
By assumption, MRSep is always positive and
MRSea is vector valued giving the MRS's of energy
for the other inputs.
Note that in general these MRS's are complex
functions of energy gain, predation risk, and alternative activities. For instance, it may often be reasonable to assume that there are diminishing returns to fitness from increasing net energy gain
(i.e. the marginal fitness of energy may decline with
net energy gain). On the other hand, the marginal
fitness of surviving predation increases with net
energy gain; 82 G/~p ~ e = ~ F/O e > 0 (if the individual has more energy it has more to lose from being
eaten). Thus, as the individual acquires more energy the MRS~p increases.
We can now reconsider expression (1) which
gave the quitting net harvest rate of an optimally
foraging individual in any arbitrary patchj. Substituting the MRS's given by expressions (5) and (6)
into (1) gives:
e/~ t~= - MRSep [~p (-)/~ tj] + MRSea [8 a (')/8 ti]
(7)
The first term of the right hand side of expression (7) is the cost of predation expressed in units
of energy (P) and the second term is the value
of alternative activities expressed in units of energy
(MOC). The net quitting harvest rate can be split
between the rate of harvest, H, and the energetic
cost of foraging, C. Thus, the quitting harvest rate
satisfies:
H=C+P+MOC
A forager should leave each resource patch
when the patch harvest rate is no longer greater
than the sum of the energetic, predation, and
missed opportunity costs of foraging.
The above result comes from applying an optimization criterion to a general model of patch exploitation. For illustration, the Appendix considers
a specific example of a fitness function for an organism which can either forage in one of two
patches or remain dormant. From the example in
the appendix (or from careful consideration of the
general formulation), a number of deductions can
be made. First, if patches share the same risk of
predation, cost of foraging and harvest rate function, then they should be foraged to the same quitting harvest rate. Second, if fitness is only a function of net energy gain and there is no predation
risk, then each patch should be harvested until harvest rate just compensates for the additional cost
of foraging over remaining dormant. Third, if two
patches share the same energetic cost of foraging
and the same harvest rate function, then any differences in quitting harvest rates should reflect differences in patch specific risks of predation. Fourth,
if two patches differ in energetic costs and in risk
of predation in such a way that one is safer but
energetically more costly than the other, then the
forager's relative quitting harvest rates in the two
patches may reflect its state. If the forager is in
a state of high energy gain then it may have a
lower quitting harvest rate in the less risky patch.
The converse may occur if the forager is in a state
of low energy gain (see McNamara and Houston
1986 for how an individual's state influences behavior).
Measuring patch use
Researchers have used four techniques for measuring patch use. These include measuring giving up
times (Krebs et al. 1974; Hubbard and Cook 1978;
Townsend and Hildrew 1980), measuring total
time spent in a patch (Cowie 1977; Hartling and
Plowright 1979), quitting harvest rates (Pyke 1978,
1980; Milinski 1979; Hodges 1981), and the giving
up density of resources (Whitham 1977; Hodges
and Wolf 1981). These authors were interested in
determining foraging decision rules and seeing
whether models of optimal foraging provide good
characterizations of foraging behavior [see Krebs
et al. (1983) and Pyke (1984) and references therein].
Here, I will advocate: 1) the use of controlled
field experiments, 2) the use of giving up densities
as a measure of patch use, and 3) the use of foraging behavior to investigate predation risk, habitat
preferences, and interspecific competitive relationships.
Controlled field experiments using artificial or
manipulated resource patches offer two important
advantages. First, the foragers remain in their natural environment and are faced with familiar alternative activities, competitive interactions, and predation risks. Second, the use of artificial patches
permits the controlled manipulation of one or several variables of interest while the available set of
alternative activities is held constant.
If harvest rates are a function of patch type
and resource density then giving up densities
(GUD's) provide an estimate of quitting harvest
rates. In many circumstances it is easier to measure
G U D ' s rather than quitting harvest rates. Measuring GUD's requires assessing the remaining density of resources following use by one or several
41
foragers. On the other hand, measuring the quitting harvest rate requires an organism whose encounter and capture of prey can be observed and
timed. It is particularly difficult to measure the
instantaneous harvest rate of a forager as it leaves
a patch.
To validate the model, all four components of
the model can be tested. To test for the effect of
harvest rate (H) on G U D ' s requires holding C,
P, and M O C constant. Adjacent manipulated resource patches in the same microhabitat should
not differ in C, P, or MOC. Thus, G U D ' s in adjacent patches which differ in substrate should reflect
differences in harvest rates. The patch with the
'slower' substrate should have the correspondingly
higher G U D which just equalizes the quitting harvest rates in the two patches.
To test for the effect of energetic costs (C) on
G U D ' s requires holding H, P, and M O C constant.
For endotherms, the metabolic costs of foraging
should be influenced by temperature when ambient
temperatures are below the forager's thermal neutral zone. Adjacent patches with the same substrate
should not differ in H, P, or MOC. If one patch
is rendered colder then it should have the higher
G U D which just compensates the forager for its
higher energetic cost of foraging in that patch.
Similar tests can be used to test for the effects
of P and M O C on GUD's. Increasing P (e.g. by
manipulating predator densities or cues of predation) should increase GUD's. Similarly, increasing
the M O C (e.g. by providing alternative resources
or foraging opportunities) should increase GUD's.
By manipulating patches in ways which are known
to increase predation risk, harvest rate, foraging
costs, or missed opportunity costs, the researcher
can test whether the forager's behavior is consistent with that required to maximize G (-).
The model can also be used to allow the forager
to reveal its preferences and assessments of the environment. The researcher assumes that the forager's G U D is a truthful revelation of its harvest
rate and foraging costs. Temporal or spatial differences in the G U D ' s of a forager reflect the effects
of different habitats on missed opportunity costs,
harvest rates, foraging costs, and predation risk.
The researcher can investigate species and habitat
specific differences in any one of these costs by
holding other costs constant among habitats.
Missed opportunity cost can be controlled for
by ensuring that several patches are available to
the same forager. If these patches are within a relatively short distance of each other, then while foraging in either patch the forager has the same set
of alternative activities and thus, experiences the
same missed opportunity cost in each patch. Harvest rate can be controlled by ensuring that the
structure of the artificial patches is the same with
regards to factors that affect harvest rate, such as
substrate and resource type. The energetic cost can
be controlled by maintaining constant climatic factors across patches. Conversely, different combinations of abiotic factors such as temperature, humidity, or wind can be used to test their effects
on foraging costs. Differences in the cost of predation can be measured if patches are manipulated
in such a way that energetic costs, harvest rates,
and missed opportunity costs remain constant.
Methods
I applied the approach of the previous section to a community
of desert granivorous rodents in September 1983. The study
site is located near Tucson, Arizona (see Brown 1986). The
dominant perennial plant species in descending order of
groundcover are creosote (Larrea tridentata), desert zinnia (Zinnia sp.), mouse ears (Coldenia canescens), and mesquite (Prosopsis juliflora). Total vegetation ground cover by perennials is
about 20%. There are four granivorous rodent species (in parentheses : numbers and mean weights of individuals trapped):
round-tailed ground squirrel, Sperrnophilus tereticaudus
(18/121 g); Merriam's kangaroo rat, Dipodomys merriarni
(16/37 g); Arizona pocket mouse, Perognathus amplus
(55/12g); and antelope ground squirrel, Ammospermophilus
harrisii, (12/104 g).
Aluminum trays, measuring 45 cm on a side and 2.5 cm
deep, filled with 3 g of unhusked-millet seed mixed into 3 l
of soil provided resource patches. A total of 60 trays were
divided over two grids. Each grid was laid out as a seven by
seven small mammal trap grid with stations 25 m apart. The
thirty seed trays on each grid were divided into pairs and assigned to 15 stations picked at random from the 49 trap stations. Live trapping and seedtrays were not run simultaneously.
At each station with seedtrays, one tray was placed directly
under the canopy of a creosote bush and the other was placed
2 4 m away from the first in the open microhabitat. Although
shadowed by the canopy, the surface of trays under shrubs
was unobstructed by branches or leaves up to a height of no
less than 20 cm. The trays in the open were placed on bare
ground at least 2 m from the nearest creosote bush.
For seven mornings (9, 10, 11, 14, 15, 17, and 18 September) and afternoons (9, 10, 11, 13, 14, 15, and 17 Sept.),
I collected foraging data from the seed trays (rain prevented
consecutive mornings and afternoons). The morning data collection was at sunrise, a period following the cessation of activity by pocket mice and kangaroo rats and preceding the initiation of activity by ground squirrels. The afternoon data collection at sunset followed squirrel activity and preceded that of
pocket mice and kangaroo rats. Thus, pocket mice and kangaroo rats had one entire night to forage the trays and squirrels
had one entire day. (The rodents were very familiar with these
seed trays from similar work done in the latter part of August.)
Data collection from trays consisted of noting any footprints in the sired soil, sifting the soil to recover the remaining
seeds, and recharging the trays with millet. The distinctiveness
of footprints permitted identification of the forager down to
species and sometimes to the exact individual based upon toeclips. The squirrels could be distinguished by the presence
42
Table 1. The mean G U D ' s (in grams) for the pocket mouse
Table 2. The results of two one-way ANOVA's (separate analy-
(P.a.), kangaroo rat (D.m.), and squirrel species (S.t. and A.h.)
ses for the bush and open microhabitats) showing the differences between species in GUD's. The group variables are the
four species and G U D is the dependent variable (log transform
of grams of millet). The column and row headings are the four
species (abbreviated scientific names). Entries are the F of improvements for each pairwise comparison. Entries above and
below the diagonal are for the bush and open microhabitats
respectively. The error mean sum of squares for the bush and
open analyses are 0.646 and 0.602 respectively. Because the
six pairwise comparisons within each analysis are not orthogonal, the error rate of each test was adjusted according to the
Dunn-Sidak method (Sokal and Rohlf 1981)
in the bush and open microhabitat. Bush/Open preference is
determined by a sign test comparing the number of times the
bush tray at a station had the lower G U D (number preceding
the comma) to the number of times the open tray had the
lower G U D (number following the comma). A " B " or " O "
indicates whether the bush or open had the significantly lower
GUD
Species
Bush
GUD
Sample
size
Open
GUD
Sample
size
Bush/open
preference
P. amplus
D. merriami
A. harrisii
S. tereticaudus
0.439
0.952
0.994
0.974
121
51
39
97
0.591
0.610
1.591
1.571
103
86
32
89
51,35
14,52
36,3
81,14
B*
O***
B***
B***
* P < 0 . 0 5 , *** P < 0 . 0 0 1
Species
P.a.
D.m.
S.t.
A.h.
P.a.
D.m.
S.t.
A.h.
-
33.17"**
64.95***
35.54***
52.82***
0.03
0.01
30.38***
0.06
0.02
-
0.08
75.73***
39.71 ***
*** P<0.001
(round-tailed squirrel) and absence (Harris's antelope squirrel)
of tail drags. Even when no or a few seeds were removed,
footprints provided p r o o f of encounter. In some cases the footprints of two rodent species were visible. In this case the foraging was attributed to both species. I attributed the giving up
density of a tray to a single rodent species if its were the only
detectable tracks even though it may not have been the only
species present in the tray that day or night. In the case of
squirrels, forages ascribed to Harris's antelope squirrels may
also include activity by the other squirrel species.
The seeds recovered from seedtrays were cleansed of debris
and weighed to measure the giving up density. I recorded an
encounter and rejection if the tray had footprints and if fewer
than 0.2 g of seeds were removed. The 0.2 threshold is approximately the number of seeds which can be harvested by simply
collecting seeds on the surface without digging or intensive
searching. Once discovered by a forager, a seedtray was invariably visited on following nights or days.
Results
The data are the giving up densities (grams of millet; millet has a mean weight of 0.0066 g per seed)
and they are specified by date, station, microhabitat, and species. The results are summarized in Table 1 as the mean G U D ' s by rodent species and
microhabitat. Mean G U D ' s are for trays foraged
by a single species. The following analyses were
performed: i) differences between species in
G U D ' s within each microhabitat, ii) differences in
G U D ' s between microhabitats for each species, iii)
correlation between bush and open G U D ' s at a
station for each species, iv) propensity for each
species to reject the opportunity to forage a tray,
and v) the effect of date and site on G U D ' s in
each microhabitat. Because the distributions of
G U D ' s are skewed to the right, analyses were performed on logarithmically transformed data (mean
G U D ' s are reported as the backtransformation of
the logarithmic means).
One-way A N O V A ' s were used to examine differences between species in each microhabitat (Table 2). The G U D ' s of the two squirrel species did
not differ in either the bush or open microhabitat.
In the bush microhabitat, pocket mice had a lower
G U D than either kangaroo rats or squirrels. In
the bush microhabitat there was no significant difference between the G U D ' s of kangaroo rats and
the two squirrel species. In the open microhabitat
both pocket mice and kangaroo rats had lower
G U D ' s than squirrels. In the open microhabitat
there was no significant difference between the
G U D ' s of pocket mice and kangaroo rats. In summary, independent of microhabitat pocket mice
had the lowest and squirrels had the highest
GUD's.
A sign test comparing G U D ' s between trays
at a station was used to test for differences between
microhabitats. The round-tailed and antelope
ground squirrels, and the pocket mice had lower
G U D ' s in the bush trays than in the open. Kangaroo rats had a lower G U D in the open microhabitat (Table i).
For each species, a Spearman Rank test was
used to test for a correlation between G U D ' s in
the open and bush trays at a station. All species;
squirrels (the two squirrel species were combined
for this analysis), kangaroo rats, and pocket mice;
had a positive correlation between the G U D ' s of
pairs of trays (r~=0.576, P<0.001 ; r~=0.682, P <
0.001; rs=0.619, P < 0 . 0 0 1 respectively). Because
of possible differences in G U D ' s between days, I
ran this correlation separately for each day. For
43
Table 3. The results of six random-block ANOVA's show the
effect of date subset of 9 18 Sept. and microhabitat (bush and
open) on GUD's (in grams, log transform) of each species (the
two squirrel species A. h./S. t., have been combined for this
analysis). The columns include F of improvement, degrees of
freedom (dr), and mean sum of squares, MS, of the error term
Species
Date
F (df)
Station
F (df)
Error
MS (df)
P. amplus
12.00"** (5)
2.50* (8)
0.400(40)
21.76"** (3)
0.35 (5)
0.292 (15)
0.48 (2)
1.37 (4)
0.268 (8)
0.32 (4)
1.66 (5)
0.211 (20)
10.26"** (4)
4.66*** (16)
0.222 (64)
3.40* (3)
5.32*** (12)
0.195 (48)
(Bush)
P. amplus
(Open)
D. merriami
(Bush)
D. merriami
(Open)
A. h./S. t.
(Bush)
A. h./S. t.
(Open)
* P<O.05, ** P<O.01, *** P<O.O01
each species, all seven days yielded a positive correlation. At a station, the G U D in the bush tray
is a good predictor of the G U D in the open
tray.
A G-test of heterogeneity was used to examine
differences between pocket mice and kangaroo rats
in their propensity to reject the opportunity to forage in one of the trays of a pair. A double forage
occurs when both trays at a station are foraged
by the species and a single forage occurs when
one tray is foraged but the other tray is rejected.
Kangaroo rats (46 double and 19 single forages)
were more likely than pocket mice (85 double and
3 single forages) to forage at only one tray of a
pair (Gh----20.79 with William's correction, P <
0.001). Of the 19 trays in which kangaroo rats rejected the opportunity to forage, 18 were in the
bush microhabitat.
Two-way random-block ANOVA's were used
to test for the effect of station (space) and date
(time) on G U D ' s in each microhabitat. To permit
consideration of each species and microhabitat, six
separate analyses were performed (Table 3). For
each species, as many dates and stations as possible
were used while still maintaining a value in each
cell. Both station and date affected the G U D ' s of
squirrels and pocket mice in the bush microhabitat.
In the open microhabitat, station and date affected
the squirrels' GUD's, but only date affected the
pocket-mice's GUD's. Neither station nor date affected the kangaroo-rats' G U D ' s in either microhabitat.
Discussion
One assumption of optimal foraging theory is the
simple axiom that an individual should engage in
an activity until the benefits and the costs are
equal. Identifying the components and fitness
values of these costs and benefits presents a significant challenge. The seemingly insurmountable
complexity of ecological systems has led to the criticism that models of optimal foraging ignore other
relevant activities besides foraging, and ignore
other relevant inputs into fitness besides energy
(but see Tilman 1982; M c N a m a r a and Houston
1986).
Here, I present a simple cost-benefit analysis
of foraging that incorporates any number of alternative activities and permits any number of inputs
into fitness. By considering the marginal rates of
substitution of different inputs into fitness, all inputs can be expressed as a single currency. Here,
I advocate using the forager's giving up density
in resource patches to gain insights into the energetic cost, predation cost, and missed opportunity
cost of foraging. When harvest rate is an increasing
function of patch resource density, G U D is an estimate of the harvest rate which balances the costs
of foraging. Laboratory or field measurements of
patch harvest rates permits conversion of G U D ' s
into harvest rates.
Without sacrificing environmental complexity,
it is possible to gain insights into species interactions, habitat preferences, patch use versus rejection, and species-specific foraging differences. The
data collected for desert rodents provide an example. In the experiment, except for microhabitat,
trays at a station were identical and the same individual had the opportunity to forage in both trays
(travel time between trays was on the order of seconds). Thus, at a station the cost of foraging, C,
and the missed opportunity cost, MOC, should
have been the same for both trays. Any differences
in G U D ' s between the trays were therefore the result of differences between microhabitats in predation risk; kangaroo rats perceived greater predation risk in the bush microhabitat; squirrels and
pocket mice perceived greater predation risk in the
open.
While many models of habitat selection predict
the acceptence and rejection of resource patches
(Fretwell and Lucas 1970; Rosenzweig 1974,
1979), tests of optimal habitat selection have measured the proportional use of patch types (Heinrich
1979; Lewis 1980; Mittlebach 1981; Baharav and
Rosenzweig 1985; Pimm et al. 1985). The experiment of this paper demonstrates that GUD's pro-
44
vide information both on the proportional use of
patch types and on the acceptance and rejection
of resource patches. It was possible to show that
kangaroo rats were much more likely to reject the
opportunity to forage a tray than were pocket
mice. The approach is suitable for testing models
of habitat selection which make predictions concerning both patch acceptance and use (Brown and
Rosenzweig 1986).
The ability to distinguish between the foraging
bouts of the different species based on footprints
makes the technique suitable for investigating differences among species. GUD's, besides being a
measure of harvest rate, are also a measure of foraging efficiency (Tilman 1982; Vance 1985; Brown
1988). The results showed that pocket mice were
the most efficient foragers in both microhabitats.
If the conditions pertaining to September were
maintained throughout the year and if foraging
efficiency on seeds was the sole determinant of
competitive ability then pocket mice would outcompete both the squirrels and the kangaroo rats.
Is the forager's behavior consistent
with fitness maximization ?
Up to this point in the discussion it has been assumed that the foragers are behaving optimally
and that the G U D is a truthful revelation of the
forager's perceived costs and benefits. Insofar as
it is possible to control for the various costs of
foraging it is possible to test directly whether the
forager's behavior is consistent with fitness maximization. Increasing the energetic cost, predation
cost, and missed opportunity cost of foraging
should result in higher GUD's. While these would
be worthwhile experiments, they were not performed in the study reported here. However, inferences regarding foraging consistency with fitness
maximization can still be drawn from the data.
At a station, independent of other differences,
the missed opportunity cost should be the same
for both the open and bush tray. As a result of
individual differences (the footprints of a single
individual were rarely found at more than one seed
tray station) and spatial variability between stations in natural resource abundances and predation risk, missed opportunity costs and the marginal valuation of energy should vary between stations. Thus, the GUD's in the open and bush microhabitats at a station should be positively correlated. The results indicate that all of the rodent
species exhibit a positive correlation (see Results).
Spatial variation in natural resources, predation risk, and edaphic characteristics should result
in relatively long term differences in foraging costs
between stations. Similarly, on a daily and nightly
basis, there should be temporal variation in climate
and predation risk. Thus, in both microhabitats
there should frequently be consistent differences
in G U D ' s between stations (spatially) and between
days or nights (temporally). For squirrels and
pocket mice (but not for kangaroo rats), the results
indicate significant effects of space and time on
GUD's.
Problems with the approach
The use of manipulated resource patches presents
at least four problems: i) the patches are not natural, ii) the resource may not be appropriate, iii)
the foragers may become satiated, and iv) the trays
may be visited by more than one forager. The first
two problems concern the realism of the foraging
situation and the last two concern possible distortions of the results by the foragers.
How accurately the artificial patches mimic
natural resource patches depends upon the questions under investigation. The type of resource and
the substrate used in the patch will influence both
the species that participate in the experiment and
their GUD's. If the questions of interest concern
differences between microhabitats or time periods
in predation risk or missed opportunity costs then
the realism of the substrate and resource are relatively unimportant. However, when the significant
parameters involved in regulating habitat selection,
predation risk, and interspecific interactions include substrate and resource type then the choice
of experimental resource patch is important and
will influence the results.
The presence of artificial resource patches may
distort the normal behavior of foragers. If foragers
devote time to the artificial patches, they will have
less time for alternative activities and they may
harvest more resources. The marginal fitness of
alternative activities should increase as less time
is spent on alternative activities, and the marginal
fitness of energy should decrease as more resources
are harvested. Together, these two effects should
increase the missed opportunity cost of foraging
above what it would be in the absence of artificial
patches. Any distortion is a concern for investigations where the G U D must reflect the animals,
normal missed opportunity cost of foraging. Fortunately, it is possible to measure this distortion.
The distortion in missed opportunity cost is
the result of the foragers taking time from alternative activities to forage in the artificial patches.
As the initial density of resources in a patch in-
45
creases, the time required to forage the tray to
the same giving up density also increases. Thus,
increasing the a m o u n t o f resources in artificial
trays increases the u p w a r d distortion o f missed opp o r t u n i t y costs. The m a g n i t u d e o f the distortion
can be examined by measuring the change in G U D
that results f r o m increasing the resources available
to a forager or g r o u p o f foragers. I f there is a
range o f initial resource availabilities that does n o t
influence the G U D then over this range o f experimental conditions the seed trays are n o t significantly distorting the missed o p p o r t u n i t y cost. Mitchell and B r o w n (ms.) showed that for up to 12 g
o f millet per station, the G U D ' s o f k a n g a r o o rats
( D . m e r r i a m i ) remained constant. The experiment
reported here used only 6 g o f millet per station
a n d no distortion o f missed o p p o r t u n i t y costs
should have occurred.
I f several individuals o f the same species forage
in the p a t c h then the researcher only measures the
G U D o f the last forager. Thus, m a n y foragers per
p a t c h m a y lead to a biased sampling o f the species'
G U D . Individuals with low G U D ' s will be overrepresented in the data.
Comparison
to other techniques
Live-trapping is a frequently used m e t h o d for investigating habitat selection, p r e d a t i o n risk, a n d
competition. Because live-trapping is necessarily
an all or n o t h i n g m e a s u r e m e n t it is a less sensitive
measure o f behavior t h a n G U D ' s . Live-trapping
distorts the animal's o r d i n a r y activities; once
trapped, it is p r e v e n t e d f r o m engaging in other activities. O n the o t h e r hand, G U D ' s in artificial
patches need not distort o r d i n a r y activity, at any
time the forager is free to engage in other activities.
O t h e r researchers have used artificial resource
patches to gain insights into m a m m a l i a n foraging
b e h a v i o r ( B r o w n 1971; M a r e s and Rosenzweig
1978; Frye a n d R o s e n z w e i g 1980; A b r a m s k y 1983;
Vickery 1984). H o w e v e r , their previous use has n o t
included e m b e d d i n g the resource in a substrate
[but see Schneider (1984) for G U D ' s in artificial
patches a n d see W h i t h a m (1977), a n d H o d g e s and
W o l f (1981) for G U D ' s in u n m a n i p u l a t e d patches].
W h e n the resources are free standing there is little
or no search time a n d so harvest rate does n o t
decline with resource density. I f the missed o p p o r tunity cost o f foraging remains undistorted, the
C U D in the p a t c h is zero a n d p a t c h use should
be all or n o t h i n g (the entire p a t c h is treated as
a single resource). I f there is a giving up point
in the p a t c h it is because the missed o p p o r t u n i t y
cost o f foraging is being distorted u p w a r d s as the
forager harvests resources. In all respects, m a n i p u lated resource patches with minimal or no search
time are less sensitive measures o f b e h a v i o r than
patches where the harvest rate is a decreasing function o f p a t c h resource density.
Acknowledgements. This work is in partial fulfillment of the
requirements for a doctorate in Ecology and Evolutionary Biology at the University of Arizona. The manuscript and the ideas
herein benefited greatly from comments by and discussions with
Burr Kotler, William Mitchell, Jean Powlesland, Sandra Rode,
Thomas Valone, and three anonymous reviewers. I thank my
committee, James Brown, James Cox, Robert Holt, Thomas
Vincent, and David Vleck for their input and guidance. Special
thanks to my advisor Michael Rosenzweig. I am grateful to
Hubert Markl for his scientific and editorial comments. The
Graduate Program Improvement Fund from the Graduate College of the University of Arizona and the Mitrani Center for
Desert Ecology (Blaustein Institute, Ben-Gurion University,
Sede Boqer, Israel) provided financial assistance. This is publication #63 OFTHEM1TRANICENTER.
Appendix
Consider a forager who alots a certain amount of time, T,
between foraging in one of two patches (tl and tz are times
spent foraging in each patch) or remaining dormant in its nest
(ta is time spent dormant). While foraging in patch 1 or 2 the
individual incurs a constant per-unit-time risk of predation
(rl or r2), harvests energy, and expends a constant rate of energy (c~ or c2). While dormant the individual incurs negligible
risk of predation, expends energy (ca), and acquires 'maintenance' (m). Maintenance can be thought of as an input into
fitness resulting from the benefits of grooming, burrow maintenance, or offspring care. I assume that the energetic cost of
dormancy is less than the energetic cost of foraging; ca < cl, c2.
For many organisms, dormancy provides a valuable alternative
activity either when the organism has a lot to lose from being
preyed upon or when resource abundances and harvest rates
are low.
Let the fitness function be given by:
G(t~,t2,ta)=p(tl,t2) {[1 -- E X P ( - - m ( t d ) )][1 + fle(t~,tz,ta)]} ~
where 0 < ~ < 1, and fl > 0 are constants (henceforth, when convenient, probability of surviving predation (p), maintenance
(m), and energy gain (e) will be written without their arguments). The fitness function assumes that there are diminishing
but positive returns to fitness from energy gain (c~<1), and
that the fitness value of energy is weighted by an efficiency
term, 1 - E X P ( - m ) , whose value increases with maintenance
performed by the individual.
The population dynamics of individuals using a particular
strategy of time allocation is: N(7) = G(-) N(0) (G(-) > 1 or
G(-)<I determines whether population size is increasing or
decreasing). To insure that G(.) is greater than zero assume
that G(.) is positive even if the individual harvests no resources;
i.e. 1-flTci>O for i=1,2.
Let probability of surviving predation, maintenance, and
energy gain be:
p - EXP[--rltl-rzt2]
(A.la)
m=yta
(A.lb)
e = Hi(t1) + H2(t2) -- c1t I -- c2t 2 -- catd
(A.lc)
where Hi(h) is the cumulative harvest rate in patch i= 1,2.
46
To solve for the optimal patch quitting harvest rate, first
calculate the MRSev and MRSem using expressions (5) and (6) :
MRSe, : (1 +fle)/~flp
[1 + f l e ] E X P ( - m )
MRSem = fl[1 - E X P ( - m)]
(If tl*, t2*, td* >0, the optimal values for time spent in patch
1, tl*, and 2, t2*, must satisfy condition (7).) Take the derivatives of (A.la-c) with respect to tl or td and substitute these
and the MRS's into (7) to yield:
aHi(.)
r~[l+fle]
at, - c ~ + ~ +
7[l+fle]EXP(-m)
fl[1--EXP(-m)]
ed
(A.2)
where i = 1,2. The optimal allocation of time can be obtained
by solving three simultaneous equations. The first two generated by (A.2) and the third generated by the constraint: tl +
t2 + ta = T.
The term on the left is the quitting harvest rate, H. The
first term on the right is the energetic cost of foraging in the
patch, C. The second term is the cost of predation, P. And,
the iast two terms are the missed opportunity costs, MOC.
The first component of MOC is the foregone benefit of additional maintenance, and the second is the foregone cost of dormancy.
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