F. R. Adler and D. M. Gordon.

15 Pages·2003·0.29 MB·English

vol. 162, no. 5 the american naturalist november 2003
Optimization, Conflict, and Nonoverlapping
Foraging Ranges in Ants
Frederick R. Adler1,* and Deborah M. Gordon2,†
1.DepartmentofMathematicsandDepartmentofBiology, (Ho¨lldobler and Lumsden 1980). Even withoutactivede-
UniversityofUtah,SaltLakeCity,Utah84112; fense,organismsmaypartitionspaceintononoverlapping
2.DepartmentofBiologicalSciences,StanfordUniversity,
foraging ranges to avoid exploitation competition. Ex-
Stanford,California94305-5020
ploitationcompetitioncandeterintrudersifanindividual
SubmittedOctober29,2002;AcceptedApril7,2003; can systematically exploit resources in its foraging range,
ElectronicallypublishedNovember6,2003 thus reducing the mean availability of resources and re-
movingattractivehigh-densitypatches(DaviesandHous-
ton 1981; Possingham 1989). Overexploitation of re-
sources near the boundary could also act as a deterrent
tointruders(PatonandCarpenter1984;LucasandWaser
abstract: Anorganism’sforagingrangedependsonthebehavior
1989).
of neighbors, the dynamics of resources,andtheavailabilityofin-
formation.Weuseawell-studiedpopulationoftheredharvesterant Space partitioning is determined by the ways that in-
Pogonomyrmexbarbatustodevelopandindependentlyparameterize dividuals allocate effort within a territory or foraging
models that include these three factors. The models solve for an range. Understandinghowtheseindividualdecisionslead
allocation of foraging ants in the area aroundthe nest inresponse
to partitioning of space in particular cases has proven
to other colonies. We compare formulations that optimize at the
difficult. As Adams (2001) points out in a recent review,
colonyorindividuallevelandthosethatdoordonotincludecosts
few theoretical models of space partitioning have com-
ofconflict.Modelpredictionswerecomparedwithdatacollectedon
anttimebudgetsandantdensity.Thestrategythatoptimizesatthe binedthemanyrealitiesthatchallengeorganisms:distance
colony level but neglects costs of conflictpredictsunrealisticlevels (Harkness and Maroudas 1985; Holder Bailey and Polis
ofoverlap.Incontrast,thestrategythatoptimizesattheindividual 1987), familiarity and information(StampsandKrishnan
level predicts realistic foraging ranges with or withoutinclusionof
2001),resourcedistribution(CristandHaefner1994),and
conflictcosts.Boththeindividualmodelandthecolonymodelthat
risks from conflict with neighbors and predators (Ham-
includesconflictcostsshowgoodquantitativeagreementwithdata.
ilton 1971).
Thus,anoptimalforagingresponsetoacombinationofexploitation
andinterferencecompetitioncanlargelyexplainhowindividualfor- In group foraging species, as in individually foraging
aging behavior creates the foraging range of a colony. Deviations ones, individual behaviors create emergent patterns.
between model predictions and data indicate that colonies might Rather than considering how space is partitioned among
allocate a larger than optimal number of foragers to areas near
groups,thisliteraturehasconcentratedonhowgroupsize
boundariesbetweenforagingranges.
and composition is determined by the costs and benefits
Keywords:foragingranges,competitionforspace,gametheory,op- of different behaviors (Hake and Ekman 1988; Livoreil
timizationmodels,territoriality. and Giraldeau 1997). Foragers must address a set of con-
flicting demands similar to those faced by individual for-
agers, including trade-offs between foraging success and
How can we explain the ways that organisms partition predation risk (Hamilton 1971; Poysa 1987), the use of
space? In the most visible cases, boundaries between ter- information (Smith et al. 2001; TempletonandGiraldeau
ritoriesareactivelydefended.Individualsmayinsteadde- 1995), and the quality of information (Valone 1993) and
fendresourcesorattackintruderswithinaforagingrange spatial effects (Ruxton 1995).
Models of territorial behavior or group foraging can
* E-mail:[email protected]. seektoexplainspatialpatternsbyfindingwhichindividual
† E-mail:[email protected]. behaviorsareconsistentwithpatternsobservedatthecol-
Am.Nat.2003.Vol.162,pp.529–543.(cid:1)2003byTheUniversityofChicago. ony and population scales or by deriving individual be-
0003-0147/2003/16205-20398$15.00.Allrightsreserved. haviorsusingoptimalforagingtheory.Understandinghow
530 The American Naturalist
optimal foraging behavior, withininformationalandcog- pendsonthesizeofthecolony,theforagingarea,andthe
nitive constraints, leads to space use requires developing distance from the nest, Adams’s (1998) model accurately
spatially explicit models that make testable predictionsat predicts areas and shapes of territories. However, models
boththeindividualandpopulationlevel.Previousstudies of this type make no predictions about the behavior or
have addressed organisms ranging from birds (Hake and locations of individuals (or roots in the case of plants),
Ekman1988)andbees(DukasandEdelstein-Keshet1998) assume that foraging ranges do not overlap, and are not
to wolves (Lewis and Murray 1993) and trees (Lopez et based on an explicit cost-benefit analysis.
al. 1994). Lewis and Murray (1993) and Lewis et al. (1997) de-
We use a well-studied population of the red harvester veloped a model of display behavior that predicts the
ant Pogonomyrmex barbatus (Gordon 1999) to build and shape,size,anddegreeofoverlapbetweenwolfterritories.
parameterize models that include the behavior of neigh- Their model derives the locations of boundaries and the
bors, resource dynamics, and a simple model of infor- degreeofoverlapbut,likepressuremodels,assumesmove-
mation sharing. The models find foraging ranges as the ment rules and has no measures of the costs andbenefits
solution of a game that takes into account the responses of different strategies.
of neighbors, and the models predict the location of A first step in building spatially explicit models of op-
boundaries between the foraging ranges of neighboring timizing competitors was Maynard Smith’s (1982) one-
colonies,thedegreeofoverlap,andthespatialdistribution dimensionalgametheoreticmodel,whereeachindividual
of foragers. We test our models with observations of in- chooses a display strategy that depends on the distance
dividual ants and counts of density. from the center of its foragingrange.Thismodelempha-
Whether through defense, avoidance, or exploitation sizedcostsofconflictbutincludesonlyphenomenological
competition, low overlap in foraging ranges is character- benefits of territory size and is restricted to two compet-
istic of harvester ants. Active defense and avoidancehave itors in a one-dimensional habitat. Lewis and Moorcroft
been observed in these ants (Harrison and Gentry 1981; (2001)usetheirspatiallyexplicitmodelofwolvesanddeer
Acosta et al. 1995; Gordon and Kulig 1996), as has the (LewisandMurray1993;Lewisetal.1997)toparameterize
deterrenteffectofexploitationcompetition(Dreisig2000). aone-dimensionalmodelofthegamebetweentwoneigh-
Thebreadthofpreviousworkontheindividualbehaviors boring wolf packs and find an evolutionarily stable level
thatcreatepatternsofforagingrangeoverlapmakesthese of avoidance of scent marks.
antsanidealsystemtobuildmodelsthatelucidatemech- The models presented in this article derive from a
anisms. Studies of seed-harvesting ants in several genera slightlydifferenttheorylineage(Getty1981;Tullock1983).
have addressed time and energy budgets (MacKay 1985; Modelsofthistypeassignavaluetoeachlocationinspace
Fewell 1988; Weier and Feener 1995), foraging behavior asafunctionofdistancetothecentralplaceforthatforager
(Davidson1977;Fewell1990;CristandMacMahon1991, andameasureofexploitationorinterferencecompetition
1992; CristandHaefner1994;FersterandTraniello1995; from other foragers. From these values, the optimal al-
Morehead and Feener 1998; Gordon 2002), and colony locationofeffortinspacecanbecomputedwhenmultiple
interactions (Davidson 1977; DeVita 1979; Harrison and individuals or colonies compete. However,intheoriginal
Gentry 1981; Gordon 1992; Ryti and Case 1992; Lopezet models (Getty 1981; Tullock 1983), costs of distance and
al. 1994; Wiernasz and Cole 1995; Yamaguchi 1995; Gor- competitionarenotderivedfromunderlyingresourcedy-
don and Kulig 1996; Dreisig 2000). namics or the quantitative effects of encounters with
Ourmodelsbuildonawell-developedmodelingframe- neighbors.
work that has been used to describe the foraging ranges Our approach is most similar to a model of wagtails
of individual organisms as well as colonies. Early models (Davies and Houston 1981). Using a fully parameterized
offoragingrangederiveoptimalterritoryareainresponse description of the resource dynamics, feeding rates, and
toresourcesandintruders(Schoener1987).Thesemodels the costs of defense characteristic of wagtails, Davies and
arespatiallyimplicitinsolvingonlyforarea.Becausethey Houston(1981)predictwhetheraterritoryownershould
treatotherforagersasaspatiallydistributedintruderpres- accept a “satellite” that aids in defense at the cost of de-
sure, they do not treat competition with neighbors pleting resources. This model assigns a value to locations
explicitly. on the basis of a mechanistic accounting of time anden-
Later models use the analogy to pressure to model the ergy budgets and makes accurate predictions of behavior
spatialdistributionofmultipleabuttingterritories(Brisson on the basis of optimization principles.
and Reynolds 1994; Korzukhin and Porter 1994; Adams The model of wagtails is simpler than ourextensionto
1998). Territorial boundaries are assumed to lie at points ant colonies because these birds can systematicallysearch
where the pressures from neighboring colonies are equal theirterritoriestominimizeinefficiency,leadingtoasim-
(Adams 1998). Using an expression for pressure that de- ple pattern of effort allocation in space. As members of a
Nonoverlapping Foraging Ranges in Ants 531
colonythatoperateswithoutcentralcontrolinacomplex
two-dimensional environment (Gordon 1999), ants can-
notsystematicallysearchtheirforagingrange.Instead,in-
dividualantsmustchooseaforagingstrategyonthebasis
of information received from other ants (Adler and Gor-
don 1992; Gordon 2002). We model information limita-
tion in two complementary ways. First, we build models
with and without the assumption that ants knowthefor-
aging ranges of neighboring colonies and can assess the
expected costs of conflict. Ignorance of the detailed lo-
cations of neighbors is plausible given the rarity of en-
counters between ants of different colonies (Gordon and Figure1:Structureoftheallocationdecisionandseeddynamicsmodel.
Kulig 1996). Second, we compare ants that maximizethe Thecolonyassignsf outofFforagerstositei,ofwhichp aresearching
i i
seed collection rate at the individual level with thosethat forseeds.
use information about the whole colony to maximize at
the colony level. The former strategy is suboptimal but study showed that the seed distribution is patchy on the
simpletoachievebyequalizingseedcollectionrateacross scale of distances between nests rather than the smaller
foragers (Bartholdi et al. 1993; Pacala et al. 1996; Dukas scale of different foraging trails. Second, the seed distri-
and Edelstein-Keshet 1998). butionisephemeral,withpatchespersistingonlyforsome
After deriving the contrasting models that optimize at weeks.Becauseourmodelsaredesignedtopredictaverage
thecolonyorindividuallevel,weusemeasurementsfrom spaceuseoveranentireseason,weassumethatthequality
earlierstudiesofthispopulationtofullyparameterizethe of different locations averages out over time.
models and predict time budgets of foragersandpatterns In this simplified resource environment, available in-
of foraging range overlap. We compare these predictions formation constrains the strategy that maximizestherate
with data collected for this study, evaluate which model ofseedcollection.Weexaminetwopossiblestrategiesthat
fitsthedatamostclosely,andexplorepossiblereasonsfor ants could use to allocate foragers in space. In the first,
deviations. which we call the colony-level model, ants find the allo-
cation that maximizes the rate of seed collection by the
colony. Achieving optimal allocation requires individual
The Models
ants to assess the foraging strategyoftheentirecolonyor
A colonyof ants mustallocateforagingeffortthroughout correctforlocaldensityofants(Pacalaetal.1996).Inthe
space in response to resource availability,traveltime,and second, which we term the individual-level model, each
conflict with other ants (Harkness and Maroudas 1985; ant seeks the site that provides it with the highest indi-
Haefner and Crist 1994; Dukas and Edelstein-Keshet vidual rate of intake. This leads to the equalization of
1998). What currency might we expect ants to optimize? intakeratesamongforagers(Pacalaetal.1996;Dukasand
Several studies have shown that energy costs of travelare Edelstein-Keshet 1998) and is simple because ants need
low in harvester ants (Fewell 1988; Weier and Feener not assess any information about the entire colony. We
1995),implyingthatratemaximizationcoincideswithef- begin by deriving and comparing the two strategies for a
ficiency maximization. Furthermore, studies of a related singlecolonyandthenextendthemodelstoaddressmul-
species have shown that ants maximize energy intake di- tiple colonies in competition.
vided by time rather than efficiency (Holder Bailey and
Polis 1987).
Harvester ants collect seeds that are heterogeneous in The Single Colony Model
size and spatial distribution. For simplicity, our models
treatallseedsasequivalentbecauseearlierstudiesindicate For a single colony, our models are similar to those pre-
thatPogonomyrmexbarbatusdoesnotchooseseedsonthe sented by Dukas andEdelstein-Keshet(1998).Considera
basisofsize(MoreheadandFeener1998).Wesimplifyby colony with F foragers to be allocated among n sites (fig.
treating the distributionof resourcesashomogeneousfor 1). Resources in site i are renewed at rate j (tables 1and
i
two reasons. First, a study of seeds collected by foragers 2havecompletedescriptionsofvariablesandparameters).
in this population(Gordon1993)showednorelationbe- Let p represent the number of foragers searching for re-
i
tweenthevegetationcurrentlygrowingintheforagingarea sources in site i. Suppose that resources are removed by
and the species of seeds collected by the ants; most seeds other agents that have an effect equivalent to b ants.The
i
are apparently distributed by wind and flooding. This total resource collection rate from site i, W, is then
i
532 The American Naturalist
Table 1: Variables To find the switching rate from searching to travel, we
useourmodelofseeddynamics.Supposethatthenumber
Symbol Meaning
of seeds R in site i follows the differential equation
i Index for sites i
n Number of sites
dR
pfi NNuummbbeerr ooff ffoorraaggeerrss asesasirgcnheindgtoinsistieteii dtipji(cid:2)h(bi(cid:1)pi)Ri,
i
R Number of seedsin sitei
i
W Rate of seed collectionin sitei where h is the per seed rate at which a forager collects
i
d Distanceto sitei resources. If the resources are in quasi steady state, they
i
ti Travel timeto sitei remain near the equilibrium of R pj/[h(b (cid:1)p)]. The
i i i i
rate at which resources are found, and, thus, the rate at
Wipjib (cid:1)pi p . (1) fwrahcictihonanoafnftorsawgietcrshetshafrtoamresseeaarrcchhininggtofotrrafvoeold,isishtRhie.nThe
i i
p 1/t
The total resource collected, W, is the sum over all sites, ip i .
or fi (1/ti)(cid:1)[ji/(bi(cid:1)pi)]
(cid:1)
n Solving for f gives
i
Wp W. (2)
i
ip1
( )
jt
f p 1(cid:1) i i p. (4)
The numberof foragersallocatedtositeiisf,withthe i b (cid:1)p i
i i i
constraint
(cid:1) The parameter h does not appear in this equation and
n need not be estimated.
Fp f, (3)
i Using these relationships, we can find the colony-level
ip1
strategy (that maximizes the colony-wide seed collection
rate)andtheindividual-levelstrategy(thatmaximizesin-
where Fisthetotalnumberofforagersavailable.Because
dividual rates). The colony-level strategy is found with
foragers spend time traveling, the number allocated, f, is
i
Lagrangemultipliers(Kaplan1984),maximizingWsubject
greater than the number searching, p. To derive the re-
i to the constraint on F and the requirement that p 10.
lationshipbetweenf andp,weneedtocomputethetime i
i i
Because both W and F are sums, the maximum occurs
spent searching and traveling by each ant.
where marginal rates of return are equal (Pacala et al.
A forager switches back and forth between travel and
1996), or
searching. The fraction oftimespentsearchingistherate
of switching from travel to search divided by the sum of
both switching rates (Adler 1998). Travel time to site i is (cid:1)Wipm .
t, equal to the travel distance d divided by the speed v. (cid:1)f c
i i i
We approximate the rate of switching from traveling to
searching as the reciprocal, or 1/t. We can then solve for p as
i i
Table 2: Parameters
Symbol Meaning Estimate
j Resourcerenewal rate .1 seed/min
F Number of foragers 1,890 (adultcolony)
v Speed 2.5 m/min
b Foraging pressurefrom other agents 1.0/m2
a Rate at whichquarrelsbegin .01/(minant2m2)
b Rate at whichquarrelsend 1.5/min
s Probabilitya quarrel escalates .07
g Rate at whichfightsend .025/min
d Probabilitya fightingant is killed .7
M Value of a lost forager 50 seeds
Nonoverlapping Foraging Ranges in Ants 533
(cid:2)
( )
1
p p jb (cid:2)t (cid:2) b. (5)
i i i m i i
c
Theindividual-levelstrategyisfoundbysettingthereturns
per ant equal, or
W
ipm .
f e
i
This equation can be solved for p as
i
( )
1
p pj (cid:2)t (cid:2)b (6)
i i m i i
e
for some value of m that need not equal m.
e c
To solve for either the colony-level or individual-level
strategy, we constrain p to be positive in equation (5) or
i
equation (6) and sum the associated values of f from
i
equation (4). This gives an equation for m or m interms
c e
of the available foragers F thatcan besolvednumerically.
Thetwostrategiesshowcharacteristicdifferencesthat
appear again in the multiple colony models. If antsfol-
low the colony-level strategy, the colony collects more
resources by allocating more ants to greater distances
(fig. 2a) and having those ants collect less resource per
unit time (fig. 2b). If ants follow the individual-level
strategy, the colony overexploits sites near the nest but
receivesonlyaslightlylowerpayoffinseedsperminute
overawiderangeofforagernumbers(fig.2c).Although
the individual-level strategy is not optimal, it uses no
globalinformation,isrobustinchangingenvironments,
and reduces colony intake rate only slightly (Bartholdi
et al. 1993; Pacala et al. 1996; Seeley 1997).
Multiple Colony Models
The model extends immediately to include exploitation
competition between colonies by including the resources
removed by other colonies in the term b. In particular,
i
for each colony, let p˜ represent the number of ants from
i
other colonies foraging in site i. The solutions found in
equations (5) and (6) give the best reply to any given
strategy by competing colonies if we substituteb (cid:1)p˜ for
i i
b. We can find a Nash equilibrium (a best replytoa best
i
reply) for this model of exploitation competition with
iteration.
Interference competition creates two additional costs: Figure 2: Single colony results. We compare the colony-level strategy
time and mortality (fig. 3). Fighting reduces the fraction (solidlines)withtheindividual-levelstrategy(dashedlines)usingrelevant
parametervaluesfromtable2.a,Numberofforagersallocatedpersquare
of time spent searching (Gordon and Kulig 1996). To ac-
meter as a function of distance from the nest. b, Total trip time as a
count for fighting time, we break fighting pairs into two
function of distance from the nest. c, Seeds collected per minute as a
categories (Gordon and Kulig 1996): brief quarrels that functionofnumberofforagers.
534 The American Naturalist
do not lead to injury (q) and escalated fights (q¯). If a Asbefore,wesolveeachequationforthevaluesofm and
i i c
focal colony with p foragers encounters other colonies m that give a total forager number of F when p is con-
i e i
with p˜ foragers, these pairs follow the differential strained to be positive. Solutions are iterated because the
i
equations value p˜ depends on the strategy of other colonies.
i
Inparticular,westarteachcolonywiththesinglecolony
dq optimum, find the best reply for each colony to that set
ipapp˜ (cid:2)bq
dt i i i ofstrategies,andrepeatuntilthestrategiesstopchanging.
To improve convergence, we do not entirely replace the
dq¯ipbsq (cid:2)gq¯, oldstrategywithitsbestreplybutinsteadwithaweighted
dt i i average of the original strategy (weighting 0.8) and the
bestreply(weighting0.2).Failuretoincludethisweighting
where a is the rate at which quarrels begin per pair of can lead to an oscillation between strategies rather than
antsinasquaremeter,bistherateatwhichquarrelsend, convergence, which is an artifact of assuming that each
sistheprobabilityofescalation,andgistherateatwhich colony responds instantaneously to the fixed strategiesof
fights end (table 2). If quarrels and fights are in quasi the others. Including the weightingbetterapproximatesa
steady state, q p(a/b)pp˜ and q¯ p(as/g)pp˜. The total system where colonies respond gradually and
i i i i i i
number of foragers allocated to the site (f) is the sum of simultaneously.
i
those foraging (p), those traveling (m), those quarreling
i i
(q), and those fighting (q¯). If we set Apa[(1/b)(cid:1)
i i Parameter Estimation
(s/g)], the average time spent quarreling or fighting per
foreign ant, then All model runs use ant parameters estimated indepen-
dently in an earlier study (Gordon and Kulig 1996) and
( ) seed parameters estimated at other desert sites. We break
jt
f p 1(cid:1) i i (cid:1)Ap˜ p. (7) parameters into three general categories: those describing
i b (cid:1)p (cid:1)p˜ i i
i i i seed dynamics, those describing foraging behavior of a
single colony, and those describing encounters between
This extends equation (4) to include a new term for ants
ants from different colonies.
engaged in fights.
If fighting also incurs mortality or severe injury with
probability d (Gordon and Kulig 1996), the normalized Seed Dynamic Parameters
rateofforagerlosscanbesubtractedfromtheintakerate.
These parameters describe seed dynamics in the absence
Let M represent the value of anantin unitsofseeds,and
of red harvester ants. We estimate the resource renewal
let KpasdM be the rate at which such value is lost as a
rate, j, as 0.1 seed/min (which corresponds to about
result of fights per ant pair per square meter. Then, the
50,000 seeds/yr). This value is consistent with that ob-
net intake W in site i is
i servedinotherNorthAmericandeserts(PriceandReich-
man 1987; Kemp 1989) but assumes a somewhat shorter
p
W pj i (cid:2)Kpp˜. residence time of seeds than estimated in those studies
i ib (cid:1)p i i (about 1 mo instead of 5 mo) because our models focus
i i
on the summer foraging season. If we assume a seed res-
Themodelwithnointerferencecompetitioncanbestud- idence time of 5 mo, a lower value ofjp0.05 seed/min
ied by setting ap0. is required to match the observed standing crop of 7,000
Colony-levelandindividual-levelstrategiesarefoundas seeds/m2 (Price and Reichman 1987; Kemp 1989). The
in the single colony case, but the solutions include addi- seedsstudiedintheseworksarecollectedbyredharvester
tional terms. With the colony-level strategy, ants (Gordon 1993).
We estimate the rate b at which seeds are removed by
(cid:2)
causesotherthanPogonomyrmexbarbatusas1.0antequiv-
j(b (cid:1)p˜)[(1/m )(cid:2)t]
p p i i i c i (cid:2) (b (cid:1)p˜). alentpersquaremeter.Thisisasmallvaluerelativetothe
i 1(cid:1)[a(cid:1)(K/mc)]p˜i i i number of ants at this location (a mean of roughly five
ants persquare meter) forseveralreasons.First,thereare
With the individual-level strategy, fewantcompetitorsforseedsatthesite,withunpublished
data showing that the density of P. barbatus at the site is
p p ji[(1/me)(cid:2)ti] (cid:2)(b (cid:1)p˜). consistently high relative to that of the other large seed-
i 1(cid:1)[a(cid:1)(K/m )]p˜ i i eatingant,Aphaenogastercockerelli.Otherseed-eatingspe-
e i
Nonoverlapping Foraging Ranges in Ants 535
(rangingfromaslowas0.8m/minupto4.0m/min).We
used the average value because our models make predic-
tions averaged over the entire season.
Ant Fighting Behavior
These parameters describe the rate and severity of en-
counters between ants. Recall that a quarrel is a brief in-
teraction that does not lead to injury and a fight is an
extendedinteractionthathasahighprobabilityofleading
to injury or death (fig. 3).
Figure3:Timebudgetforanant.Aforagingantcanbeengagedinone Weestimatetherateatwhichquarrelsbegin,a,as0.01/
of four activities: traveling, searching, quarreling (a brief nonescalated (min ant2 m2). Of 58 foraging trips madeinthedirection
encounter with an ant from another colony), and fighting (a longer
of another colony, 16 ants encountered an ant from an-
escalatedencounterwithanantfromanothercolony).
othercolony(GordonandKulig1996).Ifweassumethat
ants spend 75% of foraging time searching, or 15 min,
ciessuchasPheidolemiliticidaaresmallerandtakesmaller
thisisatotalof928antminutes.Ifallofthistimeisspent
seeds. Second, populations of seed-eating rodentssuchas
inareaswithadensityofthreeforeignants/m2(theaverage
kangaroo rats have sharply declined over the past 10–15
densitythroughoutthesitefoundinthisstudy,comparable
yr in this area. Third, local P. barbatus density affects the
to the value of five foundearlier; Gordon1995),thenthe
numbersofreproductivesindryyears(GordonandWag-
encounter rate is
ner 1997) and the survival of founding colonies(Gordon
and Kulig 1996), suggesting that depletion by P. barbatus
16 encounters
is not swamped by other species. ap
(928 ant minutes)(3 ants/m2)
Thisvalueofbproducesreasonableresidencetimesfor
seedsandsearchtimesforants.Withtheresourcerenewal 0.0057 encounters
rate of jp0.5, corresponding to longer seed residence p .
min ant2 m2
times,avalueofbp0.3antequivalentspersquaremeter
is required to produce realistic search times. Results with
We rounded up to 0.01 because these ants did not spend
these alternativeseedparametersaresimilartothosepre-
the entire 15-min search period in a region of overlap.
sented here.
The rate atwhichquarrelsend,b,isabout1.5/min,on
Because the model runs only when ants are actively
thebasisofthemeandurationofquarrelsof40sasfound
foraging,andbecauseP.barbatusforagesforapproximately
by Gordon and Kulig (1996).
20%ofthedayduringtheforagingseason(Gordon1999),
We estimate the probability that a quarrel escalates, s,
the effective values of b and j are five times larger.
as 0.07. This value is basedon thesevenfightsoutof105
quarrels observed from beginning to end by Gordon and
Kulig (1996), or 0.067. Alternatively, of the 28 quarrels
Ant Foraging Behavior
notobservedfromthebeginning,21outof28werefights
These parameters describe properties of a single colony. (Gordon and Kulig 1996). Because fights are approxi-
Weesimatethenumberofforagersactiveinanadultcol- mately 60 times longer than quarrels (and, thus, are 60
ony, F, as 1,890, which is half of the total number of times more likely to be detected), the fraction of ants
foragers presented in Gordon and Kulig’s (1996) table 5. observed to be involved in fights would be
Onlyafractionofforagersactivelyforageatanyonetime
(Gordon1999),andweusethevalueofone-halfforsim- 60s 21
p .
plicity.Someofthemodelrunsincludeyoungercolonies, 60s(cid:1)1(cid:2)s 28
and we again use half the number estimated by Gordon
and Kulig (1996): 430 for 1-yr-old, 1,000 for 2-yr-old, Solving for s gives a separate estimate of 0.047 quarrels
2,000 for 3-yr-old, and 2,600 for 4-yr-old colonies. that escalate to fights.
We estimate the speed, v, as 2.5 m/min, the average The rate at which fights end, g, is about0.025/min,on
value found by Gordon and Kulig (1996; at the peak of the basis of the mean duration of quarrels of 42 min as
foraging);thismatchesresultsfoundinthemeasurements found by Gordon and Kulig (1996).
made for this article. There is a strong effect of time of We estimate the probability a fighting ant is injured, d,
dayon foragerspeed,presumablybecauseoftemperature at 0.7. Gordon and Kulig (1996) observed 28 fights, in
536 The American Naturalist
which10endedindeathforbothants(weignoretherare
fights involving more than two ants) and 18 in injury or
deathforoneant.Thus,outof56antsinvolvedinfights,
38werekilledorinjured.Weassumethatinjuredantsno
longer forage.
The value of a lost forager, M, we estimate at50 seeds.
Ants live approximately 1 yr (Gordon and Ho¨lldobler
1987). If about 2,000 ants forage and take about 20 min
each to collect a seed, then the colony collects 100 seeds/
min while foraging. The colony is active for about 10%
of the year, or 50,000 min, so the colony collectsapprox-
imately 5#106 seeds/yr. Dividing by the roughly 10,000
ants per colony gives 500 seeds per ant. A forager can
expecttoliveroughly1mo(GordonandHo¨lldobler1987),
orone-twelfthofitslifespan.Multiplying500seedsbythis
fraction gives 42 seeds. Figure4:Mapofthestudyarea.Theopencirclesrepresentindividual
colonies active in 1999. Large circles are adult colonies (15 yr old),
medium circles are adolescent colonies (3–5 yr old), and small circles
Combined Parameters are juvenile colonies (!2 yr old). The six filled circles show the focal
coloniesfordatacollection.Thetwosquaressurroundthecrowdedregion
Only two combined parameters that summarize costs of (small square) and uncrowded region (large square) followed in the
conflict are required to compute the colony-level and models.
individual-level strategies. Mortality costs appear in the
model in the single parameter KpasdM≈0.02 seeds/ leaves substantial empty space between colonies (fig. 5a).
(minant2m2).Modelrunsindicatethatevenmuchsmaller With the colony-level strategy, the density of ants drops
valuesofKwouldbesufficienttodeterantsfromsearching offslowlywithintheforagingrangeandrapidlyatbound-
in the foraging ranges of neighboring colonies. ariesandleavesnoemptyspacebetweencolonies(fig.5c).
The parameter Apa[(1/b)(cid:1)(s/g)]p0.03/(ant2 m2) These differences translate into the distinct predictions
represents time costs of encounters. The time wasted in about the time budgets of foragers and the spatial distri-
encounters does not play a large role in the avoidance bution of ants that we tested in the field. We made two
predicted by our models. differenttypesofmeasurement.Toestablishtimebudgets,
individual foragers were tracked from six focal colonies
on four different days; we recorded travel distance,travel
Model and Empirical Results
time, search time, and success for 51 trips (ranging from
We solved the model using the estimated parametersand sevento10percolony),ofwhich44wereused.Toestablish
truecolonylocationsfromthestudysitenearRodeo,New thespatialdistributionofants,wemeasuredtotalantsper
Mexico (Gordon 1999). We focused on two regions from square meter at 87 sites on 14 d. The sites were centered
this site. The crowded region has 12 colonies in a 60# around the six focal colonies, with 12 sites near thethree
60-m square, and the uncrowded region has 16 colonies focal colonies in the crowdedregionand16–19sitesnear
in a 90#90-msquare (fig. 4). The modelwassolvedfor the three focal colonies in the uncrowded region.Speeds,
the number of ants assigned from each colony to each densities, and time budgets were found to be similar to
1.0-m2 site, with foraging ranges of the colonies on the those in other studies of ants in this genus (DeVita1979;
edges allowed to wrap around to avoid edge effects. CristandMacMahon1991;HaefnerandCrist1994;More-
Intheabsenceofconflictcosts,apopulationofcolonies head and Feener 1998).
followingtheindividual-levelstrategysegregatesintosep- We next compare the measured time budgets and ant
arate foraging areas (fig. 5a), while coloniesfollowingthe densities with values predicted by the models. We check
colony-level strategy overlap extensively (fig. 5b). A low whetherthemodelsarequantitativelyconsistentwithmea-
degreeofoverlapwiththecolony-levelstrategyispredicted sured values and assess whether the simpler individual-
only when conflict costs are included (fig. 5c). Although level model performs better than the more complex
both the colony-level strategy that includes conflict costs colony-level model that includes conflict costs.
and the individual-level strategy predictarealisticallylow Both the colony-level and individual-level models pre-
degree of overlap (Gordon and Kulig 1996), the use of dict search times accurately (fig. 6), with the two models
spacediffers.Withtheindividual-levelstrategy,thedensity having nearly identical log likelihoods. Measured travel
of ants drops off rapidly with distance from the nest and andsearchtimesdonotshowthedecreaseinsearchtime
Nonoverlapping Foraging Ranges in Ants 537
Figure5:Contoursofforagingeffort(numberofforagerssearching)inthecrowdedregion.Contourlevelsrunfrom1to10ants/m2,anddistance
isinmeters.Parametervaluesasintable2exceptasnoted.a,Individual-levelstrategy.b,Colony-levelstrategywithoutconflictcosts(ap0).c,
Colony-levelstrategywithconflictcosts.
as a function of travel distance predicted by each model. Ourmodelsaredesignedtoaverageovertheseeffects,and
Thecolony-levelmodelhasalowerslopeandqualitatively the individual-level model is thus quite successful in pre-
fitsthedatabetter.Asatest,wecreatedamodificationof dicting the average.
the individual-level model in which ants weight search Finally,eachmodelexplainsasignificantportionofthe
timemoreheavilythantraveltime(thusequalizingsearch varianceinmeasuredantdensity(fig.8).Thecolony-level
times), but this model is not supported by the additional
model produces a slope of 0.69, closer to the predicted
analyses.
valueof1.0.Althoughsomeofthevarianceisduetolower
Theindividual-levelmodelaccuratelypredictsthemean
densitiesintheuncrowdedsite,analysisperformedonthe
distance ants move (fig. 7), while the colony-level model
two sites separately gives similar results. Stepwise regres-
overestimatesthemean.Bothmodelspredictalowervar-
sion indicates that the colony-level model is better sup-
iance than observed (and are therefore rejected by a
ported by the data.
Kolmogorov-Smirnovtestatthe0.05level).Thisdeviation
If we tentatively accept the colony-level model, we can
may occur because these simplified models ignore move-
examine patterns of deviation betweenmodelpredictions
ment during the searching process itself, which could
spread ants assigned to a given site. Furthermore, ants andmeasuredvalues.Siteswithmoreantsthanpredicted
were tracked in pairs, one that left early in the day and tend to lie close to predicted boundaries, while sites with
onethatleftlater.Becausedistancestraveledincreasedur- fewerantsthanpredictedliefartherfromboundaries(fig.
ing the day, these pairs would tend to include one short 9). There is no effect of distance to the nest on the de-
and one long foraging trip, leading to increasedvariance. viationsbetweenpredictedandmeasureddensities.Results
538 The American Naturalist
colony-levelmodelthatdoesincludeconflictcostspredicts
realistic patterns of overlap and gives the closest fit of all
models to measurements of individualantsanddensities.
While we could reject the colony-level model in the
absence of conflict costs, we were unable to distinguish
the much subtler difference between the individual-level
model and the colony-level model that includes conflict
costs. Both models predict mean search times and their
weak dependence on travel distance equally well (fig. 6).
However, only the individual-level model correctly pre-
dicts the mean distance traveled byforagers (fig.7).Both
models predict a significant portion of the variance in
measured ant densities, but the colony-level model pro-
vides a better fit in a multiple regression analysis that
includes both models (fig. 8).
Figure 6: Search time as a function of distance traveledbeforesearch Theserelativelysimplemodelscannotpredicteveryde-
begins,comparingdata(circles),thecolony-levelstrategy(solidline),and
tail of the complex pattern of space use by ants. To un-
the individual-level strategy (dashed line). Cox proportionate hazards
regressionshowsnoeffectofdistanceonsearchtime(pp.24),partic- derstand which missing factors are important, we now
ularlyaftertheeffectoftimeofdayisremoved(pp.65).Includingthe consider deviations of model predictions from data and
effectofcolony,thecolony-levelandindividual-levelmodelshavenearly the model generalizations that could address them. We
identical likelihoods ((cid:2)159.2 and (cid:2)160.1, respectively)ifsearchtimes then discuss the missing factorsandconcludebydescrib-
areassumedtobeexponentiallydistributed.
ing how this modeling approach could be generalized to
other systems.
withtheindividual-levelmodelaresimilar,althoughthere
There are three main deviations of model predictions
is a weak effect of distance to nest.
from the data. First, the effect of distance traveled on
We tested sensitivity by measuring outputs for a range
search time is weak (fig. 6). This could occur for two
of parameter values (table 3). The results are highly in-
reasons: depletion might be unimportant in this system
sensitivetothecombinedparametersAandK.Thespatial
or foraging effort might be sufficiently uniformlydistrib-
distribution of ants is relatively insensitivetothevalueof
uted in space to equalize the standing crop of resources
b, but search times are sensitive because this parameter
(Davies and Houston 1981; Possingham 1989; Dreisig
determines the standing crop. We chose a value thatgave
2000;LewisandMoorcroft2001).Depletionhasbeenob-
resultsconsistentwiththeobservedtraveltimes.Allresults
served in some ant systems (Crist and MacMahon 1992)
are sensitive to the value of j and affect both foraging
ranges and travel times.
Discussion
On the basis of independently measured parameters de-
scribing resource dynamics, time budgets, and costs of
conflict,wedeveloprelativelysimplemodelsofindividual
behavior that predict the observed low levels of overlap
between foraging ranges of neighboring colonies of red
harvester ants and quantitatively fit more detailed mea-
surementsofanttimebudgetsanddensities.Wecompare
individual-level optimization with more complexcolony-
level optimization, both with and without incorporation
ofthetimeandenergeticcostsofconflict.Theindividual-
level model, with or without inclusion of conflict costs,
predictsrealisticallylowlevelsofoverlapbetweenforaging
ranges and provides a reasonable quantitative fit to mea- Figure7:Distributionofdistancestodiscoveryoffood,comparingthe
data (dotted line) with the colony-level model (solid line) and the
sured time budgets anddensities.Thecolony-levelmodel
individual-level model (dashed line). Mean distancesare7.5mforthe
that does not include costs of conflict predicts unrealist-
data, 9.0 m for the colony-level model, and 7.4 m for the individual-
ically high levels of overlap between foraging ranges. The levelmodel.