Lycaenid Butterflies and Ants: Two-Species Stable Equilibria in Mutualistic, Commensal, and
Parasitic Interactions
Author(s): N. E. Pierce and W. R. Young
Reviewed work(s):
Source: The American Naturalist, Vol. 128, No. 2 (Aug., 1986), pp. 216-227
Published by: The University of Chicago Press for The American Society of Naturalists
Stable URL: http://www.jstor.org/stable/2461546 .
Accessed: 11/06/2012 16:28
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
The University of Chicago Press and The American Society of Naturalists are collaborating with JSTOR to
digitize, preserve and extend access to The American Naturalist.
http://www.jstor.org
The AmericanNaturalist
Vol. 128, No. 2
August 1986
LYCAENID BUTTERFLIES AND ANTS: TWO-SPECIES STABLE
EQUILIBRIA IN MUTUALISTIC, COMMENSAL, AND
PARASITIC INTERACTIONS
N. E.
PIERCE AND
W. R.
YOUNG*
Museum of ComparativeZoology, Harvard University,Cambridge,Massachusetts 02138;
Woods Hole Oceanographic Institute,Woods Hole, Massachusetts 02543
SubmittedNovember 7, 1983; Revised December 14, 1984, and June28, 1985;
Accepted January6, 1986
The larvae of many species in the butterfly
familyLycaenidae associate with
ants (for review, see Hinton 1951; Atsatt 1981; Cottrell 1983). Althoughfew
studies have attemptedto assess the actual costs and benefitsof the association
forbothpartners(e.g., Ross 1966; Pierce and Mead 1981; Pierce 1983; Pierce and
Easteal 1986), lycaenid butterfliesand ants appear to exhibitall three kinds of
interaction:mutualism,commensalism,and parasitism.Considerationofthenatural historyof lycaenid-antassociations has led us to develop a model in whichthe
presence of lycaenids affectsthe equilibriumdensityof theirattendantants, and
thepresence of ants influencesboththegrowthrateand equilibriumdensityof the
butterflies.This singlemodel can describe facultativeas well as obligateinteractions. Moreover, even when the association is mutualistic,the model possesses
globallystable equilibria.
The mainemphasisof our model is theobservationthatone ofthe species in the
association exhibits two dramaticallydistinctlife stages, interactingwith its
partnerduringonlyone of those stages. Thus, in theparticularexample discussed
here, the butterfliesare holometabolous,and only the immaturestages interact
withants. This is just one example of a generalcharacteristicof manysymbioses,
includingplant-pollinator
relationships,certainhomopteran-ant
associations, and
a greatvarietyof interactionsin marinesystems.In lycaenid-antassociations, the
presence of ants influencesthe fractionof pupae that eclose into adults. The
subsequent dynamics of the butterflypopulation is not affectedby ants. In
particular,the numberof butterflies
thatsurviveto lay eggs is not alteredby the
ant population. However, the effectof lycaenids on attendantants is characterized by the more traditionalnotion that food supplies affectthe numberof ant
colonies capable of existingin a given environment.In otherwords, the equilib* Presentaddresses: N.E.P., Departmentof Biology, PrincetonUniversity,Princeton,New Jersey
08544; W.R.Y., Departmentof Earth,Atmosphericand PlanetarySciences, MassachusettsInstituteof
Technology,Cambridge,Massachusetts 02139.
Am. Nat. 1986. Vol. 128, pp. 216-227.
? 1986 by The Universityof Chicago. 0003-0147/86/2802-0004$02.00.
All rightsreserved.
TWO-SPECIES STABLE SYMBIOSIS
217
rium density of the ants is raised in mutualisticassociations and lowered in
parasiticones.
A possible example of a mutualisticinteractionbetween lycaenids and ants
is that of the Australian lycaenid Jalmenus evagoras and its attendantants,
Iridomyrmexsp. 25 (Australian National Insect Collection, anceps group; see
Pierce 1983). The immaturestages ofJ. evagoras secretesubstantialfood rewards
in the formof sugars and amino acids that the ants harvest. For example, 67
juveniles of J. evagoras livingon a singlefood plant secretedapproximately400
mg of dry biomass for theirattendantants over a 24-h period (Pierce 1983). In
returnforthisfood, ants protectlarvae and pupae againstparasitoidsand predators. Field experimentswiththisspecies showed thatparasitismof and predation
on the immaturestages are so intensethatwithoutattendantants, populationsof
J. evagoras would not survive(Pierce 1983). Antsare sufficiently
importantin the
life historyof J. evagoras that females use them as cues in selectingsites for
oviposition(Pierce and Elgar 1985).
A likely commensal relationshipbetween lycaenids and ants is that of Deloneura ochrascens (Jackson 1937). The larvae of this butterfly
are found with
ants of the species Crematogastercastenea among lichens on tree bark (Acacia
stenocarpus). The larvae feed on the lichen and do not interactwith the ants.
Nevertheless,larvae oftennestle in the entrancesof the ant galleries,probably
gainingprotectionfromtheirproximityto the ants. OtherAfricanspecies such as
Iridana incredibilisand I. perdita marina have similar life histories (Jackson
1937).
Finally, a clear case of lycaenid-antparasitism is that of the large blue,
Maculinea arion, and its attendantant, Myrmicasabuleti (Frohawk 1903; Chapman 1916; Thomas 1980; Cottrell 1983). Larvae of M. arion secrete substances
thatmimicant-broodrecognitionsignals. A developinglarva feeds on flowersof
thyme,Thymusdrucei,untilafterthethirdmolt,whereuponitdropsto theground
and awaits discoveryby a workerof M. sabuleti. As soon as a workerfindsthe
larva, she picks it up, carries it into the nest, and places it with the brood.
Undetectedby theants, thelarva becomes predaceous and feastsupon thebrood.
It grows to a large size, overwinters,and pupates in the ant nest.
ASSUMPTIONS
OF THE MODEL
Our model stresses the followingfeaturesof the naturalhistoryof lycaenid-ant
interactions.
1. Mutualisticlycaenids supply nutrientsfor the ants, therebyincreasingthe
numberof ant colonies that can survive in a given environment.Nutritionis
importantto both the survivalof queens and the productionof new queens (e.g.,
Wilson 1971; Brian 1983). Hence, the presence of mutualisticlycaenids could
raise the equilibriumdensityof the ant population.Commensalinteractionshave
no effecton the equilibriumdensityof theants,butparasiticinteractionsdecrease
the equilibriumdensity(fig. 1).
2. Our previousremarksabout holometabolysuggestthatthereare reallythree
interactingpopulations in our model: immaturelycaenids, adult lycaenids, and
218
THE AMERICAN NATURALIST
Mutualism
Equilibrium
density of Ants
A (b)
l
0000
Commensalism
Parasitism
Butterfly population density (b)
FIG. 1.-Three cases of the equilibriumdensityof the ants, A(b), as a functionof the
butterfly
populationdensity,b: ifthe interactionis mutualistic,A(b) increases withb; ifthe
interactionis commensal,A(b) is constant;ifthe interactionis parasitic,A(b) decreases as b
increases. In all cases, theeffect"saturates" and A(b) approaches a constantas b approaches
infinity.
ants. For simplicitywe have lumpedthefirsttwo intoa singlevariable,thedensity
of adult lycaenids, and assumed that this variable also reflectsthe population
densityof immaturestages: themorebutterflies
present,themorelarvae thereare
to feed the ants. It is possible to constructa three-variablemodel with one
variableforeach stage,buttheadditionalcomplexityofthemathematicsobscures
the presentation.If our goal were detailed quantitativepredictionratherthan
qualitativeunderstanding,such a constructionwould be useful.
3. The primaryeffectof the ants on the butterflies
is to protectthe immature
stages. This protectionaltersthe growthrateof thebutterfly
population(fig.2) by
increasingtherealized fecundityofindividualbutterflies:
in thepresenceofants,a
largerproportionof the eggs laid by any one butterfly
will surviveand eventually
emerge as adults. Again, the naturalhistoryfeatureemphasized here is thatthe
ecological nicheoccupied byjuveniles is different
fromthatoccupied by adults. In
this case, ants interactwiththe larvae and pupae of lycaenids,but not withthe
adults. In our model, the "birth" rate of butterflies
is equivalentto the numberof
pupae thateclose. Because ants protectthelarvae and pupae, mutualisticinteractions with ants increase the survivalof immaturestages, therebyincreasingthe
eclosion rateof thebutterflies.
The presence of ants does notaffectthe survivalof
butterflies,however, since ants do not interactwiththe adults. In the parasitic
case described above (Maculinea arion), ants provide larvae withfood and protection, and we can consider the ant nest itselfanalogous to a womb for the
The effectof theants thatnurturethelarvae withinthenest
developingbutterflies.
is to raise the eclosion rate of the butterflies.
4. Because lycaenid butterfliesare holometabolous, a large fractionof each
TWO-SPECIES STABLE SYMBIOSIS
Growthrate
of Butterflies
S(a)
a?
219
Ant populationdensity(a)
FIG. 2. -The intrinsicgrowthrateofthebutterflies,
S(a), as a functionof theantpopulation
density,a. At low ant densitiesS(a) is negativebecause of intenseparasitismand predation.
There is a critical size of the ant population,a., at which the growthrate of the butterfly
populationbecomes positive.At largevalues of a, theeffectsaturates(all larvae survive)and
S(a) approaches a constant.
individual'slifeis spent as an egg, larva, and pupa. The majorityof the foraging
responsibleforgrowthand eventualgameteproductionoccurs in the larval stage,
and thislifehistorystage is probablymostvulnerableto predationand parasitism.
Hence, the equilibriumdensity of the butterfliesis stronglyinfluencedby the
numberofjuveniles thatsurviveto eclosion, and survivalrate depends on theant
population size. In addition, the equilibriumdensityof the butterflieswill be
regulatedby density-dependent
controls,such as predationand shortagesof food
supplies,thatare independentof the numberof ants. Both of these ant-dependent
and ant-independenteffectshave been incorporatedinto our model.
THE MODEL
The model we propose is an extensionof the classical Lotka-Volterraequation
forpopulationgrowth,and is expressed as
daldt = ra - r[a2/A(b)]
dbldt = S(a)b
- SO(b21B),
(1)
(2)
wherea is thepopulationdensityof theants (in termsof numbersof ant colonies);
b, the populationdensityof the butterflies;
r, the intrinsicgrowthrateof the ants;
A(b), the equilibriumdensityof the ants, whichdepends on the butterfly
population density (fig. 1); S(a), the intrinsicgrowthrate of the butterflies,which
depends on theant populationdensity(fig.2); S0/B,a constantof theenvironment
regulatingthe populationdensityof the butterflies;
and BS(a)/SO,the equilibrium
densityof the butterflies,
which depends on the ant populationdensity.
Note that S(a) is negative if a < ac; ac is the criticalant population density
220
THE AMERICAN NATURALIST
requiredto ensure thatthe growthrate of the butterfly
populationis positive. If
thereare not enough attendantants, parasitismand predationon larvae are so
intense that the butterflypopulation has a negative growthrate and becomes
extinct.The model can also describe facultativemutualismby simplyassuming
thatS(a) is always positiveand thatthereis no criticalnumberof ants necessary
forthe butterflies
to survive.
In constructingthe model, we decided to use the formin equation (2) rather
than a superficiallysimilarform:
db/dt= S(a)b - S(a)b2/B.
(3)
The differencebetween the two is most easily appreciated by consideringthe
behavior of the butterfly
populationdensitywhen
b/B >> 1
(4)
(i.e., a butterfly
epidemic). In thiscase, equation (2) reduces to db/dt SO(b-1B
and the decrease in densityis independentof the ant population density. We
introducethe symbolsS0/Bseparatelyratherthanas a singleconstantso thatthe
analogywiththe standardLotka-Volterraequation is moreapparent.Dimensional
analysis is also made easier: So has the dimension1/time
and B has the dimension
butterfly
population size.
in thelimit(4), equation (3) gives db/dt - S(a)b2/B,and therate
Alternatively,
of decrease of the butterfly
populationis influencedby the populationdensityof
the ants throughS(a). This latter possibilityseems unrealisticin view of the
naturalhistoryof the system:theantsincreasethegrowthrateofthebutterflies
by
ensuringthat a larger fractionof the larvae survive and metamorphoseinto
butterflies.Since the ants have no effecton the mortalityof butterflies,they
cannot influencethe rate at which epidemic populationsof butterflies
decrease.
Other processes signifiedby the constant SO/Bthat are independentof the ant
populationdensity,such as predationor starvation,are the controlsthat determinethe equilibriumdensityof the butterflies
underthese situations.Hence, the
systemdescribed by equations (1) and (3) has several biologicallyunreasonable
equilibriawhose existenceis due solely to ant-dependentbehaviorof thebutterfly
population when (4) is satisfied.The systemof equations (1) and (2) is a more
faithfultranslationof biological intuitioninto mathematics.
Several further
refinements
to our model mightbe considered.It is possible that
butterfliesalso affectthe growthrates of ants by increasingthe numbers of
reproductivesproduced by ant colonies (fordiscussion, see Wilson 1971; Brian
1983). This effectcould be includedby positingthatr in equation (1) is a function
of b. In addition,the larvae and butterflies
could be treatedas separate variables
usinga three-component
model (see above). In the same vein, one could convincinglyarguethatfora species in whichsinglegenerationsare clearlyseparatedby a
seasonal cycle (as with Glaucopsyche lygdamus in Colorado; see Pierce and
Easteal 1986), a differenceequation mightbe a more appropriatemodel than
equation (1). Even for species whose generationsoverlap (as with Jalmenus
evagoras in Australia; see Pierce 1983), the timelag between egg depositionand
the emergence of the reproductivesmightbe importantas a stabilizingmechanism,and it could be modeled using delay-differential
equations.
TWO-SPECIES STABLE SYMBIOSIS
221
Some of these embellishments,such as the possible effectof butterflies
on ant
growthrate, are easily includedbut lead only to quantitativechanges: the structure of equilibriumsolutions and the separatricesin the phase planes remain
unaltered.Others, such as timelags, have complicatedconsequences thatmight
alter the conclusions of the model. Although it would be straightforward
to
constructa model thatincludes all of the above, it would not be easy to solve it.
The presentmodel is intendedto illustratea process in isolation,ratherthan to
predictthe outcome of several interactingmechanisms.The process is the stabilization of obligateand facultativesymbiosesby the noninteraction
of the partners
when one is in the reproductivestage.
We now discuss the three varieties of lycaenid-antinteractionsin turn.The
threecases are distinguishedby the behaviorofA(b) (as shownin fig.1). Figure3
shows the (a, b) phase plane in mutualism.The arrows indicatethe directionin
whichthe systemtendsto move ifit is at a particularpointin the(a, b) plane. Also
shownare the curves on whichda/dtor db/dtis zero. For instance,fromequation
(1), da/dtis zero ifa = 0 or a = A(b). Equilibriumsolutionsare pointsat which
both da/dtand db/dtare zero. These are indicatedby heavy dots and are also
numbered.
As figure3 illustrates,the behaviorof the systemis not entirelyindependentof
the functionsof figures1 and 2. The most importantquestion is whetherthe
startingnumberof ants,A(O), is largerthanac. That is, is the carryingcapacityof
theant populationin the absence ofbutterfly
larvae largerthanthecriticaldensity
requiredto ensure a positive growthrate forthe butterflies?
It seems most likely
that,in general,the answer to thisquestionis yes (and A(O) > ac, as shownin fig.
3i), simply because it is the easiest situationin which to imagine the system
evolving.Nevertheless,a conditionsuch as thatdescribedby figure3ii is conceivable, particularlyin unstable or seasonal environmentsin which A(O) mightfall
below ac. Both cases are examinedhere. First,suppose A(O) is greaterthanac (fig.
3i). There is one globally stable equilibriumsolution that eventuallytraps the
system(equilibriumpoint 1). Note how theequilibriawithb = 0 (i.e., points2 and
3) are unstable.
But suppose thatA(O) is less thanac. As figures3ii and 3iii indicate,thereare
two possibilities. In figure3ii, the growthrate of the butterfly
populationrises
sharplyonce a exceeds ac. There are now fourequilibriasolutions,two of which
are locally stable, as indicatedby the arrows. In solution1, b is nonzero and the
ants have an elevated populationdensity.In solution3, the butterflies
are extinct
(b = 0) and the ant populationdensityis A(O). The systemis eventuallytrapped
by one of these two stable equilibriumsolutions.There is a watershedor separatrix(shown by the dashed line passing throughthe unstable solution2), which
separates the domains of attractionof the two locally stable solutions. Initial
conditionsabove the watershedwould eventuallycollapse intothe valley bottom
at point 1, while those below the watershedwould go to point3.
In figure3iii, A(O) is again less than ac, but the growthrate of the butterfly
populationrises slowlyas a increases. There are two equilibria,and onlypoint1 is
stable. The butterfly
populationmustbecome extinctin this situation.
Now consider commensalism.Figure 1 illustratesthatthis case is a transition
betweenmutualismand parasitism,and a real systemmay lean towardone or the
a=A(b)
i)
/b=BS(a)/SO
b1
/ N,
2
-3
aC
A(0)
a
(ii)
a A(b)
1
'4
b
BS (a) /S
A(O)
ac
a
(iii)
b
a A(b)
//1
|
A(O)
b BS(a)/So
ac
a
FIG. 3.-The (a, b) plane when the interactionis mutualistic.Arrowsindicatethe direction
in whichthe systemtendsto move at a givenpoint.(i), Equilibrium1 is stable and 2 and 3 are
unstable;(ii), equilibria 1 and 3 are stable; the dashed line delineatesa separatrixdividingthe
domainsof attractionof points 1 and 3; (iii), equilibrium1 is stable. In all cases, ifthe system
is displaced slightlyfromthe stable equilibriumpoint,the displacementdecreases exponentiallywithoutoscillations.
TWO-SPECIES STABLE SYMBIOSIS
223
a=A(b)
(i)
ac
~~~A(O)
b
a
(ii)
a=A(b)
A(O)
b= BS(a)/S(
Ab
FIG. 4. -The (a, b) plane whentheinteraction
is commensal.(i), Equilibrium1 is stableand
2 and 3 are unstable;(ii), equilibrium1 is stable, and thebutterflies
become extinct.Perturbations about the stable equilibriadamp exponentiallywithoutoscillations.
other depending on the costs and benefits to each partner. Figure 4 shows the (a,
b) plane. In this case A(b) is a constant; that is, the changes in the ant population
are independent of the butterflypopulation. Again the fate of the butterflypopulation hinges on whether A(O) exceeds ac. If the equilibrium density of the ants
exceeds the critical density required to protect the butterflypopulation (fig. 4i),
then the butterflies survive. In the other case (fig. 4ii), the butterflies become
extinct.
Finally, suppose the interaction is parasitic. If A(O) is greater than ac, there is a
stable equilibrium solution in which the butterfliessurvive (fig. 5i). Otherwise, the
butterflypopulation becomes extinct (fig. 5ii). Figure 5i suggests that ifthe system
is perturbed, the return to the equilibrium solution may take the form of damped
oscillations. A straightforward linear stability analysis shows that this is possible if daldt, evaluated at the equilibrium point, is sufficientlynegative. This
224
THE AMERICAN NATURALIST
.)
l)
a A(b)
ac
b BS(a) /S
(0)/~~~~~~~A
a
b= BS (a) / SO
A(O)
ac
a
1is stableand2
FIG. 5.-The (a, b) planewhentheinteraction
is parasitic.
(i), Equilibrium
1 is stableand thebutterflies
and 3 are unstable;(ii), equilibrium
becomeextinct.If the
frompoint1, thereturn
systemis displacedslightly
to equilibrium
maytaketheformof
dampedoscillations.
same analysis indicatesthat small perturbationsin all othercases decay without
oscillation.
DISCUSSION
Previous models describingthe populationdynamicsof mutualisticinteractions
have been based on the assumptionthatthe presence of each species alters the
equilibriumdensity of its partner(e.g., May 1973a,b; Levins 1974). In these
models, depending on the values of the interactioncoefficients,mutualismis
unstableor destabilizing;in thewelterof mutualenhancement,thepopulationsize
of both species careens to infinity.Stable conditionsare possible only withthe
addition of density-dependentinteractioncoefficientsthat ensure intraspecific
regulation(e.g., Whittaker1975; Christiansenand Fenchel 1977; Vandermeerand
TWO-SPECIES STABLE SYMBIOSIS
225
Boucher 1978; Goh 1979; Dean 1983) or with externalregulationfroma third
species such as a predator or competitor(Heithaus et al. 1980). May (1976),
examiningthe stabilityof obligate mutualismat different
population densities,
foundthat at highpopulationdensities mutualismscould reach a stable equilibrium,whereas at low densitiestheywere likelyto become extinct.He reasoned
fromthisthatobligatemutualismswere morelikelyto evolve in thetropics,where
environmentsmightbe more constantand both mutualistscould maintainhigh
populationdensities.
Addicott (1981) demonstratedin a numerical analysis that althoughcertain
mutualisticinteractionsmay be unstable, others,perhaps the majorityof cases,
are relativelystable. He introduceda new class of models, of which ours is an
example, in which the interactionbetween mutualistsincreases the equilibrium
density of one species and the growthrate of another (a I-III interactionin
Addicott's terminology).He argued that,in mutualism,"species derive benefit
fromeach other in qualitativelydifferentways" (Addicott 1979, p. 43). This
intuitiveinsightmotivatedhis proposal of density-dependent
growthrates. He
then used computersimulationsto analyze the return-time
stabilityand persistence of six combinationsof threebasic two-speciesmodels (i.e., those of Gause
and Witt 1935; Whittaker1975; Addicott 1981). By comparingthese models of
mutualismwith the appropriatemodels withoutmutualism,he determinedthat
fourof the six combinationsshowed higherreturn-time
stability,and all models
showed persistencestability.For example, ifthe interactionbetweentwo species
increases the equilibriumdensityof one and the growthrate of another,then a
perturbedsystem returnsto equilibriummore rapidlywhen it is involved in a
mutualisticinteractionthan when it is not.
Althoughour model fallsintothe same class of models introducedby Addicott,
it differsfromhis analysis in several ways. First,we have based our model on the
naturalhistoryof a particularsystem,identifying
holometabolyas a biological
mechanismunderlyingthe "I-III" interactionsdescribed by Addicott. Second,
our model can describe obligate mutualismas well as facultativemutualism,
depending on the presence or absence of a, in figure2, and it encompasses
parasiticand commensal relationshipsas well as mutualisticones. This makes it
particularlyusefulin describinglycaenid-antassociations, since all threetypesof
interactionsapparentlyoccur betweenthese two organisms.Finally(and thisis a
methodologicaldifference),we use phase-plane geometryratherthan numerical
analysis to understandthe behaviorof the system.For our purposes, the advantage of phase-plane analysis is that it is not necessary to postulate specific
dependence of the butterflygrowthrate on the ant population density,or ant
equilibriumdensityon butterfly
populationdensity.Instead,we can show thatthe
logic of our principalconclusions are generic and depend only on the simplest
qualitative propertiesof these density-dependent
functions.Any functionthat
looks like that sketched in figure2 will lead to solutions that have the same
qualitativeproperties.This is a desirablefeaturefora biologicalmodel, since it is
often difficultin nature to measure or even estimate population parameters
accurately.
In conclusion, this model describes the population dynamics of mutualistic,
226
THE AMERICAN NATURALIST
commensal, and parasitic lycaenid-antinteractionsand shows that all three of
these two-species interactionscan be stable in nature. The model may have
general applicabilityto other two-species interactions,such as those between
homopteransand ants or between plants and theirpollinators.In the lattercase,
thegametes(pollen and eggs) are again in an ecological nichedifferent
fromthatof
the adult plants. Pollinators increase the birthrates by increasingfertilization
rates.
Our model does not address the question of how different
kindsof two-species
interactionsmightevolve. Pierce (1984) discussed some of the factorsthathave
influencedthe evolutionof lycaenid-antassociations. Several authors,mostnotably Axelrod and Hamilton(1981) usinggame theoryand Roughgarden(1975) and
Keeler (1981) using cost-benefitanalysis, have developed models that predict
conditions under which cooperative species interactionscould evolve. Wilson
(1980) adopted an approach similarto thatof May (1976) in developinga series of
models to examine the evolutionof communitywelfareas a whole.
SUMMARY
We present a two-species model that describes the population dynamics of
mutualistic,commensal, and parasitic interactionsbetween lycaenid butterflies
and ants. The lycaenids affectthe equilibriumdensityof theirassociated ants,
whereas the ants influenceboththegrowthrateand the equilibriumdensityof the
butterflies.
The model can describeobligateas well as facultativerelationshipson
the partof the lycaenids,and even when the interactionis mutualistic,the model
has globallystable equilibria.We suggestthatthismodel has generalapplicability
to othertwo-species interactions.
ACKNOWLEDGMENTS
We thank S. H. Bartz, R. D. Braddock, M. A. Elgar, R. M. May, T. R. E.
Southwood, M. F. J. Taylor, Y. Tsubaki, and G. F. Voss foradvice, encouragement,and stimulating
discussionoftheideas presentedin thispaper. M. Amphlett
kindlypreparedthefigures.N.E.P. was supportedby a National Science Foundation Doctoral DissertationImprovementGrant.
LITERATURE CITED
Addicott,J. F. 1979. A multi-speciesaphid-antassociation: densitydependence and species-specific
effects.Can. J. Zool. 57:558-569.
1981. Stabilitypropertiesof 2-species models of mutualism:simulationstudies. Oecologia
(Berl.) 49:42-49.
Atsatt,P. R. 1981. Lycaenid butterflies
and ants: selectionforenemy-freespace. Am. Nat. 118:638654.
Axelrod, R., and W. D. Hamilton. 1981. The evolution of cooperation. Science (Wash., D.C.)
211:1390-1396.
Brian, M. V. 1983. Social insects: ecology and behaviouralbiology. Academic Press, London.
Chapman, T. A. 1916. Observations completingan outline of the life historyof Lycaena arion L.
Trans. R. Entomol. Soc. Lond. (1915):298-312.
TWO-SPECIES STABLE SYMBIOSIS
227
Christiansen,F. B., and T. M. Fenchel. 1977. Theories of populations in biological communities.
Springer-Verlag,New York.
Cottrell,C. B. 1983. Aphytophagyin butterfliesand its relationshipto other unusual behaviour,
especially myrmecophily.Zool. J. Linn. Soc. 79:1-57.
Dean, A. M. 1983. A simple model of mutualism.Am. Nat. 121:409-417.
Frohawk, F. W. 1903. The earlier stages of Lycaena arion. Entomologist36:57-60.
Gause, G. F., and A. A. Witt. 1935. Behavior of mixed populations and the problem of natural
selection. Am. Nat. 69:596-609.
Goh, B. S. 1979. Stabilityin models of mutualism.Am. Nat. 113:261-275.
Heithaus, E. R., D. C. Culver, and A. J. Beattie. 1980. Models of some ant/plantmutualisms.Am.
Nat. 116:347-361.
Hinton,H. E. 1951. MyrmecophilousLycaenidae and otherLepidoptera:a summary.Proc. So. Lond.
Entomol. Nat. Hist. Soc. 1949-1950:111-175.
Jackson,T. H. E. 1937. The early stages of some AfricanLycaenidae (Lepidoptera), withan account
of the larval habits. Trans. R. Entomol. Soc. Lond. 86:201-238.
Keeler, K. H. 1981. A model of selection for facultativenonsymbioticmutualism.Am. Nat. 118:
488-498.
Levins, R. 1974. The qualitativeanalysis of partiallyspecifiedsystems.Ann. N.Y. Acad. Sci. 231:
123-138.
May, R. M. 1973a. Qualitative stabilityin model ecosystems. Ecology 54:638-641.
1973b. Stabilityand complexityin model ecosystems.PrincetonUniversityPress, Princeton,
N.J.
1976. Models for two interactingpopulations. Pages 49-70 in R. M. May, ed. Theoretical
ecology. Saunders, Philadelphia.
and ants.
Pierce, N. E. 1983. The ecology and evolutionof symbioses between lycaenid butterflies
Ph.D. diss. Harvard University,Cambridge,Mass.
and its
1984. Amplifiedspecies diversity:a case study of an Australianlycaenid butterfly
attendantants. Symp. R. Entomol. Soc. Lond. 11:197-200.
Pierce, N. E., and S. Easteal. 1986. The selective advantage of attendantants for the larvae of a
lycaenid butterfly,
Glaucopsyche lygdamus.J. Anim. Ecol. 55:451-462.
Pierce, N. E., and M. A. Elgar. 1985. The influenceof ants on host plant selection by Jalmenus
Behav. Ecol. Sociobiol. 16:209-222.
evagoras, a myrmecophilouslycaenid butterfly.
Pierce, N. E., and P. S. Mead. 1981. Parasitoidsas selectiveagentsin the symbiosisbetweenlycaenid
butterfly
larvae and ants. Science (Wash., D.C.) 211:1185-1187.
IV. The ecology and ethologyof Anatole
Ross, G. 1966. Life-historystudies on Mexican butterflies,
rossi, a myrmecophilousmetalmark(Lepidoptera: Riodinidae). Ann. Entomol. Soc. Am.
59:985-1004.
model. Ecology 56:1201Roughgarden,J. 1975. Evolutionof marinesymbiosis-a simplecost-benefit
1208.
Thomas, J. A. 1980. Why did the large blue become extinctin Britain?Oryx 15:243-247.
Vandermeer,J. H., and D. H. Boucher. 1978. Varieties of mutualisticinteractionsin population
models. J. Theor. Biol. 74:549-558.
Whittaker,R. H. 1975. Communitiesand ecosystems. 2d ed. Macmillan, New York.
Wilson, D. S. 1980. The natural selection of populations and communities.Benjamin/Cummings,
Menlo Park, Calif.
Wilson, E. 0. 1971. The insect societies. Belknap Press of Harvard UniversityPress, Cambridge,
Mass.