Models of Facultative Mutualism: Density Effects
Author(s): Carole L. Wolin and Lawrence R. Lawlor
Reviewed work(s):
Source: The American Naturalist, Vol. 124, No. 6 (Dec., 1984), pp. 843-862
Published by: The University of Chicago Press for The American Society of Naturalists
Stable URL: http://www.jstor.org/stable/2461304 .
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December 1984
The AmericanNaturalist
Vol. 124, No. 6
MODELS OF FACULTATIVE
CAROLE
L.
MUTUALISM: DENSITY EFFECTS
WOLIN* AND LAWRENCE
R.
LAWLORt
Departmentof Zoology, Universityof Texas, Austin,Texas 78712
SubmittedMay 10, 1982; Accepted May 4, 1984
Mutualism
is a commonoccurrence
innature.Notableamongsuchinteractions
and plants(Faegriand van der Pijl 1979;
are associationsbetweenpollinators
Gilbert1975;Richards1978); seed dispersersand plants(Handel 1978;Howe
and Orians1982);mycor1977;Pudloet al. 1980;van derPijl 1972;Wheelwright
bacteria
rhizaeand plants(Janos1980;Meyer1967;Tinker1975);nitrogen-fixing
and plants(Burnsand Hardy1975;Lange 1967);insectsand fungi(Batra 1979;
Cooke 1977);algaeandprotozoa(Karakshian1970,1975;Taylor1975);coelenter(Dustan1979;Goreauet al. 1979;KinzeeandChee 1979);
atesand zooxanthellae
nectaries(Bentley1977;Janzen
and Hymenoptera
and plantsbearingextrafloral
to
interactions
resultin directcontributions
1966). Some of these mutualistic
suchas
forexample,gameteor zygotedispersal.Indirectfitness
benefits
fitness,
or herbivory
occurin others.
againstpredation
nutritional
rewardsor protection
The relativeimportof such effectsvariesnotonlyfromspeciesto speciesand
but also withpopulationparameterssuch as
frominteraction
to interaction,
forexample,intensity
ofcomCommunity
parameters,
densityofthemutualists.
withor predation
on mutualistically
associatedspecies,mayalso modify
petition
theimpactof mutualistic
interaction.
In addition,mutualismmay be obligateor facultative.Models of obligate
mutualism
havequalitatively
thanthoseoffacultative
different
stability
properties
and Boucher1978).In thisstudywe consideronlyfacultative
ones (Vandermeer
themmutualism.
arethosewhichcan existandperpetuate
Facultativemutualists
selveswithor withoutthepresenceofa mutualist
partner.
A formulation
Lotka-Volterra
equaoffacultative
mutualism
basedon modified
(May 1974).This
tionshas been used to predictthatmutualism
is destabilizing
is to increasethe
ofmutualism
modelincorporates
theassumption
thattheeffect
to exceedits
(i.e., to allowthepopulation
equilibrium
densityoftheparticipants
ariseforspeciespairs,each describedbysuch
carrying
capacity).Two situations
is greater
a model.Whentheproductoftheinterspecific
interaction
coefficients
thanor equal to one, populations
bound.However,thisseemingly
growwithout
andBoucher1978),
unstablebehaviorallowspopulation
persistence
(Vandermeer
*Presentaddress: Departmentof Zoology, Universityof California,Davis, California95616.
tPresentaddress: Box 119, Gualala, California95445.
Am. Nat. 1984. Vol. 124, pp. 843-862.
All rightsreserved.
C 1984 by The Universityof Chicago. 0003-0147/84/2406-0007$02.00.
843
844
THE AMERICAN NATURALIST
and the addition of a predatorcan lead to a stable equilibrium(Heithaus et al.
1980). Smaller interactioncoefficients,on the other hand, result in a stable
equilibriumin which both species coexist at higherequilibriumdensities than
whenalone. In thislattercase, however,Lyapunov stabilityis reducedrelativeto
a systemwithnoninteracting
species, i.e., thereturntimeto equilibriumfollowing
perturbation
is increased. Goh (1979) has demonstratedthatin two-speciesLotkaVolterra models of mutualismthat local stabilityimplies global stability.The
importanceto stabilityof highlevels of intraspecificcompetitionrelativeto the
intensityof mutualisticbenefithas been noted by Dean (1983), Gause and Witt
(1935), May (1981), and Travis and Post (1979).
Not all models of mutualismincrease the participants'equilibriumdensities.
Rather than simplyincreasingeach other's equilibriumdensity,mutualistscan
increase r, the intrinsicrate of increase, or both r and the equilibriumdensity.
Simulationsdemonstratethatthese lattertwo typesofmutualism,and particularly
mutualismwhich increases the intrinsicrate of increase alone, enhance stability
(Addicott 1981).
Population models of facultativemutualismhave not yet been based on the
mechanism of action of mutualism. Here we develop mechanisticmodels of
mutualismratherthan simplyexpandingor modifying
termsof a Lotka-Volterra
type equation. Mutualism is an interspecificinteractionwhich increases births
and/ordecreases deaths of individuals.We base our models on this simple assumption.The impactof mutualisticinteractionon an individualis a function,of
among other things,recipient density. In certain natural systems mutualistic
benefitincreases withincreasingrecipientdensity(Beattie 1976; Plattet al. 1974;
Silander 1978) while in others it decreases with increasingrecipient density
(Addicott 1979; Antonovicsand Levin 1980; Gilbert1982). We presentsix possible relationsbetween recipientdensityand the effectof mutualismon birthsand
deaths of individualsof the recipientpopulationand use these to generatecorresponding population growthequations. We then analyze two-species systems
based on these models and characterizethe dynamicalbehaviorof these systems
of mutualistswithrespect to stabilityand persistence.
This approach has several advantages. The impactof mutualismon thedynamics of populationgrowth(i.e., the effecton r-actual)is clearlyevidentin each of
the models presented. This facilitatesrelatingmodels to the "real world" and
testingpredictionsgeneratedby them in real systems. By constructinga set of
models, each withdiffering
responses to recipientdensity,we can compare the
dynamicalbehavior of a varietyof two-species systems. In so doing we reach
some generalizationsabout the dynamicsof mutualisticsystemsand the conditions under which mutualismenhances systemstability.
MODELS
The startingpointforeach of the models is one in whichper capita birthsand
deaths exhibitlinearresponses to densitywhen no mutualistis present,per capita
birthsdecreasingand per capita deaths increasingas a functionof density,thatis
MODELS OF FACULTATIVE MUTUALISM
4=
bo-
845
bN,
and
(1)
do + dN1.
=
4 and d are per capita birthand death rates, respectively,bo the birthrate at zero
density, do the death rate at zero density, and b and d per capita densitydependentregulatoryfactors.From these assumptionsWilson and Bossert (1971)
derive the logisticequation,
N1
=
rNj(K-
N-
K
by definingr = bo - do and K
= rl(b + d).
Mutualisticeffectsare superimposedon this basic model. The mutualistincreases births and/or decreases deaths in a density-dependentor a densityindependentmanner.Since we consideronly facultativemutualistshere, we use
the situation in which species exist alone, described by equations (1), as a
referencepoint forcomparingthe effectsof different
models of mutualism.
Density-Independent
Model
Mutualisticeffectson birthsand/ordeaths may act in a density-independent
manner.In such cases additionofa givennumberofmutualistsproducesthesame
per capita benefitto recipientsat all recipientdensities (fig. la). For simplicity
considerthecase in whichthebirthrateis increasedas a resultofmutualismwhile
the death rate remainsunaltered:thus equations (1) become
4 = bo-
bNj + mN2
a = do + dN1.
m is theper capita effectofN2 on thebirthrateof therecipient,N1. Alternatively,
mutualismcould decrease the death rate. This produces an analogous mathematical derivationand will not be formallyanalyzed.
The per capita change in populationdensity,birthsminusdeaths, becomes
Nl11N = bo - bNj + mN2 - (do + dNj).
Substituting r = (bo - do) gives
Nl11N = r + mNN - (b + d)N1.
(2)
r and K retaintheirmeaningfromthe logisticequation in thisand the following
derivations,therebyfacilitating
comparisonsof theseresultsto thelogisticmodel,
i.e., the situationwhenthemutualisticpartneris absent. Substituting
(b + d) = ri
K in (2) gives
Nl11N = r + mN2 - rNj1K.
Rearrangingyields
Nl11N = (r + mN2)(K + mN2KIr - N&)/(K ?+ mN2KIr).
(3)
THE AMERICAN NATURALIST
846
a
b
d~~~~
b
d
c
Cd
..
//
br
b
N)..
V ../dN....N
d0NN
f
e
bo
b~N
N/
N~~~~N
birthsand deaths with no
FIG. 1.-Per capita effectsof mutualism.Dashed lines
mutualistpresent;solid lines = the same whenalteredby mutualism,and dottedlines are the
limits to possible increase in birthsor decrease in deaths as a result of mutualism.a,
effectof mutualismon the recipient;b, c, and d, highdensity
Illustratesa density-independent
effects,and e and f, low densityeffects.
Fromthisequationas wellas fromthegraphical
analysisitcan be seenthatboth
themaximum
rateofincreaseandtheequilibrium
densityincreasesas a resultof
mutualism.This model is mathematicallyequivalent to that of Gause and Witt
(1935),Vandermeer
and Boucher(1978),andto Addicott's(1981)modelinwhich
r-maxand K are increased by mutualism.
Density-DependentModels: High DensityEffects
the extentof per capita mutualisticbenefitmay varywithrecipiAlternatively,
ent density. If mutualismacts to reduce density-dependent
regulationat high
densityor simplyhas the mostimpactwhen thepopulationunderconsiderationis
at highdensitythenmodels such as the threethatfolloware applicable.
MODELS OF FACULTATIVE MUTUALISM
847
In thefirstof these (fig.lb) mutualismacts symmetrically
to increaseper capita
birthsand to decrease per capita deaths. Mutualisticeffectsincreaselinearlywith
recipientdensity.As can be seen graphicallythisresultsin an increase in equilibriumdensityabove thecarryingcapacity,butno changein the maximumintrinsic
rate of increase. A saturationin mutualisticeffectsis assumed: maximumper
capita birthrate cannot exceed bo and minimumper capita death rate cannot go
below do. However, as birthrate approaches the line 4 = bo and death rate
approaches the line a = do, equilibriumdensityincreases withoutbound.
The per capita birthand death rate in this case become
4
=
- bN1/(l
+ MbN2)
do + dN1/(1 + mdN2).
i=
For simplicitywe let mb = md = m. Per capita change in populationdensityis
given by
N1/N1=
bo-
do - (b + d)NI/(1 + mN2)
or equivalently,by
Nl11N = r - (b + d)NI/(I + mN2).
(4)
SubstitutingrIK = (b + d) into equation (4) and rearranging
yields
Nl11N = r(K + mKN2 - N&)/(K + mKN2).
(5)
This model impliesthatmutualismincreases onlytheequilibriumdensityand that
birthsand deaths are affectedsymmetrically
in a mannerthatincreases linearly
withrecepientdensity.Benefitssaturateat 4 = boand d = do. Equatinga withmK
in equation (5) resultsin the equation commonlyseen in the literatureformutualism (Addicott 1981; May 1974, 1981; Whittaker1975).
In the followingmodels, forthe sake of simplicity,only mutualisticeffectson
birthswill be formallyanalyzed. The model analogous to the one above in which
onlybirthsare alteredas a resultof mutualismis illustratedin figureIc. Again the
birthrate cannot exceed bo. Birthand death rate are given by
4 = bo - bN1/(l + mN2)
d=
do + dNj,
respectively.Population change withrespect to timeis given by
Nl11N = bo - bN1/(l + mNN9)- (do + dNj).
(6)
Substitutionfor(b + d) are rearrangement
yields
N1/N, = r[1 - (rN1 + dKmN1N2)/rK(1 + mN2)].
(7)
Again,theequilibriumdensity,butnottheintrinsicrateof increaseis increasedas
a resultof mutualism.
Figure Id presentsanothermode of action of mutualismwithprimaryeffectsat
848
THE AMERICAN NATURALIST
highrecipientdensity.Per capita increase in birthrate as a resultof mutualismis
again a linearfunctionof density.However, in this case births,expressed by
4=
bo-
(b - mN2)N1,
can exceed boiftheimpactof individualmutualistson therecipientor thenumber
of mutualistspresentis high,i.e., if mN2 is greaterthan b. Deaths are again
i=
do + dN1.
Althoughin this model the intrinsicrate of increase does not exceed thatin the
logisticmodel, it is possible for r-actualto exceed ro. Feasible equilibria,when
theyexist, are at densitiesabove the carryingcapacity.
The equation forthe populationgrowthrate is
N1/N1= r - (b - mN2 + d)N1.
(8)
Substitutioninto equation (8) for(b + d) yields
N1/N, = r(K - N1 + mN1N2)/K.
(9)
The termhere for the effectof mutualism,mN1N2,as expected in this case, is
dependenton recipientdensityand on the densityof the mutualistpartner.
Density-DependentModels: Low DensityEffects
the effectof mutualismmaybe mostpronouncedat low recipient
Alternatively,
density.This situationmay arise when (1) the numberof mutualistsis limiting;(2)
mutualismcannot compensate for limitingfactors or other density-dependent
regulatoryphenomenawhichact at highdensitiesoftherecipient;or (3) thenature
of the interspecificinteractionbetween two species is densitydependent,benewhenrecipientdensity
ficialin effectwhenrecipientdensityis low butdetrimental
is high. Two models in which maximumbenefitsfrommutualismoccur at low
recipientdensitiesare presentedbelow.
In thefirstof these models theper capita benefitderivedfroma givennumberof
mutualistsdecreases exponentiallywith the recipient's density. This may be
formulatedin termsof effectson eitherbirths(fig.1E) or deaths. For mutualism
affectingonly birthrate, the birthand death rates are
4 = bo - bNj
a=
+ mN2e-N
do + dN1.
Populationgrowthexpressed on a per capita basis is
Nl11N = r + mN2e-'XN - (b + d)NI.
(10)
Substitutingfor(b + d) in equation (10) and rearranging
gives
N11N, = r(1 - N1/K) + mN2e-xN.
(11)
It is apparentboth fromfigure1E and fromthe populationgrowthequation that
therate of increase exceeds r-maxat low densities.When atis largethe densityat
equilibriumapproaches K; when small it exceeds K.
MODELS OF FACULTATIVE MUTUALISM
849
The case in which the mutualisticbenefitdecreases linearlywith recipient
density, rather than exponentially(eq. [11]), is depicted in figureIF. While
enhancingthe recipient'sbirthrate at low densities,thismodel, unlikeall of the
other models considered, has the propertythat at densities above the carrying
capacity the birthrate is actually lower than it would be in the absence of the
"mutualist." In otherwords,mutualismitself,in thiscase, is a density-dependent
phenomena. The same species which is a mutualistat low recipientdensities
behaves as a competitoror parasite at densities above the recipient'scarrying
capacity!
The birthrate in figureIF is
4 = (bo + mNN) - (b + uN,)Nj
and the death rate
=
do + dN1.
Per capita change in populationdensityis then
Nl/NN = r + inN2 - (b + uN2 + d)NI.
(12)
When m approaches zero, this equation describes a high density mutualistic
system (as in eq. [8]), while when u approaches zero it describes a densityindependentmutualisticsystem(as in eq. [2]). In orderto insurethatmutualism
acts primarilyat low densities,we will considerthe special case where - mr/K
and hence N1 = K.
Substitutingforu and for(b + d) into equation (12) gives
Nl11N = (r + mN2)(I - N1/K).
(13)
As in the previous model (eq. [11]), in thismodel the intrinsicrate of increase is
increased by mutualism. Equating mK/rin equation (13) to (x yields one of
Addicott's (1981) models.
Dynamical Behavior of Two-Species Systems
To examine the behavior of mutualisticsystems we must consider, in the
simplestcase, pairs of interactingspecies. Paired species need not have similar
responsesto mutualismwithrespectto recipientdensity.However, forsimplicity
onlythose mutualisticpairs whichdo willbe examinedhere. Implicationsforpairs
withqualitativelydifferent
responseswithrespectto densityare consideredin the
discussion.
We characterizethe dynamicbehaviorof the two-speciessystemswithrespect
to threecriteria:Lyapunov stability,feasibilityof equilibria,and species persistence. Local Lyapunov stability,the rate of returnto equilibriumfollowingan
but smallperturbation,
is characterizedin termsoftheeigenvaluesof the
arbitrary
linearizedsystemaround the equilibriumpoint. Eigenvalues foreach of the twospecies systems are summarized in table 1. Local stabilityanalysis alone is
inadequate to characterizethebehaviorof these systemssince itdoes notindicate
feasibilityof equilibria, system behavior away fromthe equilibriumpoint, or
Z
k
=
w:
F
v:
~~~~~~~~~~A A
A
VV
S
0-~~~~~~~~~~~~~~~~~1
E
0
~~
A
A
o._u
m ~~~~~~~~~~tl tl)
tl l
V
V
MODELS OF FACULTATIVE MUTUALISM
851
species persistence.An equilibriumis mutuallyfeasibleonlyifbothspecies are at
positive densities at equilibrium. Feasibility of equilibria is determinedalgebraicallyand throughstate space diagrams(figs.2-6). A systemis persistentif
species do not become extinct,that is, species at positive densities remain at
positive densities and any species perturbedto zero density will increase in
numbersupon reinvasion. Systems in which the side solutionsare unstable and
thus mutualinvasibilityis possible are persistent.Side solutionsare equilibriain
which only one species is at positive density,while the otheris at zero density.
Unstable as well as stable systemscan be persistent.Mutualisticsystemsin which
species densitiesincrease withoutbound are unstablebut persistent;presumably
in these cases limitationto populationsize is extrinsicto themodel. Persistenceis
analyzed fromstate space diagrams.
For our model ofdensity-independent
mutualism(eq. [3])thepopulationgrowth
equations fortwo such mutualisticpartnersare
rlNl(Kl - N, + mlK1N2/rl)/KK
N1
No
=
For simplicity assume r1
=
r^N2(K2 - No + m2K2Nl/r,)/K,.
r,
=
r, K1 = K, = K, and ml = tni = in. The same
assumptions are also used in the analyses of the other models. The Jacobian
matrixthen is:
[
fi/8NlIeq 6fi/8N,1eq1
8f2/8Nlleq 6f2/8N2jeq
[ r/(I LmK/(I -
mK/r) mK/(1 - mKir)
mK/r) -r/(1 - mK/r)j
The eigenvalues (A) for the matrix are found to be - r/(1 - mKir) + mK/(1 - mKI
r). Simplifying, XI = -r and Xi = (-r - mK)/(1 - mK/r). Two cases arise,
dependingupon the parametervalues.
The firstcase is characterizedby a stable,mutuallyfeasibleequilibrium.Whenr
> mK, XI is the dominant(most positive or least negative) eigenvalue. Both
eigenvalues are negative,thus the systemis stable around the equilibriumpoint.
The logistic model provides a frame of referencefor determiningdegree of
stability.The eigenvalues for a system of noninteractingspecies governed by
logisticequations both equal - r. The measure of degree of stability,the timefor
returnto equilibriumfollowingperturbation
or T, is givenby - l/X,whereX is the
dominanteigenvalue. Returntimeforthe logisticmodel is lr. Returntimefora
pair of density-independent
mutualistsis likewise l/r,thusmutualismof thistype
does not affectthe degreeof stabilityof the system.Furthermore,
theequilibrium
is a mutuallyfeasible one. N, the equilibriumdensity,equals K(1 + mK/r)/[1(mK/r)2].N is greaterthan zero when r > mK, i.e., the conditionforstabilityis
identicalto the conditionformutualfeasibility.This situationcorrespondsto the
state space diagram in figure2a of a persistentsystemwith a stable, mutually
feasible equilibrium.
The otheroutcome is an unstable,unfeasibleequilibrium.Whenr < inK thenX2
becomes thedominanteigenvalue. Since underthese conditionsXi is positive,the
systemis unstable. However, thisequilibriumis also unfeasible.This situationis
depicted in figure2b. The conditionforthis outcome, r < mK, is equivalent to
852
THE AMERICAN NATURALIST
a.
N2
b.
N
FIG. 2.-Isoclines for mutualisticsystems with 2 density-independent
mutualistseach
describedby eq. (2) or with2 highdensitymutualistseach describedby eq. (4). In a thereis a
stable equilibriumwhile in b species are persistentalthoughthe system is unstable. For
density-independent
mutualists,a occurs when r > mK and b when r < inK. The N1 isocline
has intercepts(K, 0) and (0, - rim)and slope (b + d)/mn.Switchingvalues forN1 and N, gives
analogous parametersforthe N, isocline. For highdensitymutualistseach describedby eq.
(4), a occurs when I > mK and b when I < mK. For the N1 isoclinesthe interceptsare (K, 0)
and (0, - 1/m)and the slope is 1/mK.
(b + d)<m, i.e., mutualisticbenefitis large relativeto self-damping.Although
unstable, this systemis persistent.Side solutions[(O,K) and (K,O)] are unstable
and as can be seen in figure2b, both species have unboundedpopulationgrowth
unless limitedby some factorextrinsicto this model.
We turn next to the firstmodel for high densityeffects,the case in which
mutualismaffectsbirthsand deaths simultaneously,withgreaterimpactat high
recipientdensities(eq. [5]). The followingJacobianmatrixis obtainedfora twospecies systemof this sort:
[rtk rmK
LrmK - r-
The eigenvaluesgeneratedby thismatrixare -r(l ?i mK). Two cases, analogous
to those of the density-independent
model arise. For thelargerof the eigenvalues
to be negativemK mustbe less than 1. This correspondsto a stable and feasible
equilibrium.The systemis less stable, though,thanone withtwo noninteracting
species or two density-independent
mutualistssince returntimeto equilibrium,1/
(r - rmK), is greaterthan lhr.The linearisoclines intersectat the pointwhereN
= rl(b + d - rm),thus equilibriumdensitiesare positiveas long as the stability
conditions,mK < 1, hold. This situationcorrespondsto thatin figure2a. When
mK is greaterthan 1, the equilibriumis unstable (saddle point) and unfeasible
albeit persistent.This corresponds to figure2b. In this case populations grow
withoutbound. If mK = 1, the isoclines forN1 and N2 are parallel, thereis no
equilibriumpoint,and again populationsgrow withoutbound.
MODELS OF FACULTATIVE MUTUALISM
853
In the second model in which mutualisticeffectsare most pronouncedat high
recipientdensity(eq. [7]) the Jacobianis
L
br~m/(b
+ d
+
d~nA
br ml(b + d + dinN
+ d + dmnA)21
Ijj1;b2mnl(b
- .'
The eigenvalues forthis matrixare - r ? brl2n/(b+ d + dmN)2. The dominant
+ d + dmnN)2.The
eigenvalueis negativeand the systemis stable if - r > brl-2m/(b
eigenvaluesthusdepend on N. There are two equilibria:N = (rm - b - d)/21nd
+ L(b + d)2 + 2rin(d - b) +
'ml]"12/nd. These correspond to the two points of
intersectionof the curvilinearisoclines seen in figure3, one a stable, feasible
equilibriumand the other an unstable, unfeasiblesaddle point. In this system,
then,thereis always a stable feasibleequilibrium.The systemis less stable than
one with a pair of noninteractingspecies since Tr.> Ilr. Since the feasible
equilibriumis stable and thereare unstable side solutions,this systemis persistent.
The thirdmodel withmutualisticeffectsoccurringprimarily
at highself-density
(eq. [9]) has the possibilityof unlimitedbirthrate. Here the Jacobianis
[7r
]2
min
m&2]
- I'
and the eigenvalues are -r ? inN2, where N = (b + d)/2mn? [(b + d)2 4rin]12/2rn.
This systemwould tend toward instabilitysince wheneverInN2 exceeds the intrinsicrateof increase thedominanteigenvalueis positive.The return
timeunderconditionswhere stabilityholds, 1/(r- miN2), also exceeds thatfora
noninteracting
species pair. The parametervalues determinewhetherthereare
two real equilibria,one real equilibrium,or no real equilibrium.When (b + d)lm
> 4K, thereare two real solutions:theone at lowerdensities{N = (b + d)12mn
[(b + d - 4rm]l1/2m}is stable whilethe one at higherdensities{N = (b + d)l
2m + [(b + d)2 - 4rm]112/2m}
is unstable. In effect,as seen in figure4a, thisis a
persistentsystemin which, above a criticaldensityforboth species, population
numbersincrease withoutbound unless limitedby some factorextrinsicto the
model. The second case arises in theinfinitely
unlikelysituationthat(b + d)lm =
4K. The result is a single unstable, feasible equilibrium(fig. 4b) in which the
ifN1 and N. are above N. In thethirdcase, (b +
populationsincrease indefinitely
d)/m<4K. Here the species are also persistentand populationdensitiesincrease
withoutbound, but thereare no real equilibria(fig.4c).
Thus for all three models for which per capita mutualisticbenefitsto the
recipientspecies increase withincreased density,species are always persistent,
but stabilityis less than thatfornoninteracting
species.
In contrast,the models in which mutualismacts primarilyat low recipient
densityexhibitgreaterstability.The Jacobianformutualisticspecies pairs which
THE AMERICAN NATURALIST
854
N2
each describedby eq. (6). Thereis
FIG.3.-The isoclinesfor2 highdensitymutualists
is also unfeasible.
The N1
The unstableequilibrium
alwaysa stable,feasibleequilibrium.
N1 = rld andN, = -(b + d)l
isoclinehas intercepts
(K, 0) and(0, - I/m)andasymptotes
md.
as a function
ofincreasdecreasing
benefits
ofmutualism
eachhaveexponentially
density(eq. [11])is
ingrecipient
L
-xmN2e
N-
(b - d)
1
Nme-N
LNme-N
OwmNIe-
N-
(b + d)]
+ rN/K) ? Nie-'xN,
The eigenvalues for this matrix equal -(mN-cxeor equivalently,- rN/K - (No- + 1)Nme-IV. Since thisis a mutualisticsystem
actingprimarily
we assumeN - K. In addition,to assurethatthisis a mutualism
that
ourselvesto valuesof(x> 1/K,insuring
at low recipient
density,we restrict
effect
as K is approached.Undertheseconditions
themutualistic
dropsoffrapidly
N ( > 1 and X < - rN/K< - r. This means thatX < - r and returntimeto the
is inall cases morerapidthanthatfornoninteracting
species.
feasibleequiibrium
5 inwhichitcan be seenthatthereis alwaysa
The isoclinesare depictedinfigure
ofthissortis notonlypersistent,
but
point.Mutualism
stable,feasibleequilibrium
enhancessystemstability.
atlowrecipient
In theothermodelinwhichmutualistic
benefits
occurprimarily
effectsat densities
interaction
producesdetrimental
densities,the interspecific
exhibitstability
above thecarrying
capacity(eq. [13]). Pairsof suchmutualists
thesameas thoseforthemodeljust described.
whichare qualitatively
properties
The Jacobianmatrixforthissystemis
-rI
mK
-
0
r - mKJ
and the eigenvalues both equal (-r - mK). This systemis always stable and
returntimeto equilibrium,
1/(r+ mK), is less than1hr.In otherwords,this
like
and
the
one
discussed,
alwayshas a stable,feasibleequilibrium
just
system,
a.
N2
0I1
S
N
111
b~
~~. ~
C.
I
N2
N2
0
v
~ ~
=
10
C4
C.
~N2
=0
N N1
FIG. 4.-The isoclines for2 highdensitymutualistseach describedby eq. (8). There are 2
equilibria when (b + d)lnm> 4K, the one at lower densities stable and the one at higher
densitiesunstable,as depictedin a. When(b + d)/n = 4K thereis an unstableequilibriumas
in b and when (b + d)/rn< 4K thereis no equilibriumand the systemis unstable.as in c. In
all cases species are persistent.The interceptforthe N, isocline is (K, 0) and the asymptotes
are N, = 0 and N, = (b + d)lnm.
856
THE AMERICAN NATURALIST
N2
0
low densitymutualists,
byeq. (10). Thereis
FIG.5.-The isoclinesfor2_______\Z%/r;2=o
11 ~each described
N
always a stable, feasible equilibriumand an unstableequilibrium.The systemis thus stable
are(K,0)
is (K
FortheN1 isoclinetheintercepts
andpersistent.
(.. and - rlm),theminimum
- tua, - rKange[I-K]) andthereis an asymptote
at N =g e
b
N2
m ehnistictiey
mutualis
o isc thsriedb
m
geneato
of3)ah syset
The.
ouTcoe ofa e fore2lo
muuais on
[10] [13])ep
inoprtn fetof
h
of
models(eqql.[2,h4] [6] [8],in
oforecipaien densinty.Acnalsso
species. pairstoug
stbiriths
andordathsaseate
fntion
)
deninty
repnsseucdts
thrbenchneea geeal
pasutten
of mutualism:(i.
wthesimilabriu
a
densitybreipintmndo
ofhlo densmityrcipientsrfchingh
mtho
lse characeriti
or e
o behaviion
vimprtn
of density
efecssthe
dhensity-independentreipechnts.the
oftwo-speciesmutualistic
systemscan be seenfromtable1.
as follows.(1) In thosesystemswheremaxThese resultscan be summarized
MODELS OF FACULTATIVE MUTUALISM
857
density,
mutualism
exertsa stabilizing
imalbenefits
are incurred
at low recipient
beingequal,return
timetoequilibrium
effect.
In otherwords,all otherparameters
is morerapidforsuch two-speciessystemsthanforsystemsof noninteracting
Thisresultis
species.Furthermore,
thereis alwaysa stable,feasibleequilibrium.
of a generalnature,applyingbothto cases in whichthe impactof mutualism
decreasesexponentially
withrecipientdensity(eq. [11]) and to thosein which
effects
density,detrimental
mutualistic
benefits
decreaselinearlywithrecipient
whenrecipient
densityexceedsK (eq.
resulting
fromtheinterspecific
interaction
at highselfprimarily
benefit
[13]). (2) Mutualisticsystemsin whichrecipients
densityexhibitverydifferent
behavior.Two speciessystemsbasedon equations
and unstable,
on theparameter
values,eitherpersistent
(4) or (8) are,depending
or stablebutless so thanthecomparablesystemwitha pairof noninteracting
species.For a pairof specieseach describedby equation(6), thereis alwaysa
whichlikewise,is less stablethanthatfornoninstable,feasibleequilibrium,
mutualism
(eq. [2]) eitherdoes not
teracting
species. (3) Density-independent
affecttherelativestability
of thesystemas comparedto thebehaviorof noninon the
system,
depending
teracting
species,orresultsinan unstablebutpersistent
parameter
values.
inall ofthesemodelsresultsin speciespersistence.
In otherwords,
Mutualism
systemsarealwaysmutually
invasibleand do
two-species
facultative
mutualistic
occursforpairsof
notlead to speciesextinctions.
In thosecases whereinstability
unbounded
recipients,
it reflects
highdensityrecipients
or density-independent
The persisratherthanpopulations
decreasing
to zerodensity.
population
growth
and
has also beennotedby Vandermeer
tenceofunstablefacultative
mutualists
topopulation
sizewouldarisefromfactors
Boucher(1978).Presumably
limitation
incompetitive
instabilextrinsic
tothemodelsinsuchcases. In contrast,
systems,
on
andmutualinvasibility
is conditional
ityis associatedwithspeciesextinction,
values(MacArthur
1972).
parameter
that
systems
mutualistic
The assumption
is madeintheanalysesoftwo-species
arecontinupair.Becauseeigenvalues
r,K, andmareidentical
forthemutualistic
values
ous functions
oftheparameters
ofa model,forsmallchangesinparameter
resultshold
theeigenvalueswillchangea limitedamount.Thus,ourqualitative
overa rangeofparameter
valuesin whichmutualistic
speciesare notidentical.
withmixed
we didnotanalytically
determine
thebehaviorofsystems
Although
different
responsesto
models(i.e., wheremutualistic
partners
havequalitatively
mutualism
withrespectto self-density),
we can makesomegeneralpredictions
simulations
aboutthebehaviorofsuchsystems.The modelsusedincomputer
by
Addicott(1981) are mathematically
analogousto threeof the modelsderived
one fora highdensityrecipient,
recipient,
herein,one fora density-independent
and one fora low densityrecipient.In our workwe showthatthedynamical
behaviorof the systemdiffersbetweenthese classes of modelsand exhibits
commonattributes
withineach class ofmodels.It is thuspossibleto extrapolate
of
fromAddicott'ssimulations
patterns
withmixedmodelstogenerate
ofsystems
witha
thebehaviorof mixedmodels.A combination
of a low densityrecipient
resultsin systems
or witha highdensityrecipient
density-independent
recipient
of
whichare morestablethanthatfornoninteracting
species.The combination
858
THE AMERICANNATURALIST
and density-independent
highdensityrecipients
recipients
does notproducethis
effect.
Thus,mutualists
whichaccruethegreatest
percapitabenefits
frommutualism at low self-density
enhancesystemstability,
bothwhentheyoccurwitha
partnerwithqualitatively
similarresponsesto mutualism
withrespectto selfdensityand whentheyoccur witha mutualistic
partnerhavinga qualitatively
different
response.
How frequently
do eachoftheseclassesofmutualists
occurinnature?Thedata
at
the
existing
presenttime,although
somewhat
limited,
suggestthateachofthese
classes of mutualists
and
that
low
are perhapsmore
occur,
densityrecipients
commonthanhighdensityrecipients.
This is nottoo surprising
since,in many
mutualistic
systems,mutualism
cannotoverridetheeffectof all otherdensitydependent
factorslimiting
facilipopulation
growth.
Speciesforwhichmutualism
tatesnutrient
procurement
wouldbe a goodplace to lookfortheoccurrenceof
highdensityrecipients.
Predictions
aboutwhichdensityresponsesshouldoccur
withgreatestfrequency
in mutualistic
systemsare difficult
to makebased solely
inthispapersinceourmodelsarenotevolutionary
on themodelspresented
ones.
The conclusionthatcertainclassesofinteractions
areinfrequent
sincepopulation
dynamicmodelsare unstable,thoughsometimesasserted,can be misleading
when modelsdo not take into accounttemporalor spatialvariability,
other
interspecific
interactions,
age structure,
or evolutionary
This is
considerations.
truein the case of facultative
particularly
mutualism
sinceinstability
does not
Effectsof recipientdensityin real mutualistic
implyextinction.
systemshave
beenat leastpartially
in thefollowing
characterized
cases.
In some systemsit appearsthatinterspecific
interaction
at low
is beneficial
densitiesand detrimental
at highrecipient
densities,in a mannersimilarto that
modeledin equation(13). Addicotthas foundthatin ant-aphid
associationson
Epilobiumaugustifolium
aphidsbenefit
moststrongly
fromthepresenceofantsat
and thatantscan actuallybe detrimental
low-density
at highaphiddensitywhen
thelatterare experiencing
mimfoodshortage
(Addicott1979).CertainMullerian
at low
ics such as Heliconiusspecies are thoughtto interactmutualistically
densitiesbutnotat highdensities.Presumably
a criticaldensityofbutterflies
is
necessaryforpredators
to learnthesebutterflies
aredistasteful.
Athighdensities,
however,theinterspecific
interaction
since
is detrimental
ratherthanmutualistic
forresourcesthenoccurs(Gilbert1983).
competition
Othermutualistic
are probablyadvantageous
deninteractions
at low recipient
sitiesand less effective
butnotharmful
at higherdensities.Such a situation
can
occurwhendensity-dependent
limitations
on birthrateor survivalare not removedthroughmutualism.
It mayalso arise whencompetition
formutualistic
partnersnormally
at low recipient
occurs,promoting
higherper capitabenefits
density.One possiblecase involvesthe extrafloral
nectarybearingcomposite,
Helianthellaquinquenervis,
at subalpineelevations(Inouyeand Taylor1979).It
has also beenpostulated
thatpollinators
arelimiting
incertainpollination
systems
(Bierzychudek
1981;Heinrich1975;Levin and Anderson1970;Mosquin1971;
Reader1975;Waser1978).In thecase ofpollination
systems
thisdoes notalways
meanthatlowerplantdensitywouldincreasethe probability
to
of visitations
MODELS OF FACULTATIVE MUTUALISM
859
individualplants since higherdensitymay be needed to attractpollinatorsto a
patch or to maintainpollinatorconstancy(Heinrich 1979a; Levin 1979).
Density effectsare known to be importantin pollinationsystems. Pollinator
choice, constancy,and movementpatternsare dependenton plantdensity(Levin
and Kerster 1969; Carpenter1976; Gill and Wolf 1977; Levin 1978; Schaal 1978;
Heinrich 1979b) and these in turnaffectsuccess of gamete transferforthe plant
(Levin and Kerster1969; Levin 1972; Frankieand Baker 1974;Frankieet al. 1976;
Price and Waser 1979). Plant densitycan have positive(Platt et al. 1974; Beattie
1976; Silander 1978) or negative(Antonovicsand Levin 1980) effectson pollination success.
Plant density also affectssuccess of seed dispersal for plants. Sanguinaria
canadensis seeds, for instance, are dispersed longer distances at lower plant
densities(Pudlo et al. 1980). Data on the effectof densityon seed dispersal are
limited.The interactionof seed-dispersalbehavior,geneticstructureof the plant
population, distributionof safe sites, and plant densitywould be expected to
influencethe effectof plant densityon success of seed dispersaland subsequent
germination.
Furthercharacterizationof real mutualisticsystemson the basis of density
responses and comparisonsbetween these systemswithrespect to stabilityand
persistencepropertieswould be a richand potentiallyrewardingarea forresearch.
We conclude fromthe models presentedthatfacultativemutualisticsystemsare
persistentunder all conditionsconsidered. Furthermore,mutualism,even using
the stringent
propertiesof Lyapunov stability,need notbe destablizing,and when
greatestper capita benefitsaccrue at low recipientdensities,mutualismenhances
stability.The response to mutualismwithrespectto recipientdensityis thusone
key parameterin determiningthe dynamicalbehaviorof mutualisticsystems.
SUMMARY
Six populationmodels of facultativemutualismare formulatedin termsof per
capita births and deaths. Each explicitlyconsiders the per capita impact of
mutualismwith respect to recipientdensity.The models include one with per
capita benefitsof mutualismindependentof recipientdensity,three models of
density-dependentmutualismwith effectsmost pronounced at high recipient
density,and two models withmutualisticbenefitsmostpronouncedat low recipient density.The dynamicbehavior of systemsof species pairs withlike density
mutualism
responses is analyzed. The model incorporatingdensity-independent
resultsin a systemthat,dependingon parametervalues, is eitherunstableor has
stabilityequal to thatof systemsof noninteracting
species. The threemodels with
highdensitymutualismeach resultin systemsthatare eitherunstableor thatare
species. The two
stable, but less so thancomparable systemswithnoninteracting
models withlow densitymutualismeach resultin systemsthatare always stable,
and always more stable thancomparablesystemswithnoninteracting
species. All
of these mutualisticsystems,whetherstable or unstable,exhibitspecies persistence.
860
THE AMERICAN NATURALIST
ACKNOWLEDGMENTS
We thankP. Davis, D. Barnes, C. Galen, M. Oldfield,B. Pierson,D. Marshall,
J. Gray, and L. Williams for theirparticipationin stimulatingdiscussions. We
would also like to thank E. Pianka, M. Uyenoyama, C. Galen, N. Waser, L.
Gilbert,C. Toft,D. Boucher, and two anonymousreviewersforhelpfulcritiques
of the manuscript.This articleis based on the Master's thesis of C. L. W.
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