Parasitoid Competition and the Dynamics of Host-Parasitoid Models Author(S): Andrew D

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Parasitoid Competition and the Dynamics of Host-Parasitoid Models Author(s): Andrew D. Taylor Reviewed work(s): Source: The American Naturalist, Vol. 132, No. 3 (Sep., 1988), pp. 417-436 Published by: The University of Chicago Press for The American Society of Naturalists Stable URL: http://www.jstor.org/stable/2461991 . Accessed: 06/12/2011 18:26

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http://www.jstor.org Vol. 132, No. 3 The AmericanNaturalist September 1988

PARASITOID COMPETITION AND THE DYNAMICS OF HOST-PARASITOID MODELS

ANDREW D. TAYLOR* Imperial College at Silwood Park, Ascot, BerkshireSL5 7PY, England SubmittedMarch 10, 1986; Revised July24, 1987; Accepted November 12, 1987

Both parasitoidsand predatorscompete intraspecificallyfor prey or hosts. The natureof thiscompetition, however, is potentiallymuch more complex and varied forparasitoids than for predators.With predators, prey are generallyconsumed upon captureand thus cease to be bones of contention:competition is simplyfor discovery (or capture) of prey. In contrast,parasitoids do not consume hosts immediatelyupon discovery; the hosts remainavailable to be discovered-and possibly parasitized-again. An individualparasitoid's reproductionfrom a particularhost generallydepends, therefore,on whetherthe host has alreadybeen, or is subsequently,discovered by anotherfemale; that is, there is competitionfor exploitationof individual hosts (within-hostcompetition) as well as for their discovery(across-hosts competition). Parasitoidcompetition naturally is quite importantin host-parasitoiddynamics. As shown below, it is necessary for stabilityin manyhost-parasitoid models. In addition,the various formsof heterogeneitythat have been the focus of much currenttheory (e.g., Bailey et al. 1962; Hassell and May 1973, 1974; Beddingtonet al. 1978; May 1978; Hassell and Anderson 1984) all act largelyby increasingthe "discovery" componentof competition(Taylor, MS). In contrast,the within-host aspect of competitionand the potentialdifference it creates betweenpredator and parasitoid dynamics have been almost totallyignored. The standardmodels all assume what is in essence an extremeform of contestcompetition: the numberof (female) parasitoidprogeny produced fromeach parasitizedhost is independent of the numberof times that the host was encountered.Only two models with differentwithin-host dynamics have been proposed previously,and in neither were overall dynamicsstudied. Thompson's (1929) model completelylacked not only parasitoidcompetition (i.e., every encounterproduced the same numberof progeny)but also any numericalresponse to host density;thus, no equilibriumor even persistence was possible. The model of Griffithsand Holling (1969), in contrast,was in many ways similar to those developed below (includingthe

* Presentaddress: Departmentof Zoology, Ohio State University,1735 Neil Avenue, Columbus, Ohio 43210-1293.

Am. Nat. 1988. Vol. 132, pp. 417-436. ? 1988 by The Universityof Chicago. 0003-0147/88/3203-0004$02.00.All rightsreserved. 418 THE AMERICAN NATURALIST

2.0

i.5

cn 1.0-G - 0 L a _ _c 0.5

0.0 l l l l l 0 i 2 3 4 5 6 encounters

FIG. 1.-Estimated encounter-dependentyields of three solitaryparasitoids: a, Glypta fumiferanae,McLeod 1977; b, Apantelesfumiferanae,McLeod 1977; c, Diadromus collaris, egg-distributiondata fromLloyd 1940, larval survival data fromKalmes et al. 1983. The estimatedproportion of multipleencounters resulting in superparasitism(table 1) was combined withdensity-dependent larval survivalrates to obtain the predictedmean yields. negative-binomialdistribution of encounters),but theydid not analyze its multi- generationdynamics. The within-hostdynamics of few parasitoids-that is, the relationshipbetween the numberof encountersand the level of parasitoidrecruitment from a given host-are actually known under naturalconditions. The available evidence does suggest, though, that the "constant-yield" assumption of standard theory is probablyincorrect for some (thoughprobably not for most) species. Failure of the assumption-that is, variationin the numberof progenyfrom a host, depending on the numberof timesthe host was encountered-requires thatfemales at least sometimesoviposit in or on previouslyparasitized hosts (superparasitize,in the broad sense of Fiske 1910) and thatlarval competitiondoes not nullifythe effects of thissuperparasitism. I have been able to finddata on both these aspects (fig.1) foronly threespecies (all solitary):of these, one shows a substantialincrease in yieldwith increasing numbers of encounters.Evidence of one or the otherof the requirementsfor variable yield, however, is available for many species. Super- parasitismoccurs at least occasionally, and in some cases frequently,in the solitaryspecies in which it has been examined (table 1); it is not so clearly observable in gregariousspecies in the field,but it has been reported(Shiga and Nakanishi 1968; van Alphen and Nell 1982; Takagi 1987), and it is common in laboratorystudies (reviews in van Lenteren 1981; Taylor 1984, 1988b). Larval competitionhas been studiedextensively in gregariousspecies (see Taylor 1984, 1988a), and yield has been shown to increase continuously(e.g., Shiga and Nakanishi 1968), to rise to a plateau (e.g., Klomp and Teerink 1967), or firstto increaseand thento decrease as densityrises (e.g., Takagi 1985); in no case was a PARASITOID DENSITY DEPENDENCE 419

TABLE 1 FIELD EVIDENCEOF SUPERPARASITISMBY SOLITARY PARASITOIDS

FREQUENCYOF SUPERPARASITISM*

PARASITOID SOURCE Mean Range Nt

Diadromus collaris Lloyd 1940 0.84 1 Synomelixsp. Carl 1976 0.83 1 Eurytomarobusta Varley 1941 0.79 1 Pimplopterusdubius J0rgensen1975 0.65 0.43-0.86 2 Lathrolestesluteolator Carl 1976 0.50 0.0-1.0 2 Apantelesfumiferanae McLeod 1977 0.49 0.0-1.0 15 Glyptafumiferanae McLeod 1977 0.48 0.0-1.0 15 Exenterusamictorius McLeod 1972 0.44 0.24-0.61 5 Ibalia leucospoides Chrystal1930t 0.30 0.0-1.0 26 Collyriacalcitrator Salt 1932 0.28 1 Angitia sp. Lloyd 1940 0 1 Angitia sp. Ullyett1943 0.24 0.16-0.33 5 Exenterusdiprionis McLeod 1972 0.21 1 Aphelinusmali Thompson 1934 0.19 0.02-1.0 9 Limneriumvalidum Timberlake 1912? 0.17 1 Eurytomacurta Varley 1941 0.01 1

NOTE.-Included are all available studiesreporting the distributions of eggs or progenyacross hosts, excludingsamples withparasitism of 100% or less than5% and samples in whichmore than one egg is likely to have been laid on some encounters. Additional unquantifiedreports of superparasitism: Ooencyrtuskuvanae, Lloyd 1938; Trioxysindicus, Singh and Sinha 1982; Ceratobaeus sp., Austin 1984; Diadromus pulchellus, Labeyrie and Rojas-Rousse 1985. * Proportionof encounters with previouslyparasitized hosts that result in superparasitism,cal- culated as in Rogers 1975. t N is the numberof separate samples reported. t In Rogers 1975; originalnot consulted. ? In Salt 1934; originalnot consulted. constantyield independentof egg densityobserved. Yield apparentlyvaries with egg densityin a numberof solitaryspecies also (table 2), thoughin manyothers larval competitiondoes produce a constantyield (fig. 1; Ryan 1971; Wylie 1971; Gerling1972; Khalaf 1982; review in Salt 1961). Almost certainly,then, parasitoids exhibit a wide range of within-hostdynamics, many of which violate the constant-yieldassumption of standard theory. What follows is an investigationof some of the consequences of this in simple modificationsof standardhost-parasitoid models.

A GENERAL MODEL FOR WITHIN-HOST COMPETITION Withinthe standardframework of simple,deterministic discrete-time, discrete- generationhost-parasitoid models (Hassell 1978), recruitmentof parasitoidprogeny into the next generationcan be expressed in the followinggeneral form (Griffithsand Holling 1969):

00 P+ = Ht p(iH,Pt) h(i Ht,P,), (1) i = 1 420 THE AMERICAN NATURALIST

TABLE 2

SOLITARY SPECIES IN WHICH YIELD VARIES WITH EGG DENSITY

Species Source

QUANTIFIED DENSITY RESPONSES Trichogrammajaponicum lyatomi 1958 Diadromus pulchellus Labeyrie & Rojas-Rousse 1985

UNQUANTIFIED REPORTS Several spp. Pierce 1910 Collyriacalcitrator Salt 1932 Trichogrammaevanescens Salt 1936 T. e. minutum Narayanan & Chacko 1957; Chacko 1969 Nemeritiscanescens Simmonds 1943 Pimplopterusdubius J0rgensen1975 Caliroa cerasi Carl 1976

whereHt and P, are the densitiesof hosts and offemale parasitoids in generationt; p(iIHt,Pt) is the proportionof hosts encounteredi times,which may depend on the densitiesof both hosts and parasitoids;and h(i IH,, P,) is the numberof female progenyproduced fromeach such host. In words, thisclasses hosts by how many timesthey are encountered,and thensums the parasitoidprogeny from each such host class, weightedby its abundance. The set of p's representsthe across-hosts componentof parasitoidcompetition, and the h's the within-hostcomponent. The correspondingrecruitment of the host populationdepends on the proportion of hosts escaping parasitism,p(O), and theirper capita reproduction,f(Ht):

Ht + I = Htf(Ht)p(O IHt, Pt) . (2) In general,h mightdepend on the densitiesof thetotal populations (as writtenin eq. 1), as well as on i. The distributionof encountersacross hosts (i.e., thep(i)'s) willalso oftendepend on bothparasitoid and host densities,and hostreproduction (f) may be density-dependentas well. In orderto isolate the effectsof parasitoid competitionfrom any density dependence in the host population, however, I assume that(1) h depends only on the "encounterdensity" (i) of a givenhost; (2) the distributionof encountersis independentof host density,and only its mean varies with parasitoid density; and (3) f is density-independent.The effectof relaxingassumption 2 and the relationshipbetween the dynamicsof the present models and those of Hassell et al. (1983) and Comins and Wellings(1985), which share assumptions2 and 3 but in effectassume thath depends on P, or Ht but not on i, are consideredin the Discussion; the effectsof densitydependence inf and h(i) are analyzed in a later paper. Because of the interdependenceof the within-hostand encountercomponents of competition,the models will includeheterogeneous parasitism: the distribution of encountersacross hosts, the p(i)'s, are assumed to followa negative-binomial distribution(Bailey et al. 1962; Griffithsand Holling 1969; May 1978). This is not meant to representany particularform of heterogeneity(e.g., spatial density dependence),but ratherthe generalphenomenon of variabilityamong hosts in the PARASITOID DENSITY DEPENDENCE 421

A . .dB a d C . 0) 0 ------C a .b 1 2 3 4 5 6 1 2 3 4 5 6 encournters

FIG. 2.-Within-host dynamicsof the models used in thispaper. A, Contestmodel (eq.. 4): a, d = 0.333; b, d = 1; c, d = o?,i.e., constant-yieldmodel (eq. 6); d, d = 0, i.e., density- independent(eq. 3). B, Scramble model (eq. 5): a, y = 0.25 (d = 0.2877); b, y = 0.5 (d = 0.6931); c, y = 0.75 (d = 1.386); d, y = 0, i.e., density-independent(eq. 3). risk of parasitism(Chesson and Murdoch 1986); some of the limitationsof this model are addressed in the followingsection and the Discussion.

SPECIFIC MODELS

Within-HostRecruitment Three simple functions,encompassing a wide range of dynamics(fig. 2), are used forthe h(i). In the first,parasitoids do not compete,and progenyproduction per encounteris constant: hdi(i) = ci; (3) these dynamics should not be confused with those of standardmodels (eq. 6, below) in whichprogeny production per host is constant.In equation (3) and in the followingmodels, c representsthe progenyproduced fromhosts encountered once, that is, in the absence of interclutchcompetition; Hassell (1978, eq. 1.lb) introducedthis parameterin the context of a constantyield per host but then assumed thatit equals one. The two remainingh's incorporatemonotonically increasing and hump-shaped density-yieldmodels, whichI referto as "contest" and "scramble," respectively: hc(i) = ci(1 + d)I(1 + di); (4) h,(i) = cie-d(i-l) = ci(l - y)ilI (5) The labels "contest" and "scramble," followingHassell (1975), are introduced solely for ease of referenceand are not meant to implymechanisms. In these models, d scales the densitydependence, withlarger values representinga more rapid decrease in progenyproduction with increasing i. If d = 0 in eitherof these models, equation (3) results.The term1 + d appears in the numeratorof equation (4), and the exponentin equation (5) is i - 1 ratherthan i, in orderthat h(1) will equal c. The parametery = 1 - e -d is introducedin equation (5) fornotational convenience below. 422 THE AMERICAN NATURALIST

The crucial distinctionbetween these two models is thatin the contestmodel the yield, h(i), always increases withdensity, i, but the scramblemodel is overcompensating:yield initiallyincreases withdensity but thendecreases as density increases further.In particular,if y in the scramblemodel is greaterthan ?/2(i.e., d > In2), the numberof progenyfrom a host encounteredmore than once will be less than thatfrom a host encounteredonly once. The model of Griffithsand Holling (1969), mentionedin the introduction,used h(i)= ci, i ? b, (6) h(i) = c(b -a), i > b, withb a maximumsupportable density and a "the degree of scramble" (i.e., the decrease in yieldcaused by excess density).These within-hostdynamics, with an abrupt,possibly even stepped, transitionfrom density independence to perfect constancy, are somewhat peculiar. When combined with a negative-binomial distributionof encounters,though, they produce population-recruitmentcurves qualitativelysimilar to those of the models in thepresent paper (cf. theirfig. 6 with fig.3, below); the resultingpopulation dynamics, which have not been analyzed, presumablywould also be similar. Most previous host-parasitoidmodels, the constant-yieldmodels, imply the function h(i) = c, h(O) = 0, (7) where i > 0. Note thatthis model is the limitingcase of the contestmodel (eq. 4) withd infinite. Distributionof Encounters As noted above, I assume thatp(i) follows the negative-binomialdistribution (Bailey et al. 1962; Griffithsand Holling 1969; May 1978). I also assume that(1) the distributionis independentof host density (i.e., there is no handlingtime [Holling 1959] or any aggregativeor otherresponse to absolute host density);(2) the mean of the distributionis a constantmultiple of the numberof parasitoids, aPt (e.g., there is no mutualinterference; Hassell and Varley 1969); and (3) the aggregationof the distribution,defined by the parameterk, is constant. Althoughthey have not been investigateddirectly, the firsttwo of these as- sumptionsseem unlikelyto have substantialqualitative effects, though mutual interferenceprobably would somewhatweaken theeffects of within-hostcompetition. The negative-binomialmodel with k independentof densities has been criticized recently (Chesson and Murdoch 1986; Perry and Taylor 1986; cf. Griffithsand Holling 1969). For present purposes, however, a constant k is advantageous: it allows the effectof parasitoiddensity dependence to be analyzed in the absence of any host densitydependence. The Parasitoid Models The density-independentmodel is extremelysimple: every encounteryields c progeny,regardless of how manytimes a particularhost is encountered.Recruit- PARASITOID DENSITY DEPENDENCE 423

5 A. a B. --b0 0 V45 I\C

30O

0 5 10 15 20 0 5 10 15 20 a P tt) a P tt)

FIG. 3.-Parasitoidrecruitment curves, with host densities (caH,) foreach curvefixed at equilibriumvalues, and In f = 3, k = 1.20. Dottedlines indicate exact replacement.A, Contestmodel (eq. 9): a, d = 0.333(caH* = 4.90);b, d = I (caHI*= 8.14); C, constant-yield model(eq. I11;caHl* = 14.1). B, Scramblemodel (eq. IO): a, y = 0.25 (caH* = 18.8);b, y = 0.5 (caH* = 63.3); C, Y = 0.75 (caH* = 138);d, y = 0.2085(equivalent competition to constant-yieldmodel; caH* = 14.1). menttherefore depends only on the numberof encounters,not on theirdistribu- tion. The model is Pt+l I- caPtH,. (8) Combiningthe two density-dependenth(i)'s of equations (4) and (5) withthe negative-binomialdistribution of encounters(following Ives and May 1985) yields thefollowing parasitoid recursion equations forthe contestand scramblemodels, respectively: I Pt+ I = caPtHt d d iUl/d[I + (I - u)aPtlk] -(k+ ')du; (9)

P + I= caPtHt(l + yaPt 1k) -(k +1). (10) The comparable constant-yieldmodel is thatof May (1978):

P,+, = cHt[I - (I + aPtlk) -k]. (11) Typical recruitmentcurves forthese models are shown in figure3. The differences in theforms of these curves directlyreflect the differences in thewithin-host competition(fig. 2); the scramble model usually has overcompensatingcompetition (humped recruitmentcurves) in the populationas a whole, as well as within hosts, and both the contestmodel and its limitingcase, the constant-yieldmodel, are never overcompensating.

The Host Model The dynamicsof the host populationfrom one generationto the next do not depend on how the parasitizedindividuals are used by the parasitoids.Assuming thatno densitydependence acts directlyon the host's rate of increase, the usual model is Ht+ fHt(I + aPt 1k)-k (12) wheref is the constantfinite rate of increase forthe host. 424 THE AMERICAN NATURALIST

,d 50 -

40 /

1o ,

0 --- i 0.2 0.4 0.6 0.8B 1.0 y

FIG. 4.-Host equilibriain thescramble model (eq. 16),as a functionof the intensity of within-hostcompetition. Minimum stable y is indicatedby +. a, lnf = 1, k = 2; b, lnf = 1, k = 1; c, lnf = 2, k = 2; d, lnf = 2, k = 1.

ANALYSIS

Equilibria The parasitoidequilibrium, P*, is entirelyindependent of the h(i)'s:

p* = k(fllk - 1)/a. (13) The host equilibrium,H*, however,increases withincreasingly intense within- host parasitoidcompetition: density-independent, H* = 1/ca; (14) contest,

H* = d Ica(l + d) ul/d[l + (1 - U)(fi/k - l)]-(k+1)du}; (15) scramble,

H* = [1 + y(fl/k - 1)]k+l/ca; (16) constant-yield,

H* = kf(fl/k- 1)/ca(f - 1). (17) This increase in the host equilibriumwith increasingparasitoid competitionis shown, for the scramble model, in figure4. The parasitoidequilibrium, P*, and thus net parasitoid competition and ultimately H* all increase withf and aggrega- PARASITOID DENSITY DEPENDENCE 425

A. 8 B. - -

k ~~~~~~~~~4 --X , ~...... ~ ~~ ~ .: ~ ~ ~:: ~ ~~~~~~~-7 . . . . 2 - o.o 0 123 4 56 0 1 23 4 56 i n f In f

FIG. 5.-Local stabilityboundaries for the contestand scramblemodels (witheq. 12), with values of d as in figures2 and 3; the regionsbelow the curves are stable. A, Contest model (eq. 9); B, scramble model (eq. 10). Point X is illustratedin figure8. tion (i.e., decreased k); aggregationalso increases parasitoidcompetition and H* by increasingthe overlap, the across-hostscomponent of competition,among any givennumber of parasitoids.Host equilibrium,H*, also increases witha decrease in eithera (and thus the numberof encountersper parasitoid)or c (the parasitoid "intrinsic" rate of increase per encounter). Local Stability The effectof h(i) on the local stabilityof these equilibriais straightforward:it acts solely throughthe densitydependence withinthe parasitoidpopulation at the equilibrium(i.e., 5 = aPt+ I/aPt,evaluated at equilibrium;see the Appendix). When within-hostcompetition does not occur, such that equations (3) and (8) apply, thereis no densitydependence in theparasitoid dynamics (5 is 1); giventhe additional assumption of no direct densitydependence in the host population, stabilityis impossible (Appendix). The within-hostcompetition in the othermodels, however,allows stabilityfor some parametervalues (fig.5), and all else being equal, more-intensewithin-host competition(greater d and thus lower mean h(i)) generallyenhances stabilityby increasingnet parasitoid density dependence (reducing 5; fig. 3). Specifically, stabilitygenerally is possible withlarger k as d increases (fig.5). The contest model always has less competitionthan does the constant-yield model. Accordingly,the maximumk allowing stabilityin the contest model is always less than one (the value for the constant-yieldmodel), asymptotically approachingone as f or d becomes infinite(fig. 5A). In contrast,density dependence is generallygreater in the scramble model than in the other models: 5 generallyis negative.As a result,at all but smallfand d, stabilityis possible withk larger than one (fig. SB). When reproductionper host is a strictlydecreasing functionof the numberof encounters(i.e., when y > ?/), competitionis always moreintense than in the constant-yieldmodel, and the stabilityboundary (fig. SB) is always at k largerthan one. Stabilityis possible, in fact,for any finitek in the scramble model, given sufficientlyintense competition; with random encounters (k = oc), it is not quite possible (Appendix). A novel featureof the scramblemodel is thatstability can actuallybe lost as the 426 THE AMERICAN NATURALIST

A. B. (I) WD i6 i6

. 2i4O 8 i4 \

(Dwi2 i2 - 0O 2i o 0 2 4 6 8 jo generation generation

FIG. 6.-Example populationtrajectories for the scrambleand constant-yieldmodels, with y forthe scramblemodel chosen such thatthe mean recruitmentper host, and thus caH*, is the same in the two models. lnf = 3, k = 1.2. Solid lines, hosts; dashed lines,parasitoids. A, Scramble model, y = 0.208524; B, constant-yieldmodel.

resultof increases in any of the factorsthat generally favor stability, that is, f, d, or aggregation(low k). Althoughthis occurs only withvery large f (extremeright of fig.5B) in the presentmodel, it occurs at lowerf ifhost densitydependence is also overcompensating(Taylor, MS); it cannot occur in the contestor constant- yield models. As figure5 shows, stabilityis also generallyfavored by increasingfordecreas- ingk (except in the constant-yieldmodel; May 1978). This effectoff is indirect,by increasing P* and thereby increasing across-hosts competition. Aggregation (smallerk) also acts on across-hostscompetition, both indirectly by increasingP* and directlyby increasingthe overlap among the parasitoids. The possibilityof parasitoid recruitmentcurves with differentshapes has an importantconsequence: two models of differentform may have the same average recruitmentper parasitized host (average h) at equilibriumbut very different sensitivityto variationin density(5). They could thushave the same equilibriabut differin stability.Specifically, y in the scramblemodel can be chosen (forgivenf and k) such that equilibria are the same as in the constant-yieldmodel, but the marginaldensity dependence (6; fig.3) and stability(fig. 5) of the scramblemodel will always be greater. The population trajectoriesin figures6-8 illustratethese effectsof parasitoid densitydependence (5) on stability.Figure 6 compares stable dynamicsresulting fromstrong parasitoid competition with unstable dynamics resulting from weaker densitydependence. Strongcompetition causes the parasitoids,when abundant, to decrease rapidly,at the same time that the hosts are decreasing because of intenseparasitism; similarly, relaxation of this strongcompetition at low density allows the parasitoidpopulation to increase and keep up withthe now-increasing host population. In contrast,weak (fig.6, unstable trajectory)or absent (fig.7) parasitoidcompetition allows large parasitoidpopulations to continueincreasing even while the hosts are decreasing and, conversely,to continuecrashing even afterthe hosts have startedto recover. In essence, dampingthe fluctuations of the parasitoidpopulation decreases the lag between its cyclingand thatof the hosts; withless parasitoidcompetition, the parasitoidsdo not trackthe hosts as closely, and divergingoscillations result. The instabilityresulting from overly strong parasitoid density dependence (very PARASITOID DENSITY DEPENDENCE 427

I\1 2[_ < o I 12000 30 I i50 1r2 oo20

(f I 5 (0 5000i

. O O o 0 4 8 i2 i6 0 4 8 i2 i6 20 generation generation

FIG. 7 (left).-Examplepopulation trajectory for the density-independent model. lnf = 2, k = 0.5. Solid line, hosts; dashed line, parasitoids. FIG.8 (right).-Examplepopulation trajectory for the scramble model with instability due to excessive competition(see fig.5B). lnf = 6, k = 4, y = 0.75. Solid line,hosts; dashed line, parasitoids. negative8) in the scramblemodel is shown in figure8: a large parasitoidpopulation crashes because of intense competition,which then allows both parasitoids (now released fromcompetition) and hosts (withtheir huge reproductiverate and now littleparasitism) to reboundto even greaterlevels, and so on. The cycles of the two populations(of period two) are now exactlyin phase, and theyare similar to the cycles caused by overcompensatingcompetition in single-speciesmodels.

DISCUSSION These resultsshow thatthis previouslyignored aspect of parasitoidbiology- the opportunityfor competitionarising fromthe possibilityof a host's being discovered and exploited more than once-can indeed substantiallyaffect host- parasitoiddynamics. On the one hand, quantitativeconclusions from such simple models should not be taken literally.On the otherhand, the qualitativeeffects of within-hostcompetition on the rate and densitydependence of parasitoidreproduction,and thus on abundances and stability,seem likelyto be robust;whether theyare importantin natureis an empiricalquestion depending on which "within- host" dynamicsactually occur and on whetherother factors swamp theireffects. The effectivenessof a parasitoid in suppressingits hosts is oftenthought to depend primarilyon its search rate, the mortalityit inflictson its hosts in a generation.The above models, however,emphasize thatparasitoid reproduction is equally important.The parasitoidequilibrium-the densityof parasitoidsthat will kill enough hosts to balance theirreproductive rate-depends directlyon searchingeffectiveness. The host equilibrium,however, is the densityrequired to maintainthe parasitoidpopulation, and this obviously depends not only on how many hosts are attacked but also on how much parasitoid reproductionresults (Hassell and Moran 1976) and thus on both the density-independent(c) and density-dependentaspects of per-hostrecruitment. Similarly, the productca (in essence, the per-hostintrinsic rate of increase of the parasitoid)sets the natural scaling for refugesand carryingcapacities (Taylor, MS), and the ratio of these reproductiverates-not the ratio of the a's alone (May and Hassell 1981)- determineswhether parasitoid species can coexist (Taylor, MS). (Note thatthese 428 THE AMERICAN NATURALIST pointsapply also to continuous-timepredator-prey models, in whichc is the rate of conversionof capturedprey into predators.) Within-hostcompetition of course not onlyaffects the net reproductiverate but also contributesto overall parasitoid densitydependence. It is not surprising, then, that it helps determinestability, and indeed it may be the only source of stability,as shown by the density-independentmodel above. A less obvious point is that the instabilityin coupled consumer-resourcemodels such as these has a differentcause, and thusis affecteddifferently by competition,than the instability in single-speciescompetition models. In the lattermodels, any densitydependence at all is sufficientfor stability,with instabilityresulting only fromexcess overcompensationand thus only in scramblemodels. When the futureabundance of the resourceis affectedby consumption(as in the models above), however,the densitydependence mustcompensate for the destabilizingeffect of the timelag in the interaction(Appendix). As a result,stability in such coupled models requires greatercompetition than in single-speciesmodels, and greaterovercompensation can be tolerated before stabilityis lost (Taylor, MS). This explains both the stabilizingeffect of d and the greater stabilityof the scramble model, when compared withthe contest model in the host-parasitoidmodels above. Most other sources of stability,and in particularthose affectingthe across- hosts componentof parasitoid competition,reduce parasitoidreproduction and thus create a trade-offbetween stabilityand host suppression. In contrast,a differencein the formof parasitoidcompetition can affectstability without affect- ing equilibria, or indeed could simultaneouslylower the host equilibriumand increase stability.This importantnew possibilityimplies, among other things, that superparasitism(without which variationin per-hostyield cannot occur) can be beneficialto controlof the host, contraryto the opinionof some biological-control workers(e.g., Ullyett1943; McLeod 1972; Vinson 1977; Propp and Morgan 1984); even if, as implicitlyassumed by these authors, parasitismis limitedby the parasitoids' egg supply, the cost of superparasitismis primarilyin parasitoid reproduction(Thompson 1929).

OtherForms of Parasitoid DensityDependence Heterogeneousparasitism.-A majorpoint throughout this paper has been that bothheterogeneous parasitism (across-hosts competition) and within-hostcompe- titioncontribute to parasitoiddensity dependence, such that an increase in one can compensatefor a decrease in the other.Indeed, the effectof each depends on the other:the effectof within-hostcompetition depends on how muchit is brought intoplay by multipleencounters, and the consequences of how oftenhosts are re- encountereddepend on the within-hostcompetition. As a result,the amountof heterogeneitynecessary for stabilitycan be quite differentfrom what previous models had indicated,depending on the strengthof within-hostcompetition (for similarresults concerning true parasites, see Andersonand May 1978). The relationshipbetween heterogeneity and within-hostdynamics may oftenbe more complex than in the precedingmodels. For manyforms of heterogeneous PARASITOID DENSITY DEPENDENCE 429 parasitism,the assumptionthat the distributionof encountersdoes not depend on host densitymay not be valid (Hassell 1980; Chesson and Murdoch 1986; Perry and Taylor 1986). If it is eliminated,an increase in within-hostcompetition, by increasingH*, will change thep(i)'s as well. If thischange is towardless aggregation of encounters,as seems reasonable, the proportionof hosts escaping parasitism (for a given parasitoid density) decreases, and with it the equilibrium parasitoiddensity. These decreases in both aggregationand abundance decrease the across-hostscomponent of competitionand oppose the effectof the increase in the within-hostcomponent. A decrease in aggregationwith increasingd may also reduce densitydependence in the host population.As an extremeexample of these effectsof inverselydensity-dependent aggregation, stability can actually decrease withincreasing d in a model witha constant-numberhost refuge(Taylor, MS). This occurs partlybecause in this model, as the host equilibriumincreases (as it does withincreasing d), the host densitydependence decreases (because the refugehas proportionatelyless effect).In nature this effectwould probablybe offsetby an increase in host competition;thus, largerd mightnever actuallybe destabilizing.This example does suggest, though,that the stabilizingeffect of within-hostcompetition may oftenbe somewhat weaker, as well as more complex, than in the models presentedin this paper. Per-hostrecruitment dependent on overalldensities.-As well as, or insteadof, varyingwith the numberof times a given host is encountered,parasitoid reproductionper attackedhost could varywith overall parasitoid or hostdensities, as in the models of Hassell et al. (1983) and Comins and Wellings(1985); in the notation of this paper, c could vary withPt (and perhaps Ht). The two formsof density dependence have similar effectsfor the most part. An importantdifference, though, is that encounter-dependentwithin-host competition combines with across-hostscompetition in an interactiveway, as discussed above, whereas per- host reproductionthat depends on total population density is independentof encounterdensities and thus of the across-hosts competition.As a result, its stabilizingeffect can be strongeven withlittle overlap of encounters,and indeed, stabilityis possible even withrandom encounters (Hassell et al. 1983). Hassell et al. (1983) and Comins and Wellings(1985) intendedto model density- dependent sex ratios, arising fromeither differentialcompetitive mortality or maternalmanipulation of primarysex ratios. Both mechanisms,though, depend on variationin the numberof clutcheslaid in or on an individualhost (or at mosta local patch of hosts; forthe sex-ratiopredictions, see, e.g., Charnov 1982) so that theywould be bettermodeled as encounter-density-dependent.Clutch size might vary with overall densities,but it would increase, not decrease, with increasing density(Charnov and Skinner 1984, 1985; Parker and Courtney1984), creating destabilizingpositive feedback rather than regulation. It appears thatthe stabilizing per-hostrecruitment that varies only with overall densities,as modeled by Hassell et al. (1983) and Comins and Wellings(1985), would arise primarilyfrom sources extrinsicto the host-parasitoidinteraction per se, such as predation(e.g., Hassell 1969), whereas intrinsicfactors would primarilyproduce encounter- dependentvariation. 430 THE AMERICAN NATURALIST

Implicationsfor EmpiricalResearch To determinethe role of within-hostcompetition in the dynamics of a real system,the seeminglystraightforward approach would be to describethe h(i)'s of thispaper: to determinethat the standardconstant-yield assumption is valid or to describeparasitoid reproduction explicitly. It is also clear, however,that describ- ing the within-hostcomponent of competitionnot only may be difficult(see above), but also will be insufficient:the distributionof encountersover hosts,and its dependence on densities, must also be described. This too will be generally difficult. One way around these difficultieswould be to predictwithin-host dynamics from a parasitoid's general biology. For example, the mechanisms of larval competitiondiffer, in general,between solitaryand gregariousspecies (Salt 1961); gregariousspecies may also superparasitizemore readily. Weak within-hostcompetition,leading to low unstablehost abundances, may thusbe typicalof gregari- ous parasitoids,whereas a constantor decreasingyield, givinghigher but more- stable equilibria, may typifysolitary species. It would be unwise, however, to apply such a generalizationin a specific case withoutverification: the literal distinctionbetween gregariousand solitaryspecies is not in h(i) but in c, which affectsequilibria but not stability. A more concrete approach would be to describe parasitoidrecruitment curves analogous to those in figure3. To do thisdirectly would require,however, that the host densitybe kept constantwhile the parasitoiddensity is varied,which might be difficult.If one simplyplots parasitoidrecruitment over successive generations of a host-parasitoidinteraction, the correlationbetween host and parasitoiddensities totallydistorts the relationship(fig. 9A). Alternatively,the correlationbetween host and parasitoiddensities might actually be taken advantage of: one of the featuresof the simulationsin figures6-8 is that tighterdensity-dependent regulation of the parasitoids reduces the lag between the populations' cycles. Time-seriesanalysis, or even simple correlation, mightdetect this effect. It also shows up as a strongercorrelation between parasitismand host density(fig. 9B), thoughthis might be difficultto distinguishin noisyreal data (it is unclearwhether the slope of thisrelationship, which is whatis usually looked at, provides informationabout parasitoid densitydependence). Probablythe most usefulmethod for studying within-host dynamics is to exam- ine the mean yield per parasitizedhost, as a functionof eitherparasitoid density (fig. 9C) or the densityof attacked hosts (fig. 9D; Hassell and Huffaker1969). Overcompensating,and thus (generally)stabilizing, competition will produce a negative correlationbetween mean yield and density(figs. 9C,D). Weaker, and thereforeless stabilizing,competition, such as in the contest models above, converselyproduces a positive slope. This approach mightnot reveal the specific cause of the densitydependence (thoughcensuses at several stages mightallow this; e.g., Hassell 1969) and would not distinguishyield dependenton encounter densities(as in the models of thispaper) fromyield dependent on overalldensities (as in Hassell et al. 1983; Comins and Wellings 1985). Even in noisy data, however,this approach shouldreveal any substantialdensity dependence in yield. This relationship,combined with the relationshipbetween densities and the num- PARASITOID DENSITY DEPENDENCE 431

6-A. E B. 16 A.Itn 96 ------4J.

- + i4 9-

o i2 *'- L

11 12 13 14 15 16 12 13 14 15 16 aP (t) caH (t)

0.1.C. oI. D.

H -H 1. o - ___- - ____- -. i.0__ ___

E 1112 13 14 15 16 E 1112 13 14 15 16 aP(t) hosts attacked

FIG. 9.-Possible indicatorsof stabilizingparasitoid density dependence. Data are from figure6. Solid lines, the stable scramble model; dashed lines, the unstable constant-yield model. A, Total parasitoid recruitment;B, parasitismrate as a functionof host density;C, mean recruitmentper attacked host ("yield") as a functionof parasitoiddensity; D, mean recruitmentper attacked host ("yield") as a functionof the densityof attackedhosts. ber of hosts attacked, could then be used to constructa recruitmentcurve. Comparisons of such descriptionsof parasitoiddensity dependence should then be useful in predictingthe differencesin dynamics between otherwise similar systems,for example, a parasitoidattacking one or the otherof two similarhosts, or a single host-parasitoidpair in differentenvironments.

SUMMARY The standardhost-parasitoid models assume that each parasitizedhost yields the same number of parasitoid progeny; but in many parasitoid species, the number of progeny per host depends on the number of times that host was encountered.I presenta familyof new host-parasitoidmodels incorporatingthis encounter-density-dependentvariability in parasitoidreproduction per host. Such within-hostcompetition (which could be between ovipositingfemales, betweenlarvae, or both) interactswith the distributionof encounters(i.e., aggregation) to determinethe net parasitoiddensity dependence. Greaterdensity dependence increases host equilibriumabundances while generallyincreasing the stabilityof the interaction.Overcompensating scramble-like parasitoid competition can be more stabilizingthan contest-likecompetition, especially when the host has a moderateto highintrinsic rate of increase and encountersare moder- ately aggregated;overcompensation is destabilizingonly at veryhigh host reproductive rates. 432 THE AMERICAN NATURALIST

These results suggest that we must study parasitoid densitydependence di- rectly,both within a host and betweengenerations within a population,in orderto understandthe dynamicsof any actual host-parasitoidsystem. The mostpractical methodappears to be to (1) describe the mean yield (parasitoidrecruitment) per parasitized host as a functionof the densityof eitherparasitoids or parasitized hosts; and (2) relate the nulmber(not the percentage) of parasitized hosts to parasitoid(and perhaps host) densities.

ACKNOWLEDGMENTS I am most gratefulto M. Hassell for his advice and encouragement,and for space and computerfacilities, and to P. Chesson, A. Ives, J. Lawton, R. May, W. Murdoch, J. Reeve, and two anonymousreviewers for helpful comments on the manuscript.Part of thiswork was supportedby a NATO PostdoctoralFellowship grantedin 1985.

APPENDIX

STABILITY ANALYSES General Effectsof Parasitoid Competition Modelslike those in thetext, which assume that the proportion of hosts parasitized, the reproductiverate of survivinghosts, and parasitoidreproduction per host are all independentof hostdensity, can be representedgenerally by theequations Ht+I = fHtgH(Pt) (Al) Pt+ I = Htgp(Pt), wherefisa constant(the host per capita reproductive rate), and gH andgp are functions of Pt defining,respectively, the proportion of hosts escaping parasitism and the production of parasitoidprogeny per host. The equilibriafor this general model are H* = P*/gp(P*), (A2) P* = gij(1/f), whereg- 1 is theinverse function of gH. The hostequilibrium, H*, dependson parasitoid reproductionand competition,but P* does not.For thenegative-binomial models in the text,gH is definedby equation(11) and P* = k(f1'lk- 1)/a. (A3) Local stabilityin discrete-timetwo-species models, including the host-parasitoid models developedin thispaper, requires (May 1974)that

2 > I + otb- P- > lot+ 81, (A4) where allt+ I Allt+I apt+lIapt+ I aa 3 ~ , , and 8-

all derivativesevaluated at Ht = H* and P, =P*. In modelswith the general form of equations (Al), theonly affect of parasitoid reproduction(gp) on stabilityis the directone through&. The hostdensity dependence, ax, will PARASITOID DENSITY DEPENDENCE 433 always be exactly 1:

Ol = fgH(P*) = f(I/f) = 1. (A5) The cross-dependences1 and y each do depend on gp (or on H*, whichdepends on gp), but these appear in the stabilityconditions (A4) only as the productP3y, in which the dependences on gp cancel each other:

13Y= fH*gH'(P*)gp(P*) = f[P*/gp(P*)]gH'(P*)gp(P*) = fP*gH'(P*), (A6) where gH' is the derivativeof gH withrespect to Pt (which is independentof gp). The effectof the dependence of parasitoidreproduction on parasitoiddensity, then, is simply5:

8 =- H*gp'(P*), (A7) where gp' is the derivativeof gp withrespect to Pt. No Direct DensityDependence When there is no direct host densitydependence (i.e., a = 1), the firstinequality of condition(A4) becomes < I + 13y. (A8) In a host-parasitoidinteraction, or any other victim-consumerinteraction, 1 is almost always negative,and y positive; stabilitythus requiresthat 5 be less than 1, thatthere be some densitydependence in the parasitoid'sdynamics to counteractthe destabilizingeffect of 13y. In the model of density-independentparasitoid recruitment (eq. 8), however,5 is exactly 1: gp is simplythe constantca, and H* is the reciprocalof thisquantity (eq. 14); therefore, 5 = caH* = ca /ca = 1 . (A9) Thus, when the reproductionof both the hosts and the parasitoidslack directdependence on theirown densities,stability is not possible. The Scramble Model Withthe scrambleparasitoid model (eq. 10),

S = 1 - (k + l)y(fl/k - 1)/[l + y(fllk - 1)]. (AIO) Given thisrelationship, (A10), the conditions(A4), expressedin termsof the strengthof the parasitoiddensity dependence (y = 1 - e-d), are

y > kl(k + fl/k) (A1) and one of the following: k ' 1, (A12a) y < 21(k- l)(flk - 1), (A12b)

k(fl/k - 1) + 4f l/k (A12c) ' [(k - 2)fl/k + k](flIlk- 1)

Condition(Al 1), arisingfrom the firstinequality of (A4), is a requirementfor sufficient densitydependence to counteract13y. The second set of conditions(A12), fromthe second inequalityin (A4), prohibitsexcessive overcompensation:it is violatedwhen y is too large (and thus 5 too negative),and it limitsstability with f and y large and k moderate(i.e., the upper rightcorner of fig.5B). For k c 1 and anyf, stabilityis always possible: (A12a) is always met,and some valid y 's (i.e., between0 and 1) can be foundto satisfy(Al 1). For finitek - 1, thereexist some valid 434 THE AMERICAN NATURALIST y's thatsimultaneously satisfy conditions (Al1) and (Al2b), thusgiving stability, so longas f < {k(k + 3)1[k(k+ 1) - 2]1k. (A13) If, however, k is infinite(i.e., the distributionof encountersis random (Poisson)), the right-handside of (Al1) becomes 1, and the maximumvalue of y is 1; condition(All) cannot thenbe satisfied,and stabilityis impossible.

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