Generalist and Specialist Natural Enemies in Predator-Prey Interactions Author(s): M. P. Hassell and R. M. May Source: Journal of Ecology, Vol. 55, No. 3 (Oct., 1986), pp. 923-940 Published by: British Ecological Society Stable URL: http://www.jstor.org/stable/4425 . Accessed: 02/05/2014 15:47

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This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions Journal of Animal Ecology (1986), 55, 923-940

GENERALIST AND SPECIALIST NATURAL ENEMIES IN INSECT PREDATOR-PREY INTERACTIONS

BY M. P. HASSELL AND R. M. MAY

Departmentof Pure & Applied Biology, Imperial College, Silwood Park, Ascot and Department of Biology, Princeton University,Princeton, New Jersey, U.S.A.

SUMMARY

(1) The dynamics of a predator-prey, or parasitoid-host, interaction are considered where the predator or parasitoid is a generalist whose population is buffered against changes in the particularprey being considered. (2) The interactionis then broadened to include, in addition, a specialist natural enemy, and three questions are examined within this framework. (i) Under what conditions can a specialist 'invade' and persist in an existing generalist-prey interaction? (ii) How does the addition of the specialist natural enemy alter the prey's population dynamics? (c) How does the relative timing of specialist and generalist in the prey's life cycle affect the dynamics of the interaction? (3) The following conclusions emerge. (i) A specialist can invade and co-exist more easily if acting before the generalists in the prey's life cycle. (ii) A three-species stable system can readily exist where the prey-generalist interaction alone would be unstable or have no equilibrium at all. (iii) In some cases the establishment of a specialist leads to higher prey populations than existed previously with only the generalistacting. (iv) In some cases, a variety of alternative stable states are possible, either alternating between two-species and three-speciesstates, or between differentthree-species states.

INTRODUCTION

The dynamics of discrete, insect host-parasitoid interactions have been much studied since the early works of Thompson (1924), Nicholson (1933) and Nicholson & Bailey (1935) (see Hassell (1978) for a review). By having both populations coupled and synchronized with each other, it is implicitly assumed in these models that the parasitoids are effectively specialists on that one host species. Many natural enemies of , however, are polyphagous to some degree and will have rather different dynamical relationships with their prey; this is the case, for example, for many parasitoids, staphylinid and carabid beetles, birds and small mammals. In particular, a broad diet will tend to buffer the populations of such generalists from fluctuations in abundance of any one of their prey, and give dynamics that are largely uncoupled from that prey (Murdoch & Oaten 1975; Southwood & Comins 1976). Most insect populations are attacked by several natural enemies, some polyphagous and others more-or-lessmonophagous. Thus, while it is useful to know how each type alone can affect the dynamics of its host or prey, it is also important that their combined effect be understood. In this paper we first outline the dynamics of a particular generalist- host(=prey) interaction, and then use this as a basis for a three-species model in which

Correspondence:Prof. M. P. Hassell,Imperial College, Silwood Park, Ascot, BerksSL5 7PY.

923

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions 924 Generalist and specialist natural enemies generalist and specialist natural enemies act in concert. We shall be particularlyconcerned with three general questions. (a) Under what conditions can a specialist 'invade' and persist in an existing generalist-host interaction? (b) How does the addition of a specialist natural enemy alter the host's population dynamics? (c) How does the relative timing of specialist and generalistin the host's life cycle affect the dynamics of the interaction? Studies in which different kinds of species interaction are combined in multi-species models are an important step in unravelling how population dynamics can influence community structure. The manner in which predation has the potential to affect the coexistence of competitors is now quite well understood at a general level (e.g. Cramer & May 1972; Steele 1974; Roughgarden& Feldman 1975; May 1977; Fujii 1977; Comins & Hassell 1976). A similar understandingis now needed for other widespread and important interactions, such as the mix of different types of predator as examined in this paper, interactions of host, pathogen and predator, and multi-species interactions involving mutualists. We should emphasize that while the model populations in this paper may settle neatly at deterministicequilibria, natural population patterns are, of course, the result of random and deterministic processes acting together. Having said this, we believe that it often remains valuable to set the stochastic elements on one side and focus just on the deterministic processes in an attempt to understand how some key features of an interaction can promote population regulation. In this study, these 'key features' are the components of the functional and numerical responses of generalist and specialist natural enemies in relation to the rate of increase of their shared prey.

DYNAMICS OF A GENERALIST PREDATOR-PREY INTERACTION

We assume that the insect host population whose dynamics are the focus of our attention has discrete generations, such as found in many temperate Lepidopterapopulations, and is attacked during its life cycle by both a generalist natural enemy-be it another insect (predator or parasitoid), an arachnid, an insectivorous bird or a small mammal-and a specialist parasitoid. As a prelude to examining such a three-speciessystem, we first outline the dynamics of the generalist-host interaction alone against which the dynamics of the three-speciesinteraction can be readily compared. Essential ingredientsin modelling any predator-prey interaction are descriptions of the predators' functional and numerical responses (Holling 1959a,b). The functional response defines the per capita ability of the predators to attack prey at differentprey densities, and we have assumed this to take a typical type II form (recognizing that for some predators a type III or more complex form would be more appropriate).However, instead of modelling this on the basis of random encounters between predators and prey (Holling 1959b; Royama 1971; Rogers 1972), we assume a negative binomial distributionof encounters (May 1978; Hassell & May 1985), where

' Na Nt[ 1+ . (1) k(1 + aThNt) Here Nt is the number of prey in generation t, Na is the number of these attacked by Gt searching generalist predators,a is the per capita searching efficiency of the predators,Th is

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions M. P. HASSELLAND R. M. MAY 925 their 'handling time' (making l/Th the maximum attack rate per predator) and k is the parameterof the negative binomial distributiondetermining the degree of contagion in the frequency distributionof attacks amongst the Nt prey. Note that the particularform of this equation makes it most suitable for parasitoids spending the same time 'handling' healthy and already-parasitizedhosts (Rogers 1972). The equivalentexpression where only healthy prey can be 'handled' is a recurrence relationshipthat makes analysis more difficult. The differencebetween the two forms, however, has no significanteffect on the conclusions we draw below (see Arditi (1983) for functional response models intermediatebetween these two extremes). Equation (1), in contrast to its 'random equivalent' (which corresponds to k -* oc) has the merit of recognizing that many processes (spatial, temporal and genetic) combine to make some prey individuals more susceptible to predation than others. This unequal susceptibilitybetween individuals is enhanced as k decreases, and has been shown to have potentially important effects, at least on the dynamics of coupled host-parasitoid and predator-prey interactions (Bailey, Nicholson & Williams 1961; Murdoch 1977; May 1978; Hassell & Anderson 1984; Hassell & May 1985). The choice of the negative binomial over other clumped distributions is arbitrary, but it has been widely used in describing insect and other population distributions (e.g. Waters 1959; Southwood 1978; Keymer & Anderson 1979; Hassell 1980) and it does provide a very convenient and simple way of introducing non-random distributions into a variety of population models (e.g. Anderson & May 1978; May 1978; DeJong 1979; May & Hassell 1981; Hassell, Waage & May 1983; Waage, Hassell & Godfray 1985). The numericalresponse of generalist predators has attracted much less modelling effort than the functional response, despite several field studies indicating a fairly simple relationshipbetween the number of predators and the density of a particularprey species at a particulartime. Thus, Holling (1959a) gives examples for two species of small mammals feeding on sawfly cocoons (Fig. la,b), Mook (1963) shows a similar relationship for the bay-breasted warbler feeding on spruce budworm (Fig. lc) and Kowalski (1976) for a species of staphylinid beetle feeding on winter pupae (Fig. Id). These data show a tendency for Gt to rise with increasing Nt towards an upper asymptote, and have been describedhere using the expression explored by Southwood & Comins (1976):

Gt = h[1 - exp (-Nt/b)], (2) where h is the saturationnumber of predators and b determinesthe typical prey density at which this maximum is approached. In effect, we are assuming that our generalist predators have a fast numerical response in relation to changes in Nt, as would occur for instance by (a) rapid reproductionrelative to the time scale of their prey or (b) 'switching' from feeding elsewhere or on other prey species (Murdoch 1969; Royama 1979). Such numericalresponses, when combined with the functionalresponse of eqn (1), make predation density-dependentover a range of prey densities, above which the percentage predation will level off (Th = 0) or decline (Th > 0) as shown by the examples in Fig. 2. Support for such patterns under natural conditions comes best from one particular situation: the predation of lepidopterous pupae in the soil. Figure 3 shows four such examples where the generalist predators involved have in each case been identified as primarilycarabid and staphylinid beetles. The fact that none show a decline in percentage predation at high prey densities suggests that these species are not significantlylimited by

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions 926 Generalist and specialist natural enemies

10. 30 (a ) (b) 25 8 0 0 0 0 20 6 15 0~~~~~~ 4 10 2 5, Z;)c (1) 70 0 200 400 600 800 1000 1200 0 500 1000 1500

0 0 -o (1) 125 30- 0 a- (c 100 0~~~~~ 20- 75 0

50 0 10 0

25 0

0 50 100 150 0 100 200 300 Prey density FIG. 1. Numerical responses of four generalist predators. (a) Peromyscusmaniculatus Hoy & Kennicott and (b) Sorex cinereus Kerr (both as numbers per acre) in relation to the density of larch sawfly (Neodiprionsetifer (Geoff.)) cocoons (thousandsper acre). (Data from Holling 1959a.) (c) The bay-breasted warbler (Dendroicafusca) (nesting pairs per 100 acres) in relation to thirdinstar larvae of the sprucebudworm (Choristoneurafumiferana (Clem.)) (numbers per 10 ft2 of foliage).(Data from Mook 1963.) (d) Philonthusdecorus (Gr.) (pitfalltrap index)in relationto wintermoth (Operophterabrumata (L.)) larvaeper m2 (afterKowalski (1976)). In each case eqn (2) has been fitted to the data by a non-linear least squares procedure with parameter values estimated as follows (+95% C.L.): (a) h = 7.30 + 1.07, b = 76-87 + 49-16; (b) h = 24-12 + 14-02, b = 390-58 + 579-81; (c) h = 94.32 + 42.21, b = 32-97 + 42-27; (d) h = 18-36 + 9.45, b = 77-56 + 106-59.

(a) (b)

h=6

0C: 160 a-(L) h=2

50 0 Prey density

FIG. 2. Relationships between the level of predation and prey density from eqns (1) and (2) obtained by varying the numerical response parameters b and h. (a) a = 1, Th = 0.1, h = 2, k = 1, and b as shown; (b) a = 1, Th = 0.1, b = 12, k = 1 and h as shown. the maximum attack rates from their respective functional responses. Because of such clear-cut density dependence, predation of this general kind is widely recognized as having the potential to regulate a prey population (e.g. Holling 1959a; Murdoch & Oaten 1975; Southwood & Comins 1976).

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions M. P. H\SSELL AND R. M. MAY 927 (a b) 100 100

60 80

60 60

,,,' 4 40 0 100 200 300 0 0.05 01i 0-15 0c: a)0 (c) (d) n IC 100 DO0 8 80 30 - E 60

4 40

20

0 01i 0-2 0.3 0.4 0-5 1 2 3 4 Prey density FIG. 3. Examplesof density-dependentpupal mortality from generationto generationfor four speciesof soil-pupatingLepidoptera, ascribed primarily to predationby carabidand staphylinid beetle. Curves fitted by transformingthe originalpublished relationships between k-values (Varley& Gradwell1960) and log populationdensity (N). (a) Mortalityof wintermoth (0. brumata)pupae m-2; k = 0.22 + 0.31 log N (after Hassell 1980). (b) Mortalityof Pardia tripunctatanaSchiff pupae 0.18 m-2; k = 1.47 + 1.1 log N (afterBauer 1985).The original k-valueswere estimatedbetween the larval and subsequentegg stages. An arbitrarytenfold reproductiverate (F) has thus been assumedhere to avoidnegative mortalities. Other assumed values of F (if constant)will not alterthe generalshape of the curve.Such density-dependent predationwas supportedby independentfield manipulationexperiments. (c) Mortalityof Notoceliaroborana Den. and Schiff.pupae 0-18 m-2; k = 0-.39+ 0.97 log N (afterBauer 1985). F = 10 assumedas in (c). (d) Mortalityof Erranisdefoliaria (L.) m-2; k = 0.46 + 0.21 log N (afterEkanayake 1967).

Our model for a host-generalist interaction incorporatingeqns (1) and (2) thus becomes

N aFN 1-l h[ exp (-NTb)]] -k Nt+=FNt +aTN) 1 (3) k( k+ aTnN) Here Nt and Nt, are the host populations in successive generationst and t + 1 and F is the host's finite rate of increase. An unlimited host population in the absence of the generalists is assumed in order to make clearer the contribution of the generalists to the equilibrium and stability properties of the interaction. For ease of analysis we now introduce the rescaled quantities,X = N/b and ( = abT^,to give ah[1 - exp (-Xt)] -k Xt + 1=.:FXt[l (4) k(l + jXt) The equilibriumproperties of eqn (4) are displayed in Fig. 4 for the cases of zero (curves B, D) and finite (curves A, C) 'handling time' (i.e. ? = 0 and ? > 0, respectively) (see Appendix 1 for furtherdetails). Equilibriaoccur where the curves intersect the dotted 45? line and, with zero handling time (curve D), only occur if ah > k(Flk -_ 1) (5)

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions 928 Generalist and specialist natural enemies

(A) (B) (C) -10

FIG. 4. Map of the scaled prey population densities (X= N/b) in successive generations, t and t + 1, obtained from eqn (4). Equilibria occur when the curves intersect the dotted 45? line. Curve A: ah = 3, o = 0.04. Curve B: ah = 3, o = 0. Curve C: ah = 2, 0 = 0.04. A locally stable equilibrium occurs at X* and an unstable, 'release' point at XT. Curve D: ah = 2, o = 0. For further details see text. whereupon X* =-In {1 - [k/(ah)](Fl k- 1)}. (6) Otherwise, the host population is unregulated by the generalist (curve B), eventually increasing at a density independentrate given from Nt+1 = F[1 + ah/k -kNt. (7) With finite handling time (curves A and C) similar cases apply, but now even when an equilibrium exists (e.g. curve C) the host population, if sufficiently large, eventually increases unchecked. Thus, the host population is only regulatedto X* providedX remains below some thresholdvalue X < XT (see Fig. 4). A more detailed picture of how the host equilibrium,X* from Fig. 4, is influenc'edby changing k, F and ah is gained from Fig. 5a-c. Predictably,the host equilibriumfalls (i) as predation by the generalist becomes less clumped amongst the prey population (k -+ oo) (Fig. 5c), (ii) as the combined effect of the per capita efficiency and saturation number of the generalists increases (i.e. as a measure of overall predator efficiency, ah, increases) (Fig. 5a) and (iii) as the host rate of increase (F) gets smaller (Fig. 5b). Figure 5 also indicates equilibriathat are locally unstable, in which case the populations show limit cycles or even chaotic behaviour. Such persistent but non-steady states arise if the generalistpredators cause sufficiently severe density-dependenthost mortality (i.e. very rapid increases in the mortality shown in Fig. 2, in the region of the host equilibrium).This is promoted by large k, large ah, and intermediateF (excessively large values of F take the host equilibriumtowards the region where the numbers of generalists saturate at h and hence no longer respond to furtherincreases in prey density). Note that the parameterb in eqn (2) only sets the scale of prey density in relation to the equilibrium,since X* = N*/b.

HOST-GENERALIST-SPECIALIST INTERACTIONS

In broadening this framework to include a second natural enemy species-a specialist insect parasitoid-attacking the same host species, we consider the conditions allowing the

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions M. P. HASSELLAND R. M. MAY 929

( a ) (b)

4-

2-

- \ ...'

0 5 10 15 20 0 1 2 3 oh Log F

(c)

0.8-

006-

0-4-

0 1 2 3 k FIG.5. The dependenceof the host equilibrium,X*, fromeqn (4) on (a) the predatorefficiency term,ah, for F = 10 and k -, oo (left curve)and k = 1 (rightcurve), (b) the host rateof increase F, for ah = 6 and k = 1 (left curve)and k -- oo (rightcurve), and (c) the clumpingterm, k, for attackson hosts for F = 2 and ah = 10 (lowercurve), F = 2 and ah = 4 (middlecurve) and F = 5 and ah = 10 (upper curve). The dotted lines indicate locally unstable equilibria. specialist to 'invade' and co-occur with the generalist, and examine the effects of the specialists being there in terms of the equilibrium and local stability properties of the interaction. We assume the same form of functional response for the specialists as for the generalists in eqn (1), but now a numerical response determined by the number of hosts parasitizedin the previous generation-the usual case in host-parasitoid interactions. A problem that arises when hosts have discrete generations and suffer more than one mortality factor is that differentdynamics can occur, dependingupon the sequence of these mortalitiesin the host's life cycle (Wang & Gutierrez 1980; May et al. 1981). We consider two cases: Model 1 where the specialists act first, followed by the generalists, and Model 2 where the generalists act first, followed by the specialists. These are also appropriateto cases where both species are parasitoids acting on the same stage of the host life cycle, and where either specialist or generalist consistently 'win' should the same host individual be encountered, giving Models 1 and 2, respectively. A different model structure, not considered here, would be needed if the result of this competition depends on the order of encounter with a host individual.

Model 1 With the specialists acting before the generalistsin the host's life cycle, we have

X+ = FXtf (Yt)g[Xtf (Yt)] (8a) Yt+ = Xt1 =f (Y)] (8b)

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions 930 Generalistand specialistnatural enemies where G = { 1 + (ah/k)[ I - exp (-Xt/a)] }-k (9) and

f= (1 + Yt/k')-k'. (10) Here, Xt and Xt+ are the scaled host populations (X = a'cN*, where a' is the per capita searching efficiency of the specialist (cf. a for the generalists) and c is the average number of adult female parasitoids resulting from each host attacked); Yt and Yt+ are the scaled specialist populations (Y = a'P); a is the ratio of scaling factors applied to generalists and specialists, respectively (i.e. a = a'cb); and k' defines the degree of clumping in the distribution of parasitism by the specialists amongst the host population within any generation (cf. k for the generalists). Other parameters are as in eqn (3), but now handling time (for both specialist and generalist) is assumed negligible compared with the total time available for searching. Some support for this comes from the monotonically increasing curves in Fig. 3, showing no signs of the effect of a significant handling time (cf. Fig. 2). Thus, bothf and g stem from the zero time of the negative binomial distributionand define the fraction of hosts escaping from attack by specialists and generalists,respectively. Before considering any properties of this three-species system, we should bear in mind the dynamics of host and parasitoid alone (i.e. eqns (8a,b) with g = 1 andf as in eqn (10)). In this case, the interaction is stable for all k < 1, approaching the unstable Nicholson & Bailey (1935) model as k -+ oo (May 1978). As illustratedin Fig. 6, the dynamics of the three-species system characterizedby eqns (8a,b) can be complicated. The essential features can, however, be describedin a qualitative way; a more detailed discussion is given in Appendix 2. The generalistpredator can exclude the specialist, making a persistentthree-species state impossible, if the host rate of increase (F) is too low, or if the generalist'seffective attack rate (ah) is too high. The latter factor can be amelioratedby the specialist having an attack rate (a') big enough to make the scaling ratio a (=a'cb) large, or by significantclumping in the distribution of generalist attacks (k small). As F increases or ah decreases, either

4 - ? / /c :

2 -- A..

- --, , -.

Host rate of increase

FIG. 6. Three examples of the dependence of the host equilibrium, X*, on the host rate of increase, F, from eqns (8-10) (Model 1). Curve A: h = 20 and b = 10. Curve B: h = 30 and b = 10. Curve C: h = 5 and b = 0.4, with a = a' = 1 and k = k' - oo throughout. The broken lines indicate host-generalist interactions, the solid lines indicate locally stable host-generalist- specialist interactions and the dotted lines indicate locally unstable ones. For further explanation see text.

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions M. P. HASSELLAND R. M. MAY 931 individuallyor in combination with other favourable changes in parameterscharacterizing the interactions, there will eventually come a point when the specialist can establish itself, and a persistentthree-species state arises. In the simplest case (illustratedby curve A in Fig. 6), the host-generalist system gives way directly to a three-species state, in such a way that the system always possesses one and only one state (which may be a stable equilibrium point, a stable cycle, or 'deterministicallychaotic' fluctuations around some long-term average). In this simplest case, the criterion for the existence of the three-species state-that is, for the specialist to become established-can be written explicitly: we requireF > Fc, where

Fc = [1 + (ah/k)(1 - e-1a)]k (11) This makes plain the trends asserted in the preceding paragraph. A specialist will more easily be able to invade an existing host-generalist interaction if k and ah are small (indicating low levels of highly non-random predation by the generalists) and if a(=a'cb) is large (reflecting a high efficiency of the specialists (high a'c) and/or low density- dependencefrom the generalists (high b)). More generally, the regime of host-generalist only (for sufficiently low F) and three-species coexistence (for sufficiently high F) can be separated by a band of intermediateF-values in which two alternativestates exist. Curve B in Fig. 6 illustratesone of the possibilities, in which there are two alternative persistent states, one with only a generalistpredator and the other with all three species present. Curve C in Fig. 6 indicates another possibility, in which there is a band of F-values for which there are two alternative three-species states into which the system may settle. In all such situations, which state the system settles into will be determinedby the initial conditions. Notice, moreover,that either one or both of these alternativestates may be cyclic or chaotic, rather than a simple stable point equilibrium. In these more complicated situations where two alternativestates exist between the pure host-generalist state and a unique three-species state, analytic criteria for the specialist to be able to establish itself are not in general obtainable. Equation (11) continues, however, to give a good approximation;for more details, see Appendix 2. Although the possible existence of alternativestates of the system clearly do arise-and substantially complicate both the analysis and this qualitative discussion-such complexities only arise for rather delicately balanced values of the parameters.Specifically, the existence of these phenomena (as illustrated by curves B and C in Fig. 6) is only possible for ah significantlyin excess of unity, c = a'cb around unity (not too big, not too small), and both k and k' large (corresponding to effectively random search by both specialist and generalist).In short, the criterionF > Fc with Fc given by eqn (11) is a good basis for understandingthe conditions under which coexistence of specialist and generalist predatorsis possible.

Model 2 With the specialists following the generalistsin the host's life cycle, we have Xt+ l = FXt g(Xt) f (Yt) (12a) Yt+= Xtg(Xt) [1 - f(Yt)] (12b) with X, Y,f and g as defined for eqns (8a,b). Analysis of the equilibrium and stability properties of Model 2 is made easier by their

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions 932 Generalist and specialist natural enemies being identical to those for Model 1 with cr/F substituted for a in eqn (11), and the host equilibriumpopulation, N*2), defined as

N2) = N()/g = Ff(N()) (13) (see Appendix 3 for furtherdetails). Once again there are the same range of equilibrium states as for Model 1, with the possibility of two alternative states (host-generalist and three-species) arising in a transition region between the pure host-generalist state (at low F) and the unique three-species state (at high F). As before, such complications arise only for rather special combinations of the parametersof the model. In the basic case where there is always a unique state, the criterionfor the specialist to be able to invade and establish a three-species system is F > F', with FCgiven implicitlyby the relation F' = [1 + (ah/k)(1 - exp (-F,c/a))] k. (14) This criterion,eqn (14), is to be compared with eqn (11) for model 1; it bears out the remark made above, that results for model 2 follow by replacingcrwith c/F in those for model 1. As for model 1, eqn (14) shows that a specialist can invade more easily if ah and k are small and a is large. In addition, however, the appearance of F' in the exponent on the right-hand side of eqn (14) means that F' > Fc (see Appendix 3 for proof). Hence the conditions for a specialist to invade if acting after a generalist in the host's life cycle will always be more restrictive than if acting before. This is only to be expected since the specialists now have fewer hosts available compared with Model 1.

COMPARISON OF MODELS 1 AND 2

In comparing the predictionsof Models 1 and 2, we pick upon two special cases, where k = k' - oo and k = k' = 1, and for each we display the three-speciesequilibrium properties by plotting X* and Y* against ah for given a and F (cf. Fig. 5a-c). In other words, we shall examine how the equilibria, and their local stability, are affected by changing overall efficiency of the generalists, and do this for both random and moderately clumped distributions of mortality. Figure 7 shows two examples where k = k' -x oo, and Fig. 8 gives a further example where k = k' = 1. In each case, Models 1 and 2 are contrasted on the same figure, and the host equilibriawith only generalists present are also shown. In addition, Fig. 7c,d gives an example where there are two alternative states (one with host-generalist only, the other with all three species persistingtogether) for a narrow band of F or ah values. No such example can be found for k = k' = 1 (as in Fig. 8); for these relativelylow k and k' values there is always a unique state. The following general conclusions are illustratedby these examples. (1) A specialist can invade and coexist more easily if acting before the generalistsin the host's life cycle (Model 1) than afterwards (Model 2). This is reflected in Y* from Model 1 always lying above that from Model 2, and hence ah, > ah2 in Figs 7-8. (2) A three-species stable system can readily exist where the host-generalist interaction alone would be unstable or have no equilibriumat all. This can be true even if each of the two-species interactions on their own are unstable, as shown by the numerical example in Fig. 9 taken from curve A in Fig. 6, where the host-specialist interactionon its own would show expandingoscillations. (3) In some cases (e.g. curves A-C in Fig. 6 and Fig. 9) the establishmentof a specialist

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions ,

M. P. HASSELLAND R. M. MAY 933

(a) (b) 15 - 2-5

20 - ....Model.. 2.. I0

.... od I 0- - ModelI 0*5 Model I 0*5 Model 2 \ \ 0 0.5 0 0-5 1.5 (c) IS - Model 2 ( d )

3 I . . Model I 10 2

5 ModelModel 2 ...... Model I

IS 0 5 10 0 5 10 oh 0/7

FIG. 7. Examples of the dependence of the equilibria of the specialist parasitoid (Y*) and host - (X*) on the efficiency (ah) of the generalist for both Models 1 and 2 with k = k' oo. The solid, dotted and broken lines are as in Fig. 6. (a & b) ? = 1, F = 2. (c & d) ? = 3, F = 20. See text for furtherexplanation.

(a) (b) 1.5 3

Model 2 2

Model I ModelI

0 0.5 115I 0 0.5 0 1.5 2-0 oh

FIG. 8. As for Fig. 7a,b, but now k = k' = 1. leads to higher host populations than existed previously with only the generalist acting. Such a possibility could be relevant to classical biological control practices if specialist parasitoids were introduced to improve control where the pest is already regulated by generalist natural enemies. As with some other undesired consequences of introducing specialist parasitoids for biological control (e.g. unstable interactions),this affect can also be avoided by ensuring that the introduced species cause markedly non-random distributionsof parasitism (e.g. Beddington, Free & Lawton 1975; May & Hassell 1981), in which case (if k' < 1), addition of a specialist to a host-generalist can only further reduce the host equilibrium. (4) As discussed above, the equilibrium host population, X*, from Model 2

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions 934 Generalistand specialistnatural enemies

3-

2- Host

CL| I I/ lSpecialist - I I

Generalist /-____--p

120 40 60 80 A Generations FIG. 9. Numericalexample from Model 1 (eqns (8-10)) illustratinga stable three-species interactionwhere each of the two-speciesinteractions are locallyunstable. The generalist (drawn at 1/6th scale) is introducedat point A and the specialistis removedat B. Parametervalues: F= 10,= a' =1,k=k'- oo,h=20, c= 10.

(specialists following generalists) increases as the generalists become more efficient (i.e. as ah increases), provided that k' > 1. In other words, with distributionsof parasitism by the specialists that veer towards random, they can overcompensate for any changes in host abundance earlier in the life cycle, thus causing higher host equilibriaas predation by the generalist becomes more severe. The effect becomes less marked as k' decreases (i.e. specialists increasingly clumped), until at k' = 1 there is exact compensation for any level of host mortality inflicted by the generalist (as shown in Fig. 8b). Indeed, the host equilibriumis now completely unaffected by the generalist, and remains exactly the same as with only host and specialist interacting (see Appendix 4 for furtherdetails). With even higher degrees of contagion in the distribution of parasitism (k' < 1), the above effect is reversed and Models 1 and 2 become similar in so far as generalists and specialists now combine to depress furtherthe host equilibrium.

CONCLUSION

In contrast to two-species host-generalist or host-parasitoid interactions, each of which have rather straightforwarddynamics, the combined three-species system discussed in this paper presents a much wider range of dynamics, depending on the choice of parameter values and the stages of the host's life cycle attacked. Thus, while it is clearly importantto understand fully the properties of each type of interaction alone, this in itself is not sufficient for a complete picture of more complicated systems. Testing these theoretical ideas in the field will be challenging since manipulationexperiments provide much the best means of looking for the existence of alternative stable states and the conditions for invasion of a third species into an existing predator-prey or parasitoid-host interaction. In the first place, therefore, suitable laboratory systems are likely to be the most profitablein attemptingto demonstratethese patterns. Examining a variety of such multi-species systems, involving predators, pathogens,

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions M. P. HASSELLAND R. M. MAY 935 competitors and mutualists, should bring us closer to the goal of understanding how populationdynamics can affect community structure.

REFERENCES

Anderson, R. M. & May, R. M. (1978). Regulation and stability of host-parasite population interactions. 1. Regulatory processes. Journal of Animal Ecology, 47, 219-247. Arditi, R. (1983). A unified model of the functional response of predators and parasitoids. Journal of Animal Ecology, 52, 293-303. Bailey, V. A., Nicholson, A. J. & Williams, E. J. (1961). Interactions between hosts and parasites when some host individuals are more difficult to find than others. Journal of Theoretical Biology, 3, 1-18. Bauer, G. (1985). Population ecology of Pardia tripunctana Schiff. and roborana Den. and Schiff. (, ): an example of 'Equilibriumspecies'. Oecologia, 65,437-441. Beddington, J. R., Free, C. A. & Lawton, J. H. (1975). Dynamic complexity in predator-prey models framed in difference equations. Nature, 255, 58-60. Comins, H. N. & Hassell, M. P. (1976). Predation in multi-prey communities. Journal of Theoretical Biology, 62,93-114. Cramer, N. F. & May, R. M. (1972). Interspecific competition, predation and species diversity: a comment. Journal of Theoretical Biology, 34, 289-290. De Jong, G. (1979). The influence of the distribution of juveniles over patches of food on the dynamics of a population. Netherlands Journal of Zoology, 29, 33-51. Ekanayake, U. B. M. (1967). Parasitism offour species of Erannis. Unpublished D. Phil. thesis, University of Oxford. Fujii, K. (1977). Complexity-stability relationships of two-prey-one-predator species systems model: local and global stability. Journal of Theoretical Biology, 69, 613-623. Hassell, M. P. (1978). The Dynamics of Predator-Prey Systems. Princeton University Press, Princeton, New Jersey. Hassell, M. P. (1980). Foraging strategies, population models and biological control: a case study. Journal of Animal Ecology. 49, 603-628. Hassell, M. P. & Anderson, R. M. (1984). Host susceptibility as a component in host-parasitoid systems. Journal of Animal Ecology, 53,611-621. Hassell, M. P. & May, R. M. (1985). From individual behaviour to population dynamics. Behavioural Ecology (Ed. by R. Sibley & R. Smith), pp. 3-32, British Ecological Society Symposium No. 25, Blackwell Scientific Publications. Oxford. Hassell, M. P., Waage, J. K. & May, R. M. (1983). Variable parasitoid sex ratios and their effect on host parasitoid dynamics. Journal of Animal Ecology, 52, 889-904. Holling, C. S. (1959a). The components of predation as revealed by a study of small mammal predation of the European pine sawfly. Canadian Entomologist, 91, 293-320. Holling, C. S. (1959b). Some characteristics of simple types of predation and parasitism. Canadian Entomologist. 91, 385-398. Keymer, A. E. & Anderson, R. M. (1979). The dynamics of infection of Tribolium confusum by Hymenolepis diminuta: the influence of infective stage density and spatial distribution.Parasitology, 79, 195-207. Kowalski, R. (1976). Philonthus decorus (Gr.) (Coleoptera: Staphylinidae): its biology in relation to its action as a predator of winter moth pupae (Operophtera brumata) (Lepidoptera: Geometridae). Pediobiologia, 16, 233-242. May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261,459-467. May, R. M. (1977). Predators that switch. Nature, 269, 103-104. May, R. M. (1978). Host-parasitoid systems in patchy environments: a phenomenological model. Journal of Animal Ecology. 47, 833-843. May, R. M. & Hassell, M. P. (1981). The dynamics of multiparasitoid-host interactions. American Naturalist, 117, 234-261. May, R. M., Hassell, M. P., Anderson, R. M. & Tonkyn, D. W. (1981). Density dependence in host-parasitoid models. Journal of Animal Ecology, 50, 855-865. Mook, L. J. (1963). Birds and the spruce budworm. The Dynamics of Epidemic Spruce Budworm Populations (Ed. by R. F. Morris). pp. 268-271. Memoirs of the Entomological Society of Canada. No. 31. Murdoch, W. W. (1969). Switching in general predators: experiments on predator and stability of prey populations. Ecological Monographs, 39, 335-354. Murdoch,'W. W. (1977). Stabilizing effects of spatial heterogeneity in predator-prey systems. Theoretical Population Biology. 11, 252-273. Murdoch, W. W. & Oaten, A. (1975). Predation and population stability. Advances in Ecological Research, 9, 2-131.

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions 936 Generalist and specialist natural enemies Nicholson, A. J. (1933). The balance of animal populations. Journal of Animal Ecology, 2, 132-178. Nicholson, A. J. & Bailey, V. A. (1935). The balance of animal populations. Part. 1. Proceedings of the Royal Society of London, 551-598. Rogers, D. J. (1972). Random search and insect population models. Journal of Animal Ecology, 41, 369-383. Roughgarden, J. & Feldman, M. (1975). Species packing and predation pressure.Ecology, 56,489-492. Royama, T. (1970). Factors governing the hunting behaviour and selection of food by the great tit (Parus major L.). Journal of Animal Ecology, 39,619-668. Royama, T. (1971). Evolutionary significance of predator's response to local differences in prey density: a theoretical study. Proceedings of the Advanced Study Institute on Dynamics of Numbers in Population (Oosterbeek 1970), 344-357. Southwood, T. R. E. (1978). Ecological Methods with Particular Reference to the Study of Insect Populations. Chapman & Hall, London. Southwood, T. R. E. & Comins, H. N. (1976). A synoptic population model. Journal of Animal Ecology, 45, 949-965. Steele, J. H. (1974). The Structure of Marine Ecosystems. Harvard University Press, Cambridge, Massachusetts. Thompson, W. J. (1924). Le Theorie mathematique de l'action des parasites entomophages et le facteur du hassard. Annals of the Faculty of Science of Marseille, 2, 69-89. Varley, G. C. & Gradwell, G. R. (1960). Key factors in population studies. Journal of Animal Ecology, 29, 399-401. Waage, J. K., Hassell, M. P. & Godfray, H. C. J. (1985). The dynamics of pest-parasitoid-insecticide interactions. Journal of Applied Ecology, 22, 825-838. Wang, Y. H. & Gutierrez, A. P. (1980). An assessment of the use of stability analyses in population ecology. Journal of Animal Ecology, 49,435-452. Waters, W. E. (1959). A quantitative measure of aggregation in insects. Journal of Economic Entomology, 52, 1180-1184.

(Received 19 October1985)

APPENDIX 1

The dynamics of our host-generalist association is described by the first-orderdifference equation, eqn (4). In the limit < = 0 (Th = 0), the equilibriumvalue X* is easily found, and is given by eqn (6). Clearly the argumentin the logarithmin eqn (6) is positive only so long as the inequality (5) is satisfied. If eqn (5) is not satisfied, then F(1 + ah/k)-k> 1, (1.1) and eqn (4) will eventually describe density independentgrowth at this rate. In the general case when 6 * 0, an analytic solution for the non-trivial equilibrium solution of eqn (4) is not in general possible. It is clear, however, that a sufficiently large value of Xt will result in the handling time becoming so important that the functional response saturates, and the host population will run away exponentially (multiplyingitself by roughly F each generation). This threshold value, XT, is given as the largest solution of Xt+1 = Xt in eqn (4):

( ah/)[ 1- exp (-XT)] -k ( 1 + XT) For 0 << 1, as will usually be the case, we may neglect the term exp(-XT) compared to unity (because XT > 1). It follows that ah XT^T k(Fl/k - 1) . (1.2)

This threshold host density, above which the host population escapes control by the generalistpredator, is indeed much greaterthan X* when ( is very small.

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions M. P. HASSELLAND R. M. MAY 937 The stability of the equilibriumpoint at X* may be studied in the standard way. If the slope of the map described by eqn (4) at X* lies between 45? and -45?, then X* will be locally stable. If not, numerical studies are needed to find whether the overcompensating density dependenceresults in stable cycles or in deterministicchaos (May 1976).

APPENDIX 2

This appendix elucidates the dynamical properties of the three-species (host-generalist- specialist) system of model 1, as defined by eqns (8a,b) with eqns (9) and (10) for g andf, respectively. The equilibriumvalues of the host and specialist parasitoid densities, X and Y, can be obtained by putting Xt+ I = Xt = X* and Yt+1 = Yt= Y* in eqns (8a) and (8b) in the usual way. The system is, however, somewhat easier to understandif we work with the 'dummy variable', z = f(Y*); z represents the fraction of the host population that escape the specialist predator (1 > z > 0, with z = 1 when the specialist is absent). Given z, Y* follows from eqn (10), and the X* is determinedfrom eqn (8b): Y* = k'(-l/k' - 1), (2.1) X*= Y*/(1 - z). (2.2) The equation relating z to the parametersof the model is now eqn (8a), which gives 1 = Fzg[V(z)]. (2.3) Here y/(z) = X*z, which from eqns (2.1) and (2.2) resolves to rz = k'z(z-lk' - 1)/(1 - z). (2.4) In turn, g(V) is definedby eqn (9). Equation (2.3) now can be solved to find z for specified values of F and the other parameters involved in g[q(z)] (namely ah, a, k and k'). Taking logarithms in eqn (2.3), and writingg(q/) explicitly, we have z determinedfrom ln F = G(z). (2.5) Here the fraction G(z) is defined as G(z) = -ln z + k In [1 + (ah/k)(1 - exp [-V(z)/cra])]. (2.6) -- Figure A.1 shows the range of functional forms G(z) can take. As z 0, G(z) -+ oo, and at the other extreme of admissiblez-values, for z = 1, V(1) = 1 and G(1) = k In [1 + (ah/k) (1 - e-/a)]. (2.7) The G(z) curve between these extreme values at z -+ 0 and z = 1 may be monotone decreasing (as shown by curve A in Fig. A. 1), or it may have stationary points (as shown by curves B and C in Fig. A. 1). In the former curve, eqn (2.5) gives a unique solution for z (correspondingto a three-species state) if ln F > G(1). (2.8) That is, the three-species state exists if F > F?, with F, given (via eqns (2.8) and (2.7)) by eqn (11) of the main text. On the other hand, if G(z) has stationary points between z = 0 and z = 1 (as in the case for curves B and C in Fig. A. 1), then: (i) there will be a range of F-values for which eqn (2.5) has more than one solution; and (ii) solutions of eqn (2.5) may

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions 938 Generalist and specialist natural enemies 10 -

8-

6 - C

4

2 -

0 0-2 0-4 0-6 0-8 1I0 z FIG. A. 1. Examples of the range of functional forms of G(z) from appendix eqn (2.6). Curve A: a = 10. Curve B: ca= 0.2. Curve C: ca= 0.7. (ah = 7 throughout). be possible for F-values somewhat below the value Fc defined by eqn (2.8) or eqn (11). The alternative states in turn may arise in two distinguishablydifferent ways, as illustratedby curves B and C in Fig. A.1: if G(z) initially decreases as z decreases below z = 1 (as in curve C), then there is a band of F-values (just below Fc) for which eqn (2.5) has two solutions (a persisting three-species state and -an unstable 'watershed' state, with an alternative stable state of host-generalist only, corresponding to z = 1); if G(z) initially increases as z decreases below z = 1, yet has stationary points (as in curve B) then there may be two alternativethree-species states separated by an unstable 'watershed' state, or the alternative attractors may be a three-species state and a host-generalist state (dependingon whether G(z) at its lower stationary point lies above or below Fc). These various possibilities can be distinguishedby focusing on the slope of G(z), which we write as G'(z) = dG(z)/dz. If G'(1) > 0, the behaviour exemplified by curve C of Fig. A.1 must ensue. But a routine calculation gives k' - I (ah/c)e- '/ G'(1) =-I + 1 (2.9) -2k' + (ah/k) (1-e-1/.)] Thus, G'(1) will be positive, giving a narrow band of F-values for which there are two alternative states of the kind characterized by curve C, only if k' > 1, ah is large (in conjunction with k large), and crhas values around unity (too large or too small c-values lead to the second term in eqn (2.9) tending to be small). If G'(1) < 0, we must further ask whether G(z) remains negative (in which case we have curve A in Fig. A. 1, and the criterionF > Fc of eqn (11) is the exact condition for specialist and generalist to coexist), or whether G'(z) can increase to become positive for lower z-values (as typified by curve B). It can be seen that if G'(1) < 0, G'(z) will remainnegative provided the second derivative obeys G"(1) > 0. In the critical case where G'(1) - 0, the requirementG"(1) > 0 is satisfied if 3(k' - 1) [1 + (ah/k)] > 2(k' - 2) [1 + (ah/k)(l - e-/a)] (2.10)

For k and k' both large, eqn (2.10) reduces to c > 3/2.

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions M. P. HASSELLAND R. M. MAY 939 In short, the monotonically decreasing G(z) exemplifiedby curve A in Fig. A. 1 will be realized if G'(1) of eqn (2.9) obeys G'(1) < 0, and also eqn (2.10) is satisfied. In this event, there is always a unique state of the system, and eqn (11) gives an exact criterion for the critical F-value above which the specialist can invade. More complicated situations will arise if G'(1) > 0, and can arise if G'(1) < 0 but eqn (2.10) is not obeyed. All this can be pieced together to get a good analytic understandingof the numericalexamples presented in Figs 6-8 of the main text. Once the equilibriumstates X* and Y* have been obtained along the above lines, their local stability properties can be determined by routine computations (see, e.g. Hassell & May 1973).

APPENDIX 3

We first show that the analysis of the dynamical properties of model 2 are as for model 1, but with ain model 1 replaced by a/F and X* in model 1 replaced by X*/g. As in Appendix 2, we choose z = f (the fraction of hosts escaping parasitism by the specialists) as a 'dummy variable'. Again, Y* is given by eqn (2.1), and X* is given from eqn (12b) as X* Y*/[(1 - z)g]. (3.1) As in Appendix 2, the system of equations is now closed by eliminatingall variablesexcept z in eqn (12a), to get 1 = Fzg[O(z)]. (3.2) Here g(O) is given by eqn (9), and O(z)= X* can be written O(z)= Fk'z(z-1/k'- 1)/(1 - z). (3.3) The equation (3.2) can be rewritten(in analogy with eqn (2.5)) as ln F = H(z). (3.4) Now H(z) is defined as

H(z) = -In z + k In { 1 + (ahlk) (1 - exp [-F ,(z)/a])}. (3.5) Here V(z) is exactly as defined by eqn (2.4), and thereforethe only differencebetween H(z) (which determinesthe equilibriumz-values for model 2) and G(z) (for model 1) is that the parameterain eqn (2.6) for b(z) is replaced by a/F in eqn (3.5) for H(z). Hence, the dynamics of model 2 follow by repeating the analysis of Appendix 2, with a everywherereplaced by a/F (and X* replaced by X*/g). In particular,the critical F-value, Fc, that allows the specialist to coexist in model 2 in given (in the basic case when H(z) is monotone increasing)by eqn (14). We now prove that Fc > Fc, (3.6) as assertedin the main text following eqn (14). First note that 1 - exp (-F'/a) > 1 - exp (-1//a), (3.7) for all F' > 1. Thence, the right-handside of eqn (14), which defines Fc, must exceed the corresponding right-handside of eqn (11), which defines Fc. This establishes the result of eqn (3.6); namely, that it is harderfor a specialist to invade if it acts later ratherthan earlier in the life cycle of the host.

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions 940 Generalist and specialist natural enemies APPENDIX 4

This Appendix establishes the result given in section (4) of the Comparison of Models 1 and 2 (p. 922), namely that for specialists that act after generalists (model 2) and have k' = 1, the host equilibriumis completely unaffected by the presence of the generalist. Consider first the pure host-specialist system in the event where k' = 1. That is, the functional response of the specialist is f(Y,k'= 1)- 1/[1 + (1 + Y)]. (4.1) In the absence of the generalist, we put g = 1 in eqns (8a,b) or (12a,b), with eqn (4.1) for f( Y); this gives the equilibriumsolution for X*: X* = 1 + Y* =f (4.2) Now consider model 2, with specialists following generalists, in the case where k' = 1. Equations (12a,b) reduce to Xt+l= FXtg(Xt)/(1 + Yt), (4.3) Yt+i = Xtg(Xt)Yt/(1 .+ Y). (4.4) At equilibrium,we divide eqn (4.4) into eqn (4.3) to get again X* = F. The factor g(X*) simply does not affect this calculation in the special case k' = 1. This is the result we set out to prove. Notice that, of course, the presence of the generalist does make a difference to the equilibriumhost density when the parametervalues are such that the specialist is excluded (correspondingto F < F', with F' given by eqn (14)). Figure 8b illustratesthese points.

This content downloaded from 130.132.123.28 on Fri, 2 May 2014 15:47:21 PM All use subject to JSTOR Terms and Conditions