Generalist and Specialist Natural Enemies in Insect Predator-Prey Interactions
Author(s): M. P. Hassell and R. M. May
Source: Journal of Animal Ecology, Vol. 55, No. 3 (Oct., 1986), pp. 923-940
Published by: British Ecological Society
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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 & AppliedBiology, 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 naturalenemy alter the prey's populationdynamics? (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
higherprey 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-speciesstates.
INTRODUCTION
The dynamics of discrete, insect host-parasitoid interactionshave 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 insects, 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 generalisthost(=prey) interaction, and then use this as a basis for a three-species model in which
Prof.M. P. Hassell,ImperialCollege,SilwoodPark,Ascot,BerksSL5 7PY.
Correspondence:
923
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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 widespreadand 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 populationsin this paper may settle
neatly at deterministicequilibria,natural population patterns are, of course, the result of
random and deterministicprocesses 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 relationto 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 temperateLepidopterapopulations, 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 preludeto examining such a three-speciessystem, we first outline
the dynamics of the generalist-host interaction alone against which the dynamics of the
three-speciesinteractioncan 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+
k(1 + aThNt)
.
(1)
Here Nt is the number of prey in generation t, Na is the number of these attacked by Gt
searchinggeneralistpredators,a is the per capita searchingefficiencyof the predators,This
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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 distributiondeterminingthe 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 equivalentexpressionwhere only healthy
prey can be 'handled' is a recurrencerelationshipthat 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 individualsis 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 numberof predatorsand 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 moth 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 expressionexplored 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
predatorshave a fast numerical response in relation to changes in Nt, as would occur for
instance by
(a) rapid reproductionrelativeto 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
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926
Generalist and specialist natural enemies
10.
30
(a )
8
0
0
0
(b)
25
0
20
6
0~~~~~~
15
4
10
2
Z;)
c
(1)
70
0
0
-o
(1)
a-
0
5,
200
400
600
800 1000 1200
125
0
1000
500
30-
1500
0
(c
100
0~~~~~
75
50
20-
10
0
25
0
0
0
0
50
100
150
0
100
200
300
Prey density
PeromyscusmaniculatusHoy &
Kennicott and (b) Sorex cinereus Kerr (both as numbers per acre) in relation to the density of
FIG. 1. Numerical responses of four generalist predators. (a)
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 thirdinstarlarvaeof the sprucebudworm(Choristoneurafumiferana
(Clem.))(numbersper
10 ft2 of foliage).(Data fromMook 1963.) (d) Philonthusdecorus(Gr.) (pitfalltrap index)in
brumata(L.)) larvaeper m2 (afterKowalski(1976)). In
relationto wintermoth (Operophtera
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.
(b)
(a)
h=6
0C:
16
0(L)
a-
h=2
0
50
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).
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927
M. P. H\SSELL AND R. M. MAY
(a
100
100
60
80
60
60
,,,' 4
c:
0
0a)
0
100
b)
200
300
40
0
0.05
(c)
n IC
01i
0-15
(d)
100
DO0
8
80
30 -
E
60
4
40
20
0
01i
0-2
0.3
0.4
0-5
Prey density
1
2
3
4
FIG. 3. Examplesof density-dependent
pupalmortalityfrom generationto generationfor four
speciesof soil-pupating
Lepidoptera,ascribedprimarilyto predationby carabidandstaphylinid
beetle. Curves fitted by transformingthe originalpublishedrelationshipsbetween k-values
(Varley& Gradwell1960) and log populationdensity (N). (a) Mortalityof wintermoth (0.
brumata)pupaem-2; k = 0.22 + 0.31 log N (after Hassell 1980). (b) Mortalityof Pardia
tripunctatanaSchiffpupae0.18 m-2; k = 1.47 + 1.1 log N (afterBauer 1985).The original
k-valueswere estimatedbetweenthe larval and subsequentegg stages. An arbitrarytenfold
rate (F) has thus been assumedhereto avoidnegativemortalities.Otherassumed
reproductive
valuesof F (if constant)will not alterthe generalshapeof the curve.Such density-dependent
predationwas supportedby independentfield manipulationexperiments.(c) Mortalityof
NotoceliaroboranaDen. andSchiff.pupae0-18 m-2; k = 0-.39+ 0.97 log N (afterBauer1985).
F = 10 assumedas in (c). (d) Mortalityof Erranisdefoliaria(L.) m-2; k = 0.46 + 0.21 log N
(afterEkanayake1967).
Our model for a host-generalist interactionincorporatingeqns (1) and (2) thus becomes
exp (-NTb)]] -k
h[
(3)
+aTN)k+1
aTnN)
k(
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 unlimitedhost population in the absence of the generalists
is assumed in order to make clearer the contribution of the generaliststo 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 equilibriumpropertiesof 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 handlingtime (curve D), only occur if
N
aFN
Nt+=FNt
1l
ah > k(Flk -_ 1)
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(5)
928
Generalist and specialist natural enemies
-10
(A)
(B)
(C)
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
furtherdetails see text.
whereupon
X* =-In {1 - [k/(ah)](Fl k- 1)}.
(6)
Otherwise, the host population is unregulated by the generalist (curve B), eventually
increasingat 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 saturationnumber 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 persistentbut non-steady states arise if
the generalistpredators cause sufficientlysevere 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 relationto 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
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M. P. HASSELLAND R. M. MAY
(a )
929
(b)
4-
2-
-
0
\
5
...'
10
oh
15
20
0
1
2
3
Log F
(c)
0.8-
0060-40
2
1
3
k
FIG.5. The dependenceof the host equilibrium,
X*, fromeqn (4) on (a) the predatorefficiency
term,ah, for F = 10 andk -, oo(leftcurve)and k = 1 (rightcurve),(b) the host rateof increase
F, for ah = 6 and k = 1 (leftcurve)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 determinedby 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)]
Yt+ = Xt1 =f (Y)]
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(8a)
(8b)
930
Generalistandspecialistnaturalenemies
where
G = { 1 + (ah/k)[ I - exp (-Xt/a)] }-k
(9)
f= (1 + Yt/k')-k'.
(10)
and
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 appliedto 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 parametersare 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, approachingthe 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 detaileddiscussion is given in Appendix 2.
The generalistpredatorcan exclude the specialist, making a persistentthree-speciesstate
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 significantclumpingin
the distribution of generalist attacks (k small). As F increases or ah decreases, either
4- ?
2 --
-
/
:
/c
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-generalistspecialist interactions and the dotted lines indicate locally unstable ones. For further explanation
see text.
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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-speciesstate 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 criterionfor the existence of the three-species state-that is, for the specialist to
become established-can be writtenexplicitly: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 densitydependencefrom 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, ratherthan 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 criteriafor 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
complexitiesonly arise for ratherdelicately 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)
Yt+= Xtg(Xt) [1 - f(Yt)]
(12a)
(12b)
with X, Y,f and g as definedfor eqns (8a,b).
Analysis of the equilibriumand stability properties of Model 2 is made easier by their
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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), definedas
N2)
= N()/g = Ff(N())
(13)
(see Appendix 3 for furtherdetails).
Once again there are the same range of equilibriumstates 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
combinationsof 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 replacingcrwithc/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 comparingthe 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-speciesequilibriumpropertiesby
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
distributionsof 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 generalistspresent 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 generalconclusions 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 reflectedin Y* from Model 1
always lying above that from Model 2, and hence ah, > ah2in Figs 7-8.
(2) A three-speciesstable 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 numericalexample 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
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933
M. P. HASSELLAND R. M. MAY
(a)
15 -
(b)
2-5
20 -
....
Model..2..
I0
.... od
I 0-
0*5
Model I
Model 2
0
ModelI
-
0*5
\
\
0
0.5
(c)
IS -
3
..
Model I
1.5
0-5
(d )
Model 2
I
10
2
5
Model
Model 22
10
5
0
Model I
..........
..
0
IS
10
5
0/7
oh
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.
(b)
(a)
3
1.5
Model 2
2
Model I
ModelI
0
0.5
115I
0
0.5
0
oh
1.5
2-0
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
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Generalistandspecialistnaturalenemies
934
3-
Host
2-
CL
| I/
lSpecialist-
I
/-____--p
20
1
A
I
I
Generalist
40
60
80
Generations
FIG. 9. Numericalexamplefrom Model 1 (eqns (8-10)) illustratinga stable three-species
interactionwhereeachof the two-speciesinteractionsarelocallyunstable.Thegeneralist(drawn
at 1/6th scale) is introducedat pointA 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 parasitismby 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 distributionof 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-speciessystem 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 manipulationexperimentsprovide 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,
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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.
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Comins, H. N. & Hassell, M. P. (1976). Predation in multi-prey communities. Journal of Theoretical Biology,
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(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 eventuallydescribe 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 sufficientlylarge
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)]
( 1 + XT)
-k
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.
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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-generalistspecialist) 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),
X*= Y*/(1 - z).
(2.1)
(2.2)
The equation relatingz 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
parametersinvolved 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 definedas
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 extremeof 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-speciesstate) 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
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Generalist and specialist natural enemies
938
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
alternativestates in turn may arise in two distinguishablydifferentways, 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
G'(1) =-I
+
k' - I
-2k'
(ah/c)e- '/
1 + (ah/k) (1-e-1/.)]
(2.9)
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 furtherask 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 derivativeobeys G"(1) > 0. In the critical case where G'(1) - 0, the
requirementG"(1) > 0 is satisfied if
>
[1 + (ah/k)]
3(k' - 1)
2(k' 2) [1 + (ah/k)(l - e-/a)]
For k and k' both large, eqn (2.10) reduces to c > 3/2.
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(2.10)
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 criterionfor 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 propertiesof model 2 are as for model 1,
but with ain model 1 replacedby a/F and X* in model 1 replacedby 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 definedas
H(z) = -In z + k In { 1 + (ahlk) (1 - exp [-F ,(z)/a])}.
(3.5)
Here V(z) is exactly as definedby eqn (2.4), and thereforethe only differencebetweenH(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 replacedby a/F in eqn (3.5) for H(z).
Hence, the dynamics of model 2 follow by repeatingthe analysis of Appendix 2, with a
everywherereplacedby 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
correspondingright-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.
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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
functionalresponse 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 differenceto 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.
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