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Unformatted text preview: Host—ParasntOId Interactions Arthropod predator—prey and host~parasitoid interactions are ex
ceptionally well studied for several reasons. These are systems
that can often be studied on a relatively small spatial scale. These
also are interactions that can be reasonably thought ,of as rep
resenting tightly coupled systems of two species. Finally, these
are systems of great economic importance, because often the hosts are signiﬁcant pests of crops, and the parasitoids may act
as control agents. the host that leads to a tight coupling between hosts and para
sitoids; often a parasitoid must lay its eggs in only one species
of host, to which it is adapted. The typical dynamics of the host— parasitoid interaction are similar to predator—prey dynamics, as
illustrated in Figure 9.1. In this brief chapter, We examine some of the most striking features of host—parasitoid dynamics. Because the interaction is 182 This is the same Nicholson who performed
the classic experiment with blowﬂies that
we examined earlier. Discrete time is used because hosts often
have one or a small ﬁxed number of generations per year, and parasitoids must time their generations to match their hosts. 9. Host—Parasitoid Interactions 1 000
800 60.0  4:.
C)
O Papulation density 200 Generations FIGURE 9.1. Dynamics of the host parasitoid interaction between a wasp, Heterospilus pros— pidis, and its host, the bean weevil, Callosobruchus chinensis (data are from Utida 1957).
—o— Parasitoid; 0— Host so similar to the predator—prey interaction, an extensive treatment
is not needed, and only some of the differences between the
two interactions are stressed. It is important to emphasize that
the host—parasitoid interaction is not rare — parasitoids represent
about 8.5% of described insect species, and estimates of the true
fraction of insects that are parasitoids run as high as 25% (Godfray, 1994). 9.1 Nicholson—Bailey model Once again, our goal is to determine what allows host and par—'
asitoid to persist. The starting point for our discussion is a basic
model introduced to ecology in a classic paper by Nicholson and
Bailey (1935). Many models of host—parasitoid interactions take
this model and its modiﬁcations as their starting point. i We begin our analysis by looking at a general model for a host—
parasitoid system in discrete time. Let N, be the number of hosts
at time t and P, be the number of parasitoids at time t. We assume
that, the host population grows exponentially (geOmetrically) in 9.1 NicholsOn—Bailey model 1 83
M the absence of the parasitoid at a per capita rate A per year. Let
f (M,P,) be the probability that a host egg laid in year t, with
host pOpulation size N, and parasitoid population size Pt, escapes
being parasitized. The number of hosts next year is then given by the product of the per capita growth rate of the population, the number of hosts this year, and the probability of escaping
parasitism. The model basically assumes that we count given by the product of the number of parasitoids emerging from “the eggstage'”theh°5‘5?°‘herw'5“ . 0 would appear in the equation for the
each parasrtlzed host, the number of hosts this year, and the prob parasitoids. ability of a host being parasitized. The model we begin with is thus If the probability of not being parasitized is f, what is the probability of being
[VH1 == AN‘fUVh Pt) (9,1) parasitized? PH] = CM“. _f(1V:,Pt)]. distribution, which is f .___ e—aP, , (9.3) where a is a parameter (with units of 1 over parasitoid num
bers) that measures the efﬁciency of the parasitoid. With this assumption our model becomes the classical Nicholson—Bailey
model 184 First ﬁnd P from the equation for NH and then find N from the equation for PM . is
there another, trivial equilibrium? Part of the reason that the stability analysis
is more difﬁcult here is that the condition
that a matrix have both eigenvalues less
than one (in magnitude) is more complex
than the condition that both eigenvalues
have a negative real part. A second reason
for the complexity of this analysis is simply
the presence of exponentials in the
NicholsonBailey model. 9. Host—Parasitoid Interactions 10000
_ 1000
100
{’3
8
E 10
:3
C
:9 1
3
3 0.1
(U
o.
0.01 10 100 1000 104 Host numbers FIGURE 9.2. Dynamics ofthe Nicholson—Bailey model with A = 11.61 = 0.001,and c = 3. Note the loglog scale, and that when the simulation was stopped the parasitoid population
size had dropped approximately seven orders of magnitude. The dynamicsare clearly unstable. We ﬁrst ﬁnd the equilibria of the model by setting N = N, = NH;
and P = P, = PM. We obtain the equilibrium _ ' A In A
N ‘ m (96)
A
P = 131—. (9.7)
a We can immediately see that if this equilibrium is to make bio
logical sense the growth rate of the host in the absence of the
predator, A, must be greater than 1. Also, as the efﬁciency of the
parasitoid goes up — as or increases — the equilibrium level of both
Species goes down. But what is the biological implication of the
equilibrium of this model? Is the equilibrium stable? We could then proceed with a stability analysis of this model.
However, the algebra, is, shall we say, formidable. Thus, rather
than go through the steps in the analytical techniques in this chap
ter, we instead just present the results of stability analyses, and
illustrate the behavior of the models with numerical solutions. The results of the stability analysis (which we have omitted)
of the Nicholson—Bailey model are unequivocal: the nontrivial 4._._.._._. __. .. ... 9.2 Simple stabilizing features 185
EN ls it really enough to present results for one
. . . . . . . ?
illustrated in Figure 9.2, not only 18 the equ111br1um unstable but semfpa'ame‘e'va'm‘ We are now left with a quandary. Experimental host—parasitoid Before reading on, think of stabilizing
systems, and natural systems such as the interaction between fea‘"’°‘°fa “°“”p‘"a"‘°'d mm” that could be included.
red scale (Aon idiella dammit) and its parasitoid Apbytis meli nus, show that longterm coexistence of host and parasitoid is possible. Thus, important stabilizing biological features must be
missing from our simple model. 9.2 Simple stabilizing features There have been numerous suggestions of potential stabilizing
mechanisms for host—parasitoid systems. We begin by listing a few of the mechanisms that might stabilize the host—parasitoid
interaction. 0 Density dependence in the host species (Beddington et al.,
1975). o Interference among parasitoids (Beddington et al., 1978).
If female parasitoids avoid each other, or avoid laying eggs where others have laid eggs, then the rate of parasitism rises more slowly with the number of parasitoids than assumed
in the Nicholson—Bailey model. Can you think of other ways to include
density dependence? 1.86 9. Host—ParaSEtoid Interactions 1 0000 1 000 Parasitoid numbers 100 1 00 1000
Host numbers FIGURE 9.3. Dynamics ofthe Nicholson—Bailey model modiﬁed to include density dependence, with A = 1.1, a = 0.001, c: = 3,,and K = 1000. Here host and parasitoid coexist in a
stable cycle. NM = lMeKI—Nr/KFaPJ (98)
10,.1 = cN,[1 — (“3]. (9.9) The stability of this model can again be determined analytically,
but we do not do so here. We note that with this modiﬁcation
there is a wide range of reasonable parameters for which the
model allows the host and parasitoid to persist, as shown in Fig
ures 9.3 and 9.4. Depending on the parameters, the persistence
can either be cyclic (Figure 9.3) or approach a stable equilibrium
(Figure 9.4). Thus, the model can lead to the cyclic coexistence
illustrated for the laboratory population in Figure 9.1. From the ﬁgures one can see that the parasitoid still does have
an effect on the density of the host — the equilibrium (or the av—
era ge population level for the cyclic persistence) is below the
population level the host would reach in the absence of the From your experience with predatorer parasitoid.
What might 5°” destabilizing Similar numerical analyses show that the other two stabilizing
features we have listed also can produce stable equilibria. 9.3 What stabilizing features operate in nature? 1000
92
(D
.0
E
2
E 100
.9.
'5
e
(U
0. 10 .
100 , 1000 Host numbers FIGURE 9.4. Dynamics ofthe Nicholson—Bailey model modiﬁed to include density dependence. with A = 1.1, a = 0.001.c = 3, andK == 750. Here host and parasitoid coexist at a
stable equilibrium. 9.3 What stabilizing features Operate in nature? Models can be used to determine potential stabilizing inﬂuences One of the best longterm studies attempting to understand
what allows host and parasitoid to persist at a stable equilibrium
has been that of red scale (Aom'diella aurantz’z‘) and its parasitoid
Apbytis melz'nus, conducted by Murdoch and his colleagues (sum—
marized in Murdoch, 1994). A large number of potential stabilizing
mechanisms have been proposed, including many beyond what
we have discussed here, but careful experimental examination of
all these mechanisms has failed to show that any single one of them is in fact responsible for producing the apparently stable
equilibrium observed in the ﬁeld. 188 9. Host—Parasitoid Interactions
W regulating, or regulated by the parasitoid? Justify your an swer, and discuss how you would answer this question for
a ﬁeld system. 2. Host—parasitoid systems are used to look at biological con
trol, where a parasitoid is introduced to reduce the host
population of a pest on a crop. How would the models
developed here help to choose a parasitoid to use for bi ological control, and what qualities should be looked for?
What are some potential pitfalls?  a 3. By numerically solving the model with density dependence,
equations (9.8) and (9.9), for the values of a, c, and A used in the chapter, but varying K, determine the effect of density
dependence on stability. ' Suggestions for further reading A summary Of host—parasitoid models is contained in Hassell’s
1978 book The Dynamics of A n‘bropod Predator—Prey Systems. A recent, extremely comprehenswe revrew of parasrtord biology IS Applications of Nonlinear Dtﬁerence Equations to Population Biology 79 Host Infected host Adult female Parasitoid Figure 3.2 Schematic representation of a ‘ host. Infected hosts die, giving rise to parasitoid
hostparasitoid system. The adult female parasitoid progeny. Uninfected hosts may develop into adults
deposits eggs on or in either larvae or pupae of the and give rise to the next generation of hosts. PM = number of hosts parasitized in previous generation X fecundity of
parasitoids (c). Noting that 1  f is the fraction of hosts that are parasitized, we obtain NM =~i AN.f(N., R). (140)
PM = cN,[1 — f(N., 11)]. (14b) TheSe equations outline a general framework for host—parasitoid models. To
proceed further it is necessary to specify the term f(N,, P.) and how it depends on the
two. populations. In the next section, we examine one particular form suggested. by
Nicholson and Bailey (1935). ...
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This note was uploaded on 02/09/2011 for the course EEP 115 taught by Professor Waynem.getz during the Fall '10 term at University of California, Berkeley.
 Fall '10
 WayneM.Getz

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