IMA Journal of Mathematics Applied in Medicine & Biology (1984) 1,123-133
Parasitism in Patchy Environments: Inverse Density Dependence
can be Stabilizing
MICHAEL P. HASSELL
Department of Pure and Applied Biology, Imperial College, Silwood Park,
Ascot, Berks SL5 7PY
There are now many examples in the literature where the spatial distribution of
per cent parasitism by insect parasitoids is either directly or inversely dependent
on host density per patch. While it is well known that direct density dependent
relationships can contribute markedly to the stability of a host-parasitoid interaction, inverse relationships have been more-or-less ignored. Using difference
equation models, the dynamics of host-parasitoid interactions are described
where parasitism per patch varies across the range from direct to inversely density
dependent. These models demonstrate for a variety of host distributions that
inverse relationships can also strongly promote stability.
1. Introduction
of spatial heterogeneity in the population dynamics of preypredator and host-parasitoid interactions has been repeatedly stressed (e.g.
Hassell & May, 1973, 1974; Maynard Smith, 1974; Hilborn, 1975; Beddington,
Free, & Lawton, 1978; Hassell, 1978, 1980). In particular, it is now well
established that direct density dependent relationships from patch to patch, of the
kind shown in Fig. l(a, b), can be a powerful stabilizing influence on interactions.
However, there are at least as many examples of the opposite, inverse density
dependent relationships where percent predation or parasitism is greatest in the
lowest density patches, as shown in Fig. l(c, d) and discussed by Morrison &
Strong (1980,1981). In a recent survey of studies giving the spatial distribution of
parasitism by insect parasitoids, Lessells (1984) lists 15 examples of density
dependent parasitism (as in Fig. l(a, b)), 17 samples of inverse relationships (as in
Fig. l(c, d)) and a further 13 showing no relationship. To focus only on the
dynamic effects of density dependent parasitism is thus to ignore an alternative
pattern that is widespread under natural conditions. The conditions under which
such very different responses may arise are discussed by Hassell (1982) and
Lessells (1984).
THE IMPORTANCE
This paper aims to correct the balance by examining the dynamic effects of
parasitism across the range of density dependent relationships shown in Fig. 1.
2. The basic model
Let us commence with a familiar general difference equation model for a
host-parasitoid interaction (e.g. Hassell, 1978):
Nt+1 = FNJ(NOP,)
(la)
123
© Oxford University Press 1984
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[Received 20 February 1984]
124
MICHAEL P. HASSELL
(a)
(b)
100
75
40
50
20
25
0
(c)
100
2
4
6
(d)
> •
50
75
$
40
50
4
m
30
20
25
<35 « • «
• mm» • • • •
£&*•
•
n
6
200
400
600
800
Host density per patch
FIG. 1. Four examples showing either direct (a & b) or inverse (c & d) density dependent parasitism
per patch in the field, (a) Parasitism by Aspidiotiphagus citrinus (Craw.) (Eulophidae) attacking the scale
insect, Fiorina externa Ferris on the lower crown of 30 hemlock trees. Data described by: Y =
20-45 + 0-06X (p<0-001) (from McClure, 1977). (b) Parasitism by Exenterus abruptorius (Thunb.)
(Ichneumonidae) attacking cocooned larvae of the sawfly, Neodiprion sertifer (Geoff.) from different
15 x 30 m plots. Curve fitted by eye (after Sharov, 1979). (c) Parasitism by eulophid and trichogrammatid spp. attacking Cephaloleia consanguinea Baley eggs. Data described by Y = 50-4~3-3X
(p = 0-l) (from Morrison & Strong, 1981). (d) Parasitism by Ooencyrtus kuwanai (Howard)
(Encyrtidae) attacking gypsy moth (Lymantria dispar (L.)) eggs in different sized masses. Data
described by: Y = 229-5+ 33-56 In X ( p < 0 0 0 1 ) (from Brown & Cameron, 1979).
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60
INVERSE DENSITY DEPENDENT PARASITISM
125
P, +1 = dV,[l-/(N t ,P,)].
(lb)
and
Here N,, N, +1 and Pt, P1+1 are the numbers of hosts and parasitoids, respectively,
in successive generations t and t + 1 , F is the finite rate of increase of the host, c is
the average number of surviving female parasitoid progeny per host attacked
(henceforth c = 1) and the function / gives the probability of a host escaping
parasitism. In the manner of Hassell & May (1973) the environment is now
divided into n patches so that the function / for host survival becomes
°y' )1
(2)
where a{ and ft are the fractions of the total hosts and parasitoids, respectively, in
the ith patch, a is the per capita searching efficiency and Th is the handling time
as a proportion of the total time (henceforth, for simplicity, Th = 0). The
parasitoids thus distribute themselves only once in each generation in relation to
the host distribution prevailing at that time. The precise stability conditions for
this model are derived in Hassell & May (1973) culminating in the condition
F t [a,(aftP*) exp ( - a f t P * ) ] < ^
(3)
where P* is the equilibrium population of parasitoids.
Rather than compute (3) for arbitrarily chosen (cO and (ft), it is more
convenient in the first place to focus on particular forms of host and parasitoid
distribution that permit analytical solutions. For a very simple host distribution,
and again following Hassell & May (1973), let us assume that there is a single
'high density' patch containing a fraction a hosts, with the remaining n - 1
patches all containing the same fraction (1 - a)/(n — 1). For the parasitoid distribution, we assume
ft = x«r
(4)
where x is a normalization constant such that the ft values sum to unity, and pt is
the parasitoid 'aggregation index'. Two examples where equation (4) has "been
used to describe actual distributions of parasitoids are shown in Fig. 2(a, b). But
instead of considering /J, only in the range 0 to °° as done in the past, we now let it
take the full range, including negative values, as shown in Fig. 3(a). The resulting
patterns of density dependence are shown in Fig. 3(b), and (since Th = 0) reflect
closely the distribution of parasitoids. Positive and negative values of /x in
equation (4) will thus be used as a means of generating direct and inverse density
dependent parasitism.
3. Equilibrium and stability properties
The equilibrium host and parasitoid populations, N* and P*, for equations
(la, b) with / defined in equation (2) and Th = 0 are obtained by setting Nt+1 =
N, = N*, giving:
F[ae-^p* + (1 - a)e" gaeP *] = 1
(5a)
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/(Nt,Pt)=lLexp(-
126
MICHAEL P. HASSELL
(a)
0 6r
0-4
00
0.0
0-2
0-4
0-6
'o
00
Hosts per patch
FIG. 2. The distribution of searching adult parasitoids in relation to host density per patch, both
expressed as proportions of the total number (i.e. ft and ajs respectively in equation (4)). (a) 32 female
Trichogramma pretiosum Riley searching for eggs of Plodia interpunctella (Hubn.); ft = 1-05CK*04. (b)
32 female Venturia canescens (Grav.) searching for larvae of Ephestia cautella (Walk.): ft =0-77af' 81 .
(Further details in Hassell, 1982).
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0-2
127
INVERSE DENSITY DEPENDENT PARASITISM
0-4 n
00
(b)
0-2
Hosts per patch
FIG. 3. (a) The relationship between the proportion of adult parasitoids, 0, and hosts, Oj, from
equation (4) with different values of the aggregation index, ji, in the range 2 to - 2 . (b) The resulting
patterns of percent parasitism per patch obtained using the term in square brackets in equation (2)
with P, = 10, a = 0-5 and T h = 0.
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-0-5
128
MICHAEL P. HASSELL
100
go
60
20
e
°2
-1
3
o
X
(b)
15
100 r
11
9
80
60
40
20
-
2
-
1
0
1
Aggregation index
FIG. 4. Examples showing how the host equilibrium, JV*, varies with the aggregation index, ji,
calculated from equations (5a, b). (a) Different values for the fraction of hosts in the high density patch
a, as shown and a = 0-5, n = 10, and F = 2. (b) Different numbers of host patches, n, as shown and
a = 0-5. a = 0-4. and F = 2.
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40
INVERSE DENSITY DEPENDENT PARASITISM
129
N* = FP*I(F-1)
(5b)
and
1
F-l
4
(6)
(b)
(a)
3
2
1
0
-1
-2
S "3
1-4
Host rate of increase
FIG. 5. Stability boundaries between the aggregation index, \L, and the host rate of increase, F, using
condition (6). The hatched areas shown the regions of local stability for positive and negative values of yu
for four different values of a (n = 11 throughout) (a) a = 0-3, (b) a = 0-4, (c) a = 0-5, and (d) a = 0-6.
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where g = [(1 - a)/(n - I)]"*** and /3 = [1 + g(n -1)] . Some examples of how the
host equilibrium depends on the distribution of parasitism, <x, and the host
distribution, a and n, are shown in Figs 4(a, b). In general, the host equilibria are
lowest when /x » 1 , and tend to increase as /x gets either greater or smaller. Using
the terminology of MacArthur & Levins (1964) and MacArthur (1968), n = 1
(i.e. ft = oti) corresponds to a perfectly 'fine grained' situation, that becomes
progressively more 'coarse grained' as fi increases or decreases (Soberon 1982).
In short, the host equilibrium, N*, becomes progressively lower as the distribution
of parasitism corresponds more closely to the distribution of hosts. Note that this
model has no limitation on host population size other than parasitism. Thus,
including a density dependent host rate of increase could, depending on the level
of the resulting carrying capacity, reduce the steep increases in N* shown in Fig. 4.
The local stability criterion for the above model is
130
MICHAEL P. HASSELL
4. A more realistic host distribution
The host distributions in the above examples were chosen primarily for analytical convenience. Much more realistic would be to use a statistical distribution such
as the negative binomial, which is known- to describe the spatial distribution of a
wide range of populations (e.g. Southwood, 1978). Equation (2) (with Th = 0) now
becomes
f(Nt, Pt) = ^ t \.P(J)i exp (-oPtf))]
(7)
•W j=o
Here N is the mean host density, p(/) is the probability of having j prey in a patch
(in this case generated by the negative binomial), and P(j) are the number of
parasitoids in a patch with j prey (denned again by equation (4)). The negative
binomial is characterized by two parameters alone: the mean (N = NJn) and an
index of the degree of clumping (fc). The host distribution becomes more clumped
as fc decreases, and at the limits k —* °°, k = 1 andfc= 0, a Poisson, a geometric
and a log series are obtained, respectively.
While an analytical treatment of this model is possible given some constraints
(e.g. k —* 0) (R. M. May, personal communication), numerical simulation has been
used here to determine the stability boundaries in terms of values of /x (positive
and negative) over a range of host distributions from very clumped to random.
Figure 6(a) shows the results over the range k = 0-25 (highly clumped) to k = 100
(c.random) using particular values for the remaining parameters, a, F and n. For
comparison, Fig. 6(b) shows a comparable figure, obtained from the stability
condition in equation (6), representing a slice through Figs 5(a, d) where F = 2.
The asymmetry in the two figures about the line /x = 0 arises primarily from the
non-linear relationship between the distribution of adult parasitoids, generated by
/x, and the resulting pattern of parasitism (cf. Figs 3(a, b)). The most important
conclusion from the two figures is simply that, irrespective of fine details of the
host distribution, inverse as well as direct density dependent parasitism can create
locally stable host populations varying widely in their degree of clumping.
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By obtaining apP* from equation (5a), Hassell & May (1973) displayed the
stability boundaries between p. and F for a variety of host distributions, but only
for ju, > 0. They thus only considered the effects of direct density dependence. To
show the effects of inverse density dependent parasitism, Fig. 5 displays the
stability boundaries for four different host distributions where n takes the range
- 4 to 4.
The results emphasize that inverse density dependence, where parasitism is
greatest in the 'low density' patches, can be just as powerful a mechanism for
stability as the opposite direct relationships. Which contributes more so depends
on the host distribution. In these examples, the areas of stable space arising from
positive and negative /x-values are identical when there are the same proportion
of hosts in the 'high density' patch as the 'low density' ones (a = 0-5; Fig. 5(c)). As
a decreases inverse density dependent parasitism exerts the greater effect, and
vice versa as a increases.
131
INVERSE DENSITY DEPENDENT PARASITISM
Stable
Unstable
-2
Stable
Unstable
-3
(a)
-4
01
0-2 0-3 0.4 0-6 0-8
3 4
K
6 8
20
30 40 60
3
2
Stable
1
Unstable
0
Unstable
-1
Stable
-2
-3
(b)
-4
0-9
0-8
0-7
06
05
0-4
Hosts per patch
03
02
FIG. 6. Local stability boundaries between the aggregation index, IA and the degree of host clumping,
(a) Obtained by numerical simulation of equation (7) substituted in equations (la, b), with a = 1, F = 2,
n = 30, and p(j) generated from the negative binomial distribution. The degree of host clumping is
thus given by the parameter k, and the stability boundary is shown for k in the range 0-25 to 100. (b)
Obtained from equation (6) with F = 2. Host clumping is thus given by the fraction, a, of hosts in the
high density patch.
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-1
132
MICHAEL P. HASSELL
Discussion
REFERENCES
J. R., FREE, C. A., & LAWTON, J. H. 1978 Characteristics of successful
natural enemies in models of biological control of insect pests. Nature, London 273,
513-519.
BROWN, M. W. & CAMERON, E. A. 1979 Effects of disparlure and egg mass size on
parasitism by the gypsy moth egg parasite, Ooencyrtus kuwanai. Environmental
Entomology 8, 77-80.
GRIFFITHS, K. J. 1969 Development and diapause in Pleolophus basizonus (Hymenoptera:
Ichneumonidae). Canadian Entomologist 101, 907-914.
HASSELL, M. P. 1969 A study of the mortality factors acting upon Cyzeniz albicans (Fall.),
a tachinid parasite of the winter moth (Operophtera brumata (L.)). Journal of Animal
Ecology 38, 329-340.
HASSELL, M. P. 1978 The Dynamics of Arthropod Predator-Prey Systems. Princeton, New
Jersey: Princeton University Press.
HASSELL, M. P. 1980 Foraging strategies, population models and biological control: a case
study. Journal of Animal Ecology 49, 603-628.
HASSELL, M. P. 1982 Patterns of parasitism by insect parasitoids in patchy environments.
Ecological Entomology 7, 365-377.
HASSELL, M. P. & ANDERSON, R. M. 1984 Host susceptibility as a component in
host-parasitoid systems. Journal of Animal Ecology 53, (in press).
HASSELL, M. P. & MAY, R. M. 1973 Stability in insect host-parasite models. Journal of
Animal Ecology 42, 693-726.
HASSELL, M. P. & MAY, R. M. 1974 Aggregation in predators and insect parasites and its
effect on stability. Journal of Animal Ecology 43, 567-594.
HTLBORN, R. 1975 The effect of spatial heterogeneity on the persistence of predator-prey
interactions. Theoretical Population Biology 8, 346-355.
LESSELLS, C. M. 1984 Parasitoid foraging: should parasitism be density dependent?
Journal of Animal Ecology 53, (in press).
MACARTHUR, R. 1968 The theory of the niche. In Population Biology and Evolution
(Lewontin, R. C , Ed.) Syracuse, New York. Syracuse University Press.
BEDDINGTON,
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The underlying explanation for this dynamic effect of inverse parasitism from
patch to patch is common to the effects of most forms of heterogeneity. Stability
depends upon the extent that the host population is unevenly exploited, and thus
arises from a partial refuge effect in which some hosts are more at risk than
others. This differential exploitation of the hosts may arise in several ways; for
example, by parasitoids concentrating their searching effort in some patches,
leaving others more neglected (as in this study), by temporal asynchrony between
host and parasitoid populations (Griffiths, 1969; Hassell, 1969), or by host
individuals varying in their defences against parasitism (Hassell & Anderson,
1984). Despite their different provenance, all of these mechanisms have the same
effect of generating clumped rather than random probability distributions of a
host being attacked (May, 1978). With heterogeneity coming from so many
sources, it is hard to escape the conclusion that it is a widespread mechanism
contributing significantly to the stability of natural host-parasitoid interactions.
I am very grateful to Kate Lessells and Robert May for their helpful comments
on the manuscript. This work was supported by a grant from the Natural
Environment Research Council.
INVERSE DENSITY DEPENDENT PARASITISM
133
R. & LEVINS, R. 1964 Competition, habitat selection and character displacement in a patchy environment. Proceedings of the National Academy of Sciences
51, 1207-1210.
MAYNARD SMITH, J. 1974 Models in Ecology. Cambridge: Cambridge University Press.
MCCLURE,
M. S. 1977 Parasitism of the scale insect, Fiorinia externa
(Homoptera: Diaspididae), by Aspidiotiphagus citrinus (Hymenoptera: Eulophidae) in
a hemlock forest: density dependence. Environmental Entomology 6, 551-555.
MORRISON, G. & STRONG, P. R., JR. 1980 Spatial variations in host density and the
intensity of parasitism: some empirical examples. Environmental Entomology 9, 149152.
MORRISON, G. & STRONG, P. R., JR. 1981 Spatial variations in egg density and the
intensity of parasitism in a neotropical chrysomelid (Cephaloleia consanguinea).
Ecological Entomology 6, 55-61.
SHAROV, A. A. 1979 Effect of spatial structure of the interacting populations of Neodiprion sertifer and its parasite Exenterus abruptorius on the dynamics of their numbers.
Zoologicheskii Zhumal 58, 356-365.
SOBERON, J. M. 1982 Insect Population Dynamics in Heterogeneous Environments. Unpublished Ph.D. thesis, University of London.
SOUTHWOOD, T. R. E. 1978 Ecological Methods, with particular reference to the study of
insect populations (2nd Edition). London: Chapman & Hall.
MACARTHUR,
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