Host-Parasitoid Systems in Patchy Environments: A Phenomenological Model
Author(s): Robert M. May
Source:
Journal of Animal Ecology,
Vol. 47, No. 3 (Oct., 1978), pp. 833-844
Published by: British Ecological Society
Stable URL: http://www.jstor.org/stable/3674 .
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Journal of Animal
Ecology (1978), 47,
833-843
HOST-PARASITOID
SYSTEMS
IN PATCHY
ENVIRONMENTS:
A PHENOMENOLOGICAL
MODEL
BY
ROBERT M. MAY
Biology
Department, Princeton
University, Princeton,
N.J. 08540, U.S.A.
SUMMARY
(1) A host-parasitoid
model is presented
which is intermediate
in complexity between
the
Nicholson-Bailey model (in which the
parasitoids search
independently randomly
in
a
homogenous environment)
and complicated
models
for incorporating environmental
patchiness (in
which the overall distribution
of
parasitoid attacks
is derived from detailed
assumptions about
their searching behaviour
and about
the spatial distribution
of the
hosts). The model
assumes the overall
distribution
of
parasitoid
attacks per host to
be of
negative binomal
form.
There
are consequently three biological
parameters: two
are the
usual
parasitoid
'area
of
discovery', a,
and
the host 'rate
of
increase', F;
the third is the
negative
binomial
clumping parameter,
k. Such intermediate-level
models have
proved
useful
in
sorting
out ideas
in
the
related disciplines
of
epidemiology
and
parasitology.
(2) Empirical
and
theoretical
arguments
for
using the
negative binomial
to give a
phenomenological
description
of
the
essential consequences
of
spatial patchiness
in
models are surveyed.
(3)
A
biological
interpretation
of
the
parameter k in host parasitoid
models is offered.
If
the
parasitoids
be distributed
among
patches according to
some arbitrary distribution
which has a coefficient
of
variation CVp,
and if the parasitoid
attack distribution
within a
patch
be
Poisson,
then the
ensuing
compound distribution
can be approximated
by a
negative binomial
which
will
have the same
variance as the exact
distribution provided
k
is
identified
as
k
=
(I
I/CVp)2.
(4) Expressions
are obtained
for the
equilibrium
values
of
host and
parasitoid
popula-
tions. These
equilibria
are stable
if,
and
only if,
k
<
1;
that
is, provided
there
is
sufficient
clumping.
(5)
The
dynamical
effects
of
parasitoid
aggregation
in
some respects
mimic those
introduced
by
mutual interference
among parasitoids; the
appropriate coefficient
of
'psuedo-interference'
is
calculated.
INTRODUCTION
Spatial
heterogeneities
in
the
distribution
of
prey
and
predator populations
are of
central
importance
in
stabilizing prey-predator
interactions. On the
empirical
side,
the
stabilizing
effects
of a
patchy
environment
have
been
exhibited
in
quantitative
detail
in
the
experiments
of
Huffaker
(1958)
and later
workers
(see
Luckinbill
1973,
and references
therein).
Other
examples may
be
argued
to arise
in
many
natural
prey-predator
systems,
and
in
some
biological
control
situations, although
here the
quantitative proof
is
lacking
(see e.g.
the reviews
by
Krebs
1972,
ch. 17 and Huffaker
1971).
On
the
theoretical
side,
0021-8790/78/1000-0833$02.00
)1978
Blackwell Scientific
Publications
833
834
Host-parasitoid
systems
several
recent studies
(Maynard Smith
1974; Hilborn
1975;
Hastings
1977; Zeigler
1977;
Gurney
& Nisbet
1978;
and references contained
in
these
works)
have
considered
a
hypothetical
environment
made up of
many
discrete
patches,
and
shown how
the dyna-
mic
interplay among
empty
patches,
patches
containing only
prey,
and
patches
contain-
ing both
prey and
predators
could under
certain
circumstances
lead
to
overall
stability
of
the system. A
more
general class of
prey-predator
models
that
incorporate
spatial
heterogeneity are
reviewed by
Levin
(1976). One basic mechanism
is
common to
essen-
tially all these
laboratory and theoretical
studies:
if the
prey
are patchily
distributed,
and
if
the
predators tend to
aggregate
in
regions
of
relatively high
prey
density,
then the
regions
of low
prey density
constitute a kind of
implicit
refuge,
whereby the
prey
population
is
maintained;
conversely,
the
predator
population
flourishes
in
the
regions
of
relatively
high prey
density. Too
much
implicit
refuge
for
the
prey tends to
lead
to
'runaway',
with
the
prey
population
ultimately controlled
by
factors other than
predation.
Too little
implicit
refuge tends
to
produce the
diverging
population
oscillations
that
characterize
simple
and
spatially
homogeneous
prey-predator
models.
Arthropod
host-parasitoid
systems
constitute an
important
subclass of
prey-predator
interactions. In
these
systems,
each host
in
any
one
generation
either
is
parasitized,
thus
producing
a
next-generation
parasitoid, or
escapes parasitism
to give
rise to F
progeny
that
become the
next
generation
of
hosts. As
discussed
more
fully
by
Hassell
(1978),
host-parasitoid
systems
possess two
features
that
make them
especially amenable to
detailed
study.
First,
because the
life
cycles
of host
and parasitoid are
so
intimately
intermeshed, the usual
complications
of
the
predator's 'numerical
response'
(sensui Solo-
mon
1949;
Holling
1959)
are
absent; one
only has
to
deal with the
predator's
'functional
response',
which
here is
expressed by
the
probability for a
host to
escape
parasitism.
Secondly,
the small size
and
relatively
short
generation
time of
arthropod hosts and
parasitoids
permits
laboratory
studies
of
a kind
that
are
simply
not
feasible
for
vertebrate
predators
and their
prey.
(Unlike the
first, the
second
advantage is
shared
with other
arthropod
prey-predator
systems.)
In
the
classic
Nicholson-Bailey
(1935) model,
the
parasitoids
search
independently
randomly
in
a
homogeneous
environment, with
the
consequence
that both
host and
parasitoid
populations exhibit
diverging
oscillations.
Recently, several
authors
(Hassell &
May 1974;
Murdoch & Oaten
1975,
and
references
therein)
have
analysed models
in
which
the hosts
are
distributed
non-uniformly
among
many
patches, and where
the
parasitoids have
searching
behaviour that
leads to their
aggregation
in
patches with
relatively
high host
density;
these model
systems
may be stable or
unstable,
depending
on
the
values
of
the relevant
biological parameters.
Such
studies
have the merit of
being
relatively
realistic, with
the
overall
host-parasitoid
dynamics
being
explicitly
related to the
behavioural
mechanisms
whereby the
parasitoids
aggregate. The studies suffer the con-
comitant
disadvantage
of
containing
many
parameters,
which
makes
lucid
exposition
difficult and
extension to
multispecies
situations
(such
as
those
with
several
parasitoid
species,
or
hyperparasitism)
intractable.
This
paper presents
a model
of intermediate
complexity.
All
the
spatial
and
behavioural
complications
that lead
to
patterns
of
parasitoid
aggregation
are subsumed
in
the
single
phenomenological
assumption
that
the
net
distribution
of
parasitoid attacks
upon hosts
is of
negative
binomial
form.
That
is,
all
these
complications are
summarized
in
the
single
parameter
k,
which characterizes the
degree
of
clumping
or
over-dispersion
in a
negative
binomial
distribution. The
model thus has
three
parameters:
the
conventional a
and
F
(respectively
representing the
'area of
discovery' for
a single
parasitoid,
and the
number of
R. M. MAY
835
surviving progeny produced
by
an
unparasitized
host), and the new
k (summarizing the
effects of spatial heterogeneity
and consequent
parasitoid aggregation).
Clearly the model
bridges the gap between
the overly simple
two-parameter Nicholson-Bailey
model, and
the proliferation
of
parameters
that
are
required
in
relatively realistic
models.
The paper is divided
into two main parts.
First, some comments
are made about the
negative binomial.
A simple, yet general, biological
interpretation of k
is given in terms of
the
variance in the distribution
of
parasitoids
among patches.
Some
data in
support
of a
negative binomial distribution
is marshalled,
and a survey is made of
the theoretical and
empirical
evidence
that
underpins
the
use
of
similar
phenomenological
models
in
broadly
analogous contexts
in epidemiology and parasitology.
Secondly, the
host-parasitoid
model itself
is
presented,
and its
equilibrium
and
stability properties
are laid
bare.
Following
the
suggestion
of
Free, Beddington
&
Lawton
(1977),
it is
shown
that the
dynamical consequences
of
overdispersion
in
the distribution
of
parasitoid
attacks
are in
some
respects
indistinguishable
from those
produced by
mutual
interference
among
parasitoids, and
the coefficient
of
'pseudo-interference'
is
calculated.
In
conclusion,
some
potential applications
of
the model are discussed. Throughout
the
main
text, attention is
focussed
on the
biology;
mathematical details
and
proofs
are
segregated
in
appendices.
WHY THIS MODEL?
The
negative
binomial
distribution
In
the
Nicholson-Bailey model,
P
parasitoids
search
independently,
and in
a
random
fashion,
each
discovering
hosts
at a rate
given
by
the 'area
of
discovery',
a. The
probabili-
ties
for
a
given
host to
be discovered
0, 1, 2,
3
.
.
.
times
are therefore
given
by
the terms
in
a
Poisson
distribution,
with the
mean
discovery
rate
being
aP. In
particular,
the
probability
for
a host
to
escape
parasitism
is
given by
the zero
term
in the
Poisson
series, namely
exp (-aP).
We
now
replace
this
random
distribution
of
parasitoid
attacks
with
an
overdispersed
distribution,
described
by
a
negative
binomial
with
clumping parameter
k. The
other
biological assumptions
remain
as in the
Nicholson-Bailey model,
so that
the mean attack
rate is
aP. The
probability
of
escaping parasitism
is
now
the zero
term
in this
negative
binomial, namely (1
+aP/k)k.
A more full
account
of the
properties
of the
negative
binomial
distributions
are
given
from a mathematical
standpoint by
Anscombe
(1950),
and
from
a
biological standpoint
by
Southwood
(1966,
pp. 24-35).
The
mathematical
meaning
of the
parameter
k
may
be
appreciated by
noting that,
for a
negative
binomial
with
mean
m,
the
coefficient
of
variation
(CV)
is
CV2=variance
I
I
(mean)2
m
k
In
the limit
k-+oo,
the
random
or
Poisson
distribution is
recovered,
with the variance
equal
to
the
mean. As
k
becomes
smaller,
the
CV
gets larger,
with
the effect
becoming very
pronounced
for
very
small
k.
The
geometric
series
corresponds
to
k
=
1,
and
the
log
series
to the
limit
k-+O.
A
biological interpretation
of
k
within this model
In
a
host-parasitoid
context,
a
biological
meaning may
be
attached
to
k
by considering
the
following
model.
836
Host-parasitoid
systems
Suppose
the parasitoids
are distributed
among a large
number of
patches according
to
some specified
(but, as yet,
arbitrary)
distribution; the
number of parasitoids
in any
one
patch
has mean P and
variance c2p.
Within any patch,
the parasitoids
search indepen-
dently randomly,
in
Nicholson-Bailey
fashion, each
having an area
of discovery
a; the
attack distribution
within
a patch is
Poisson. The
overall distribution
of parasitoid
attacks in
any one patch
is thus given
by compounding
the specified
among-patch
distribution
of
parasitoids
with the Poisson
within-patch
distribution
of attacks.
This
overall, compound
distribution
will have
some well
determined form,
dependent
on the
specified
among-patch
distribution.
The compound distribution
will
not,
in general, be
negative binomial.
But it is necessar-
ily overdispersed,
and may
be approximated
by a negative
binomial
distribution
with the
same
mean
and same
variance
as the
exact distribution.
If this is
done,
the approximating
negative binomial
will
have a mean equal
to aP, and
a clumping parameter
k given
by
k
=
P2/U2
(2a)
This
result
is proved in
Appendix
1.
Note that k
depends only
on parameters
of the
among-patch
distribution
of
parasitoids.
By introducing
CVp
to denote
the coefficient
of
variation
of
the among-patch
distribution
of parasitoids,
we can rewrite
eqn (2a)
as
k
=
(
1/CVp)2.
(2b)
If the
among-patch
distribution
is
a
Pearson
type
III
(i.e. gamma)
distribution, then
the
compound
distribution
is
exactly
a
negative
binomial
(see, e.g.
Anscombe
1950, pp.
360-361).
Regardless
of whether the overall
negative
binomial is exact
or an
approxima-
tion, eqn (2)
interprets
k in a
way
that is direct and
biologically
meaningful.
Data
on
parasitoid
attack
distributions
Griffiths
&
Holling (1969)
have carried out an extensive series
of
experiments
in
which
ichneumon
Pleolophus
basizonus (Grav.)
parasitoids
interacted
with
sawfly
Neodiprion
sertifer (Geoff.)
hosts
in
1-22
x
2-44
m(4
x
8
ft) cages.
The distribution
of
attacks
per host
is
well described
by
a
negative
binomial,
as
is the
number
of
eggs
laid
in
hosts,
with k
around
0 8.
Surveying
fourteen
other laboratory
host-parasitoid
systems, Griffiths
and
Holling
find
three
have
negative
binomial attack
patterns;
the other
eleven
manifest the
'avoidance'
behaviour discussed
immediately
below.
In
their
laboratory
studies
of
hyper-
parisitism,
Kfir,
Podoler &
Rosen
(1976)
show
that one of
their
two
hyperparasitoid
species,
Cheiloneurus
paralia
(Walk.),
has an
attack
distribution
that
conforms
to
a
negative
binomial
with
k in
the
range
0.5-1.3.
Their other
species,
Marietta
exitiosa,
shows 'avoidance'
(again,
see
below).
Table
1
summarizes some field data from
Hassell
(unpublished),
showing
that two
measures
of
the distribution of attacks
of
Cyzenis
albicans
upon
its winter moth hosts
are
in
significant
agreement
with
a
negative
binomial
distribution.
In
laboratory studies,
it
is, however,
usually
found that
attacks are
underdispersed;
that
is,
are distributed
more
evenly
than
random
(see,
e.g.
the
experiments
and reviews
by
Rogers (1975)
and
by
Benson
(1973)).
This
is not
surprising.
The
laboratory cages
constitute
relatively
small and
homogeneous
single
patches,
and
many parasitoids
have
behavioural
mechanisms
whereby they
avoid
attacking
hosts
that are
already parasitized,
leading
to
underdispersion
within
a
single patch.
Even under
these
circumstances,
which
are
not
representative
of the
patchy
environments
found in the
field,
Rogers (1975,
p. 632)