Quantitative Methods for Ecology and Evolutionary Biology (Cambridge, 2006) - Chapter 4 pot

Chapter 4

The evolutionary ecology of parasitoids

Insect parasitoids – those insects that deposit their eggs on or in the
eggs, larvae or adults of other insects and whose offspring use the
resources of those hosts to fuel development – provide a rich area of
study for theoretical and mathematical biology. They also provide a
broad collection of examples of how the tools developed in the previous chapters can be used (and they are some of my personally
favorite study species; the pictures shown in Figure 4.1 should help
you see why).
There is also a rich body of experimental and theoretical work on
parasitoids, some of which I will point you towards as we discuss
different questions. The excellent books by Godfray (1994), Hassell
(2000a), and Hochberg and Ives (2000) contain elaborations of some of
the material that we consider. These are well worth owning. Hassell
(2000b), which is available at JSTOR, should also be in everyone’s
library.
It is helpful to think about a dichotomous classification scheme for
parasitoids using population, behavioral, and physiological criteria
(Figure 4.2). First, parasitoids may have one generation (univoltine)
or more than one generation (multivoltine) per calendar year. Second,
females may lay one egg (solitary) or more than one egg (gregarious)
in hosts. Third, females may be born with essentially all of their eggs
(pro-ovigenic) or may mature eggs (synovigenic) throughout their
lives (Flanders 1950, Heimpel and Rosenheim 1998, Jervis et al.
2001). Each dichotomous choice leads to a different kind of life
history.
133

134

The evolutionary ecology of parasitoids

(a)

(b)

(c)

(d)

(e)

Figure 4.1. Some insect parasitoids and insects that have life histories that are similar to parasitoids. (a) Halticoptera
rosae, parasitoid of the rose hip fly Rhagoletis basiola, (b) Aphytis lingannensis, parasitoid of scale insects, and
(c) Leptopilinia heterotoma, parasitoid of Drosophila subobscura. (d, e) Tephritid (true) fruit flies have life styles that are
parasitoid-like: adults are free living, but lay their eggs in healthy fruit. The larvae use the resources of the fruit for
development, then drill a hole out of the fruit and burrow into the ground for pupation. Here I show a female rose
hip fly R. basiola (d) ovipositing, and two males of the walnut husk fly R. compleata (e) fighting for an oviposition site
(the successful male will then try to mate with females when they come to use the oviposition site). The black trail
under the skin of the walnut is the result of a larva crawling about and creating damage between the husk and the
shell as it uses the resource of the fruit.


The Nicholson–Bailey model and its generalizations

(a) Generations per year:
Multivoltine

(b) Eggs per host:

>1

1

Univoltine
1

Figure 4.2. A method of
classifying parasitoid life
histories according to
population, behavioral and
physiological criteria.

>1

Solitary
Gregarious

(c) Egg production after emergence:

(d) Combining the characteristics:
Multivoltine

=0

Univoltine

>0

Solitary
Gregarious

Pro-ovigenic
Synovigenic

Pro-ovigenic
Synovigenic

The Nicholson–Bailey model and its generalizations
The starting point for our (and most other) analysis of host–parasitoid
dynamics is the Nicholson–Bailey model (Nicholson 1933, Nicholson and
Bailey 1935) for a solitary univoltine parasitoid. We envision that hosts are
also univoltine, in a season of unit length, in which time is measured
discretely and in which H(t) and P(t) denote the host and parasitoid populations at the start of season t. Each host that survives to the end of the season
produces R hosts next year. The parasitoids search randomly for hosts, with
search parameter a, so that the probability that a single host escapes parasitism from a single parasitoid is eÀa. Thus, the probability that a host escapes
parasitism when there are P(t) parasitoids present at the start of the season is
eÀaP(t). These absolutely sensible assumptions lead to the dynamical system
Ht ỵ 1ị ẳ RHtịeaPtị
Pt ỵ 1ị ẳ Htị1 À eÀaPðtÞ Þ

(4:1)

Note that in this case the only regulation of the host population is by the
parasitoid. Hassell (2000a, Table 2.1) gives a list of 11 other sensible
assumptions that lead to different formulations of the dynamics.
The first question we might ask concerns the steady state of Eq. (4.1),
obtained by assuming that H(t ỵ 1) ẳ H(t) and P(t ỵ 1) ¼ P(t). These are

easy to find.

Exercise 4.1 (E)
Show that the steady states of Eqs. (4.1) are
" 1
P ẳ logRị
a

"
Hẳ

R
logRị
aR 1Þ

135

(4:2)


136

The evolutionary ecology of parasitoids

(a)

(b)

100

300

90

Hosts

Hosts

250

80

Parasitoids

Population size

60
50
40

Parasitoids

30
20

200
150
100
50

10
5

10

15
Generation

20

25

0

30

0

5

10

15
Generation

20

(c)
60

60

40
30
20
10
0

Hosts

50

50
Population size

0

40

Parasitoids

30
20
10

0

10

20

30

40

50

0

60

0

5

Week

10
Week

15

(d)
500
Hosts

400

Population size

0

Hosts

Population size

70

300

200

Parasitoids

100

0
0

20

40

60

80
Week

100

120

140

160

20

25

30


The Nicholson–Bailey model and its generalizations

which shows that R > 1 is required for a steady state (as it must be) and that
higher values of the search effectiveness reduce both host and parasitoid steady
state values.

The sad fact, however, is that this perfectly sensible model gives
perfectly nonsensical predictions when the equations are iterated forward
(Figure 4.3): regardless of parameters, the model predicts increasingly
wild oscillations of population size until either the parasitoid becomes
extinct, after which the host population is not regulated, or both host and
parasitoid become extinct. To be sure, this sometimes happens in nature,
usually this is not the situation. Instead, hosts and parasitoids coexist with
either relative stable cycles or a stable equilibrium.
In a situation such as this one, one can either give up on the theory or
try to fix it. My grade 7 PE teacher, Coach Melvin Edwards, taught us
that ‘‘quitters never win and winners never quit,’’ so we are not going to

give up on the theory, but we are going to fix it. The plan is this: for the
rest of this section, we shall explore the origins of the problem. In the
next section, we shall fix it.
As a warm-up, let us consider a discrete-time dynamical system of
the form
N t ỵ 1ị ¼ f ðN ðtÞÞ

137

N
f (N )
45

ο

N

N

Figure 4.4. The stability of the
steady state of the one
dimensional dynamical system
N(t ỵ 1) ẳ f(N(t)) is determined
by the derivative of f(N)
"
evaluated at the steady state N.

(4:3)

where f (N) is assumed to be shaped as in Figure 4.4, so that there is a

"
"
"
steady state N defined by the condition N ẳ f N ị. To study the stability
" ỵ ntị where n(t), the perturbaof this steady state, we write N tị ẳ N
tion from the steady state, is assumed to start off small, so that
"
jnð0Þj ( N . We then evaluate the dynamics of n(t) from Eq. (4.3) by
Taylor expansion of the right hand side keeping only the linear term

df 
 ntị
"
"
"
N ỵ nt ỵ 1ị ẳ f N ỵ ntịị % f N ị þ
dN N
"

(4:4)

Figure 4.3. Although the Nicholson–Bailey model seems to be built on quite sensible assumptions, its predictions are
that host and parasitoid population sizes will oscillate wildly until either the parasitoids become extinct (panel a,
H(1) ¼ 25, P(1) ¼ 8, R ¼ 2 and a ¼ 0.06) and the host population then grows without bound, or the hosts become
extinct (panel b, H(1) ¼ 25, P(1) ¼ 8, R ¼ 1.8 and a ¼ 0.06), after which the parasitoids must become extinct. (c) Some
host–parasitoid systems exhibit this kind of behavior. On the left hand side, I show the population dynamics
of the bruchid beetle Callosobruchus chinesis in the absence of a parasitoid (note that this really cannot match
the assumptions of the Nicholson–Bailey model, because there is regulation of the population in the absence
of the parasitoid); on the right hand side, I show the beetle and its parasitoid Anisopteromalus calandre. In this case, the
cycles are indeed very short. (d) On the other hand, many host–parasitoid systems do not exhibit wild oscillations and

extinction. Here I show the dynamics of laboratory populations of Drosophila subobscura and its parasitoid Asobara
tabida. The data for panels (c) and (d) are compliments of Dr. Michael Bonsall, University of Oxford. Also see Bonsall
and Hastings (2004).


138

The evolutionary ecology of parasitoids

"
"
Since N ẳ f N ị and setting f N ¼ df =dN jN we conclude that n(t)
"
approximately satisfies
nt ỵ 1ị ẳ f N ntị

(4:5)

and we conclude that the steady state will be stable, in the sense that
"
perturbations from it decay, if j f N ðN Þj51.

Exercise 4.2 (E)
For more practice determining when a steady state is stable, do the computation
for the discrete Ricker map

!
NðtÞ
N ðt ỵ 1ị ẳ Ntịexp r 1
K

and show that the condition is |1 À r| < 1, or 0 < r < 2.

But we have a two dimensional dynamical system. Since what
follows is going to be a lot of work, we will do the analysis for the
more general host–parasitoid dynamics. Basically, we do for the steady
state of a two dimensional discrete dynamical system the same kind of
analysis that we did for the two dimensional system of ordinary differential equations in Chapter 2. Because the procedure is similar, I will
move along slightly faster (that is, skip a few more steps) than we did in
Chapter 2. Our starting point is
Ht ỵ 1ị ẳ RHtị f Htị; Ptịị
Pt ỵ 1ị ẳ Htị1 f ðHðtÞ; PðtÞÞÞ

(4:6)

" "
which we assume has a steady state H; Pị. We now assume that
" ỵ htị and Ptị ẳ P ỵ ptị, substitute back into Eq. (4.6),
"
Htị ẳ H
Taylor expand keeping only linear terms and use o(h(t), p(t)) to represent terms that are higher order in h(t), p(t), or their product to obtain
"
"
" "
H ỵ ht ỵ 1ị ẳ RH ỵ htịịẵ f H; Pị ỵ f H htị ỵ f P ptị ỵ ohtị; ptịị
"
"
" "
P ỵ pt ỵ 1ị ẳ H ỵ htịịẵ1 f H; Pị f H htị f P ptị ỵ ohtị; ptịị
(4:7)

where f H ẳ q=qHịf H; PịjH;Pị and fP is defined analogously.
" "
Now, from the definition of the steady states we know that
"
" " "
" "
H ¼ RHf ðH; Pị, which also means that Rf H; Pị ẳ 1, and that
" ẳ H1 f H; Pịị. We now use these last observations concerning
"
" "
P
the steady state as we multiply through, collect terms, and simplify
to obtain


The NicholsonBailey model and its generalizations

"
"
ht ỵ 1ị ẳ htị1 þ RHf H Þ þ RHf P pðtÞ þ oðhðtÞ; ptịị
"
"
pt ỵ 1ị ẳ htị1 1=Rị Hf H ị Hf P ptị ỵ ohtị; ptịị

(4:8)

Unless you are really smart (probably too smart to find this book of any
use to you), these equations should not be immediately obvious. On the
other hand, you should be able to derive them from Eqs. (4.7), with the
intermediate clues about properties of the steady states in about 3–4

lines of analysis for each line in Eqs. (4.8). If we ignore all but the linear
terms in Eqs. (4.8) we have the linear system
ht ỵ 1ị ẳ ahtị ỵ bptị
pt ỵ 1ị ẳ chtị ỵ dptị

(4:9)

with the coefficients a, b, c, and d suitably defined; as before, we can
show that this is the same as the single equation
ht ỵ 2ị ẳ a ỵ dịht ỵ 1ị ỵ bc adịhtị

(4:10)

by writing h(t ỵ 2) ẳ ah(t þ 1) þ bp(t þ 1), p(t þ 1) ¼ ch(t) ỵ dp(t) ẳ
ch(t) ỵ (d/b)(h(t ỵ 1) ah(t)) and simplifying. (Once again you should
not necessarily see how to do this in your head, but writing it out should
make things obvious quickly.) If we now assume that h(t) $ lt (there is
actually a constant in front of the right hand side, as in Chapter 2, but
also as before it cancels), we obtain a quadratic equation for l:
l2 À ða ỵ dịl ỵ ad bc ẳ 0

(4:11)

which I am going to write as l2 À
l ỵ
ẳ 0 with the obvious identification of the coefficients. Also as before, Eq. (4.11) will have two
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
roots, which we ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p will denote by l1 ¼ ð

=2ị ỵ
À 4
=2Þ and
2 À 4
=2Þ. The steady state will be stable if perturl2 ¼ ð
=2Þ À ð

bations from the steady state become smaller in time, this requires that
|l1,2| < 1. We will now find conditions on the coefficients that makes
this true. The analysis which we do follows Edelstein-Keshet (1988),
who attributes it to May (1974). We will do the analysis for the case in
which the eigenvalues are real (i.e. for which
2 ! 4
); this is our first
condition. Figure 4.5 will be helpful in this analysis. The parabola
l2 À
l ỵ
has a minimum at
/2, and because we require
À1 < l2 <
/2 < l1 < 1 we know that one condition for stability is that
|
/2| < 1, so that |
| < 2. The parabola is symmetric around the minimum. Now, if the roots lie between –1 and 1, the distance between the
minimum and either root, which I have called D1, must be smaller than
the distance between the minimum and –1 or 1, depending upon

139

140

Figure 4.5. The construction
needed to determined when
the solutions of the equation
l2 À
l ỵ
ẳ 0 have absolute
values less than 1, so that the
linearized system in Eq. (4.9)
has a stable steady state.

The evolutionary ecology of parasitoids

D2
D1
y = λ – βλ + γ

β
2
–1

λ_

0

λ+

1

whichever is closer. Thus, for example, for the situation in Figure 4.5 we
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
must have 1 À j
=2j >
2 À 4
=2; if we square both sides of this
expression the condition becomes 1 À |
| ỵ (
2/4) > (
2/4)
and this
simplifies to 1 ỵ
> |