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for example, adult parasitoids lay their eggs in the pupae of hosts, but not in the eggs
of their hosts or with the larvae or adults.
Denote, respectively, H(t) and P(t) as the host and parasitoid densities in time
period t, and F(H(t), P(t)) as the fraction of hosts that is not parasitized. Then
H
t
H
(
t
+
1
)=λ
∗
H
(
t
)
∗
F
(
)
,
P
(
t
))
(13.2)
P
(
t
+
1
)=
C
∗
H
(
t
)
∗
[
1
−
F
(
H
(
t
)
,
P
(
t
))]
(13.3)
where
is the host growth rate and C is the parasitoid fecundity.
Let us assume that the fraction of hosts that become parasitized depends on
the density-dependent rate of encounter of parasitoids and hosts. Encounters oc-
cur randomly, allowing us to invoke the law of mass action that we used extensively
throughout this topic to model the spread of disease from contagious to susceptible
populations. Accordingly, the number of encounters of hosts HE with parasitoids is
λ(
H
(
t
))
HE
(
t
)=
A
∗
H
(
t
)
∗
P
(
t
)
(13.4)
where A is the searching efficiency of the parasitoids.
Unlike the models of the spread of a disease from an infected to a nonimmune
population, subsequent encounters of individuals in the two populations do not al-
ter the rate at which parasitoids are propagated. Therefore, we need to modify the
law of mass action to account for the fact that only the first encounter of hosts and
parasitoids is significant in propagating the parasitoid. Once a host carries the par-
asitoid's eggs, subsequent encounters with parasitoids will not change the number
of parasitoid progeny that hatch from the host. We need only to distinguish be-
tween hosts that had no encounter and hosts that had at least one encounter with
parasitoids.
The Poisson distribution describes the occurrence of such discrete, random events
as encounters of hosts and parasitoids. We can make use of the Poisson probability
distribution to calculate the probability that there is no attack of parasitoids on a host
within a certain time period. In general, therefore
EXP
HE
(
t
)
H
X
HE
(
t
)
−
H
(
t
)
(
t
)
P
(
X
)=
(13.5)
X
!
is the probability of X attacks. This probability depends on the average number of
attacks in the given time interval, HE/H. From equation (13.4) we know
HE
(
t
)
/
P
(
t
)=
A
∗
P
(
t
)
(13.6)
Thus, for zero attacks by the parasitoids, equation (13.5) yields
