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In-Depth Information
0
EXP
(
−
A
∗
P
(
t
)) (
A
∗
P
(
t
))
EXP
(
−
A
∗
P
(
t
))
∗
1
P
(
0
)=
=
=
EXP
(
−
A
∗
P
(
t
))
(13.7)
0!
1
Equations (13.2) and (13.3) can therefore be re-written as
H
(
t
+
1
)=
H
(
t
)
∗
λ
∗
EXP
(
−
A
∗
P
(
t
))
(13.8)
P
(
t
+
1
)=
C
∗
H
(
t
)
∗
[
1
−
EXP
(
−
A
∗
P
(
t
))]
(13.9)
Let us also assume that without parasitoids, the hosts will grow toward a carrying
capacity K set by the environment. To capture growth of the host population up to
a density H(t) = K and decline of the host population for H(t)
>
K, we replace in
equation (13.8) the growth rate
λ
(H(t)) with
1
H
(
t
)
λ =
EXP
(
R
∗
−
(13.10)
K
where R is the maximum host growth rate. Thus, the equation governing the size of
the host population in time t+1 becomes
EXP
R
1
H
(
t
)
H
(
t
+
1
)=
H
(
t
)
∗
∗
−
−
A
∗
P
(
t
)
(13.11)
K
and after subtracting the respective state variables in time period t from equations
(13.9) and (13.11), we have a set of differential equations that capture the change of
host and parasitoid densities from time period t to t
+
1:
EXP
R
1
H
(
t
)
∆
H
(
t
)=
H
(
t
)
∗
∗
−
−
A
∗
P
(
t
)
−
H
(
t
)
(13.12)
K
∆
P
(
t
)=
C
∗
H
(
t
)
∗
[
1
−
EXP
(
−
A
∗
P
(
t
))]
−
P
(
t
)
(13.13)
We can now see the dynamics exhibited by this model (Figure 13.8). These equa-
tions describing changes in the host and parasitoid densities can yield a variety of
results, from the production of steady state conditions for the host and parasitoid, to
their lock in a limit cycle, to chaos.
The following graphs result from the parameters and initial conditions in the
table, and a DT = 1:
13.2.2 Questions and Tasks
1. We have modeled in this section of Chapter 13 one type of species interaction that
is almost exclusively found among insects. Typically, both the parasitoid and host
