Biology Reference
In-Depth Information
dS
dt
¼
0
:
002SI
þ
0
:
1S
¼
S
ð
0
:
002I
þ
0
:
1
Þ
(2-6)
dI
dt
¼
0
:
002SI
0
:
4I
¼
I
ð
0
:
002S
0
:
4
Þ:
In a phase plane diagram, we have a collection of arrows that describe
how the trajectories travel. To construct a phase plane, we begin by
finding the null clines for S and I; that is, where
dS
0 and
dI
dt
¼
dt
¼
0. For the
first equation in our example,
dS
dt
¼
S
ð
0
:
002I
þ
0
:
1
Þ¼
0
;
0
:
1
If
dS
¼
¼
002
¼
:
dt
¼
the null clines are S
0, there is no
horizontal movement, so the arrows will be vertical. Likewise,
0orI
50
0
:
dI
dt
¼
I
ð
0
:
002S
0
:
4
Þ¼
0
0
:
4
implies I
¼
0orS
¼
002
¼
200
:
Along the null clines for I, the arrows
0
:
will be horizontal.
In Figure 2-9, we have sketched the null clines and arrows indicating
horizontal or vertical travel. Along the line S
¼
200, the first equation
from Eq. (2-6) implies that
dS
dt
<
0whenI
>
50. Thus, above the line
I
¼
50, the arrows will point to the left. For I
<
50, the same equation
shows that
dS
dt
>
0, which means that below the line I
¼
50 the arrows
50, we get
dI
will point to the right. Similarly, along the line I
¼
dt
<
0 for
200 and
dI
S
<
dt
>
0 for S
>
200. Therefore, along the line I
¼
50, the
S
= 200
I
I
= 50
0
S
FIGURE 2-9.
Direction of travel near the null clines for the example described by Eq. (2-6).
