Biology Reference
In-Depth Information
arrows point down for S
<
200 and up for S
>
200. When S
¼
0, Eq. (2-6)
implies that
dI
dt
<
0, and the arrows along the vertical axis point down. If
0, we get
dS
I
¼
dt
>
0, and the arrows along the horizontal axis point right.
Points at which two null clines intersect have the property that both
dS
dt
¼
0 and
dI
0 at those points. This indicates that there is no motion
at those points, and the trajectory will not move from there. Such points
are called equilibrium points or equilibrium states. In this model, the
equilibrium points are (0,0) and (200,50).
dt
¼
Because S and I represent members of a population, we are only
interested in the part of the phase plane where S
0. The null
clines divide this region into four parts. We determine the direction of
travel in each region by checking the sign of
dS
0 and I
I
S =
200
dt
and
dI
dt
at one point in
dS
dS
<
0
<
0
dt
dt
that region. For example, in the region where 0
<
I
<
50 and 0
<
S
<
200,
dI
dI
<
0
>
0
dt
dt
we could take (10,20) as a test point. Then:
ð
10
;
20
Þ
¼
I =
50
dS
dt
dS
dS
>
0
>
0
10
ð
0
:
002
ð
20
Þþ
0
:
1
Þ >
0
dt
dt
dI
dI
ð
10
;
20
Þ
¼
<
0
>
0
dt
dt
dI
dt
20
ð
0
:
002
ð
10
Þ
0
:
4
Þ <
0
A
0
S
I
S =
200
indicates a direction to the right and down. We leave it to the reader to
check that the signs of the derivatives for the other regions are as shown
in Figure 2-10(A), meaning the directions are as shown in Figure 2-10(B).
The phase plane contains a wealth of information about how the process
evolves in time, but with what we have done so far, we have an
important unanswered question: If you study Figure 2-10(B), it seems
that the equilibrium point (200,50) has particular importance (and it
does). The question is, how do the trajectories for our example behave
regarding this point? Different behaviors, satisfying the conditions from
Figure 2-10, are possible for the trajectories. For instance, they could
spiral away from the equilibrium point [Figure 2-11(A)], spiral in
[Figure 2-11(B)], or orbit around [Figure 2-11(C)].
I =
50
B
0
S
FIGURE 2-10.
Phase plane for Eq. (2-6). Panel A: Regions in the
phase plane where the derivatives dS/dt and dI/dt for
Eq. (2-6) do not change sign; panel B: The directions
of movement in these regions.
In the next section, we describe how to determine the behavior of a
trajectory near an equilibrium point.
IV. STABILITY OF EQUILIBRIUM POINTS
A. Equilibrium Points
Many processes in nature come to equilibrium, and many do not. For
those that have a selection of equilibrium states, some of those states
