Biology Reference
In-Depth Information
dP
f
(
P
)
=
dt
0
P
p
1
p
2
FIGURE 1-9.
Plot of dP/dt versus P. The graph of dP/dt versus P helps to visualize how the rate of change of the
population depends upon the population size.
The derivative is zero where the graph of f (P) crosses the horizontal axis
at points p
1
and p
2
. These are equilibrium states. The derivative is
positive when the graph of f (P) is above the horizontal axis and negative
when it is below. From the graph, f (P) is positive, and the population is
growing when P is less than p
1
or greater than p
2
. When P is between
p
1
and p
2
, f (P) is negative.
Let's analyze how P changes if it is near the equilibrium states.
State
p
1
:
The graph of f (P) for P near p
1
is shown in Figure 1-10.
0, which means
dP
Suppose P is slightly smaller than p
1
. Then f (P)
>
dt
>
0
;
so P is increasing toward p
1
. On the other hand, if P is slightly larger
than p
1
, then f (P) < 0, so
dP
dt
< 0 and P decreases, again moving toward
p
1
. In either case, if P is slightly different than p
1
, then P moves toward
p
1
. We refer to a point such as p
1
as a stable equilibrium point.
State
p
2
:
The graph of f (P) for P near p
2
is shown in Figure 1-11.
dP
f
(
P
)
=
dt
0
P
p
1
FIGURE 1-10.
Interpreting dP/dt versus P. When the graph of dP/dt versus P crosses the horizontal axes at a point p
1
while decreasing near the point of crossing, the value p
1
is a stable equilibrium.
