Biology Reference
In-Depth Information
dP
f
(
P
)
=
dt
0
P
p
2
FIGURE 1-11.
Interpreting dP/dt versus P. When the graph of dP/dt versus P crosses the horizontal axes at a point p
2
while increasing near the point of crossing, the value p
2
is an unstable equilibrium.
Now if P is slightly less than p
2,
then f (P) < 0 and the derivative is
negative, so P will decrease and move away from p
2
. Similarly, if P is
slightly greater than p
2
, then f (P)
0, and the derivative is positive, so
P will increase and again move away from p
2
. In either case, if P is
slightly different than p
2
, then P will move away from p
2
. A point such as
p
2
is called an unstable equilibrium point.
>
A physical example of stable and unstable equilibrium points is shown
in Figure 1-12. If a roller coaster cart is stopped at the positions
indicated, it will remain there. If the cart is at positions 2 or 3 and is
nudged gently, it will return to its original position. On the other hand, if
the cart is at positions 1 or 4 and is nudged, it will roll down the track
and away from the position at which it was balanced.
FIGURE 1-12.
A roller coaster model of equilibrium points. Positions 2 and 3 represent stable equilibria while
positions 1 and 4 represent unstable equilibria. (Scorpion Roller Coaster Modeling System
photograph from
www.coasterdynamix.com.
Used by permission.)
