Biology Reference
In-Depth Information
equilibrium state. The MSY is then determined as the largest yield that
will guarantee the existence of a nonzero equilibrium.
E
XERCISE
1-10
Suppose we have a farm that can sustain a herd of 400 head of cattle and
that the cattle population is modeled by the logistic equation:
dP
dt
¼
0
:
002
ð
400
P
Þ
P
;
(1-22)
where
dP
dt
is the rate of change in the number of head per year. The graph
of 0.002(400 - P)P versus P is given in Figure 1-18 (solid line).
(a) Give the equilibrium state(s) for the model, and classify each as
stable or unstable.
If we decide to sell one animal per week (or 52 per year), then the
new equation governing the population would be:
dP
dt
¼
0
:
002
ð
400
P
Þ
P
52
:
(1-23)
The graph of 0.002(400 - P)P
52 versus P is shown in
Figure 1-18 (dashed line). The equilibrium states are approximately
82 and 318 cattle.
(b) Classify the new equilibrium states as stable or unstable.
dP
f
(
P
)
=
dt
52
0
P
82
318
400
FIGURE 1-18.
Models showing the logistic curves with and without harvesting. The graphs of 0.002(400 - P)P (solid
line) and 0.002(400 - P)P-52 (dashed line) versus P. The graph of 0.002(400 - P)P-52 is obtained by
shifting downward the graph of 0.002(400 - P)P by 52 units.
