e
e
c
f
c
f
x
f
d
d
d
a
f
a
f
y
f
b
b
b
+
=
=
+
=
+
−
=
=
−
=
−
µ
µ
0
4
y
15
0
x
Shark (
y
) and sardine (
x
) trajectories
The trajectories are closed curves representing periodic motion of both sharks and
sardines. The trajectories look like the trajectories of the unfished case in Example 3
except the equilibrium point has moved to the right (more prey) and down (fewer
predators).
(b)
With the parameters in part (a) and
0.5
f
=
the equilibrium point is (7, 1.5). This
compares with the equilibrium point (6, 2) in the unfished case.
As the fishing rate
f
increases
from 0 to 2, the equilibrium point moves
along the line from the unfished equilib-
rium at (6, 2) to (10, 0). Hence, the fish-
ing of each population at the same rate
benefits the sardines (
x
) and drives the
sharks (
y
) to extinction. This is illustrated
in the figure.
8
4
y
x
12
6
2
10
2
(6, 2)
(10, 0)
(sharks)
(sardines)
1
(7, 1.5)
(c)
You should fish for sardines when the sardine population is increasing and sharks when
the shark population is increasing. In both cases, more fishing tends to move the
populations closer to equilibrium while maintaining higher populations in the low parts
of the cycle.
(d)
If we look at the insecticide model and assume both the good guys (predators) and bad
guys (prey) are harvested at the same rate, the good guys will also be diminished and the
bad guys peak again. As
1
f
→
(try
0.8
f
=
) the predators get decimated first, then the
prey can peak again. If you look at part (a), you see that the predator/prey model does not
allow either population to go below zero, as the
x
- and
y
-axes are solutions and the
solutions move along the axes, thus it is impossible for other solutions to cross either of
these axes. You might continue this exploration with the IDE tool, Lotka-Volterra with
Harvest, as in Problem 24.
Full file at

188
CHAPTER 2
Linearity and Nonlinearity
Analyzing Competition Models
11.
(1200
2
3 )
dR
R
R
S
dt
=
−
−
,
(500
)
dS
S
R
S
dt
=
−
−
Rabbits are reproducing at the astonishing rate of 1200 per rabbit per unit time, in the absence of
competition. However, crowding of rabbits decreases the population at a rate double the
population. Furthermore, competition by sheep for the same resources diminishes the rabbit
population by three times the number of sheep!
Sheep on the other hand reproduce at a far slower (but still astonishing) rate of 500 per sheep per
unit time. Competition among themselves and with rabbits diminishes merely one to one with the
number of rabbits and sheep.
Equilibria occur at
0
0
600
300
,
,
, and
0
500
0
200
⎡ ⎤ ⎡
⎤ ⎡
⎤
⎡
⎤
⎢ ⎥ ⎢
⎥ ⎢
⎥
⎢
⎥
⎣ ⎦ ⎣
⎦ ⎣
⎦
⎣
⎦
.
The equilibria on the axes that are not the origin are the points toward which the populations
head. Which species dies out depends on where they start. See Figure, where
x
and
y
are
measured in hundreds.

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