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In-Depth Information
Show that
D
.2
"/t
Z
t
0
ln.F .t //
ln F
0
S./d
(3.90)
and
D
Z
t
0
ln.S.t //
ln S
0
F./d
.1
C
ı/t:
(3.91)
(d) Assume again that we have a periodic solution such that
F.T/
D
F.0/
and
S.T /
D
S.0/
(3.92)
for some T>0.Use(
3.90
), (
3.91
), and (
3.92
) to show that
Z
T
1
T
F./d
D
1
C
ı;
(3.93)
0
and
Z
T
1
T
S./d
D
2
":
(3.94)
0
(e) Use (
3.93
)and(
3.94
) to conclude that fishing increases the average amount of
fish and decreases the average amount of sharks. This helps to explain the effect
observed in the upper Adriatic Sea during World War I (see page
76
).
˘
3.6
Project: Analysis of a Simple System
In the text above we observed that we are able to derive quite a number of properties
for a certain system of ODEs. The purpose of this project is to teach you to do the
same steps on your own. In order to do this, we will study a very simple 2
2 system
and try to derive many properties, both numerically and analytically. Throughout
this project we will consider the system
u
0
.t /
D
v
.t /
u
.0/
D
u
0
;
v
0
.t /
D
u
.t /
(3.95)
v
.0/
D
v
0
;
where
u
0
and
v
0
are given.
