Information Technology Reference
In-Depth Information
which states that the
relative growth
of fish decreases as the number of fish increases.
When we introduce sharks, it is reasonable for us to assume that the relative growth
rate of fish is reduced linearly with respect to S .Thisgives
F
0
F
D
˛.1
F=ˇ
S/ ;
(3.8)
where >0is a constant. We rewrite this as
F
0
D
˛.1
F=ˇ
S/F:
(3.9)
Next, we consider the growth or decay of sharks. If there are no fish in the sea,
the relative change in sharks can be expressed as
S
0
S
D
ı;
(3.10)
where ı>0is the decay rate. Since the sharks need fish to survive, F
D
0 will
eventually lead to no sharks as well. On the other hand, the presence of fish will
increase the relative change of sharks and we assume that this can be modeled as
S
0
S
D
ı
C
"F;
(3.11)
where ">0is a constant.
We can now summarize the 2
2 system as follows,
F
0
D
˛.1
F=ˇ
S/F;
F.0/
D
F
0
;
(3.12)
S
0
D
."F
ı/S;
S.0/
D
S
0
:
(3.13)
When the parameters ˛, ˇ, and " are known, by e.g. estimation, and when F
0
and
S
0
are given, system (
3.12
)and(
3.13
) can predict the number of fish and sharks in
the sea.
Volterra studied a version of the system above and found that it could generate
periodic
3
solutions. When there was no fishing activity, the number of fish increased.
This in turn led to an increase in the number of sharks. But with a large number of
sharks, the need for food increased dramatically and thus the number of fish was
reduced, which in turn reduced the number of sharks.
3
A periodic solution repeats itself. If f.t
C
T/
D
f.t/ for some T
¤
0 and for all t ,thenf is
periodic.
