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In-Depth Information
3.2
A Numerical Method: Unlimited Resources
In order to discuss numerical solutions of system (
3.12
)and(
3.13
), we simplify the
problem by choosing simple values of the parameters involved. For illustration we
choose
1
2
;
˛
D
2;
D
"
D
1;
and
ı
D
1:
Let us also start by setting ˇ
D1
. This means that the number of fish can grow
without any limit. Since fish eat plankton, this assumption is valid at least when the
number of fish is not too large. To summarize, we have the simplified model
F
0
D
.2
S/F;
F.0/
D
F
0
;
(3.14)
S
0
D
.F
1/S;
S.0/
D
S
0
:
(3.15)
In order to solve this system numerically, we introduce the time step t > 0 and
define t
n
D
nt .LetF
n
and S
n
denote the approximations of F.t
n
/ and S.t
n
/,
respectively. Since
F.t
nC1
/
F.t
n
/
t
S.t
nC1
/
S.t
n
/
t
F
0
.t
n
/
S
0
.t
n
/;
and
we introduce the numerical scheme
F
nC1
F
n
t
D
.2
S
n
/F
n
;
(3.16)
S
nC1
S
n
t
D
.F
n
1/S
n
:
(3.17)
The scheme can be rewritten in a computational form,
F
nC1
D
F
n
C
t.2
S
n
/F
n
;
(3.18)
S
nC1
D
S
n
C
t.F
n
1/S
n
:
(3.19)
We observe from this scheme that if F
0
and S
0
are given, then F
1
and S
1
can be
computed from (
3.18
)and(
3.19
), respectively, using n
D
0.WhenF
1
and S
1
have
been computed, we set n
D
1 and compute F
2
and S
2
. In this manner, we can
compute F
n
and S
n
for all n>0.
For illustration, we choose t
D
1=1;000, F
0
D
1:9 and S
0
D
0:1. This models
a situation with a large number of fish and a small number of sharks initially. We
have plotted the numerical approximation in Fig.
3.1
. Note that, initially, the number
of fish increases almost exponentially. This leads to a strong growth in the shark
population that subsequently leads to a sharp reduction of the fish, and then of the
sharks. We note that this is what Volterra tried to model.
