Information Technology Reference
In-Depth Information
10
9
8
7
6
5
4
3
2
1
0
0
2
4
6
8
10
12
14
16
18
20
t
Fig. 3.1
The numerical solution for system (
3.14
)and(
3.15
), produced by the explicit scheme
(
3.18
)and(
3.19
)usingt
D
0:1. The solution for F is represented
by the
solid curve
, whereas the solution for S is represented by the
dashed curve
1=1;000, F
0
D
1:9,andS
0
D
3.3
A Numerical Method: Limited Resources
In the simple model (
3.14
)and(
3.15
) above, we used ˇ
D1
, which means that we
assumed unbounded resources for the fish. If we replace this by ˇ
D
2,whichisa
fairly strict assumption on the amount of plankton available, we get the system
F
0
D
.2
F
S/F;
F.0/
D
F
0
;
(3.20)
S
0
D
.F
1/S;
S.0/
D
S
0
:
(3.21)
Following the steps in Sect.
3.2
, we define a numerical scheme:
F
nC1
D
F
n
C
t.2
F
n
S
n
/F
n
;
(3.22)
S
nC1
D
S
n
C
t.F
n
1/S
n
:
(3.23)
The solutions can be computed in the same manner as in Sect.
3.2
.InFig.
3.2
we
have plotted the numerical solutions using F
0
D
1:9, S
0
D
0:1 and t
D
1=1;000.
Note that, in the presence of limited resources, the solution quickly converges
toward the equilibrium
4
solution represented by S
D
F
D
1.
4
Equilibrium solutions stay constant forever. In system (
3.20
)-(
3.21
), F
0
D
S
0
D
0 is an example
of an equilibrium solution.
