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so we get the linear system
10 56:5
56:5 319:5
A
B
D
14:1
79:6
;
(5.96)
which has the solution
A
D
2:7455;
B
D
0:2364:
(5.97)
Consequently, the data indicate that
100
p
0
.t /
p.t/
2:7455
0:2364 p.t/;
(5.98)
or
p
0
.t /
0:027 p.t/
0:00236 p
2
.t /:
(5.99)
So, the logistic model takes the form
p
0
.t /
0:027 p.t/ .1
p.t/=11:44/ :
(5.100)
Here, we note that, according to this model, 11.44 billion seems to be the carrying
capacity of the Earth. We also note that the analytical solution of (
5.100
) with the
initial condition
p.0/
6:08;
where t
D
0 corresponds to the year 2000, is given by
69:5
6:08
C
5:36 e
0:027 t
p.t/
;
(5.101)
see (
5.75
). Note that the model (
5.101
) predicts that there are
p.100/
10:79 billion
people on the Earth in the year of 2100. This should be compared with 25.2 billion,
which was predicted by the purely exponential model. In Fig.
5.14
we graph the
analytical solution of the exponential model (
5.87
) and the logistic model (
5.101
),
respectively.
Let us conclude this section with a warning. Do not put your money on these esti-
mates. They are just meant to briefly sketch how such computations can be done.
There are serious and continuous research efforts to find out how the population
evolves in various parts of the world. In some of the fastest-growing regions, serious
