Biology Reference
In-Depth Information
mathematics, whereas Eq. (1-2) uses the language of continuous
mathematics.
Equation (1-2) is in the form of a differential equation; that is, it contains
information about the derivative of the unknown function P
¼
P(t),
¼
which we hope to find. Rewriting Eq. (1-2) as dP/P
rdt and integrating,
we obtain:
Z
Z
rdt
dP
P
¼
;
so that
ln
ð
P
Þ¼
rt
þ
C
;
where C is the constant of integration. Thus:
e
ln
ð
P
ð
t
ÞÞ
¼
e
rt
þ
C
e
rt
e
C
C
1
e
rt
P
ð
t
Þ¼
¼
¼
;
(1-3)
e
C
is a constant.
where C
1
¼
Usually, we know the initial population P(0), and we can thus determine
C
1
. From Eq. (1-3), using t
C
1
e
r0
C
1
,soC
1
is P(0).
This gives us the solution of Eq. (1-2) for the unknown function P(t):
¼
0, we obtain P(0)
¼
¼
e
rt
P
ð
t
Þ¼
P
ð
0
Þ
:
(1-4)
Equation (1-4) is the fundamental equation of unfettered growth. We
want to estimate r from the data in Table 1-1 as we estimated k earlier.
Now
e
rt
e
r
ð
t
þ
1
Þ
;
P
ð
t
Þ¼
P
ð
0
Þ
and P
ð
t
þ
1
Þ¼
P
ð
0
Þ
so:
e
r
ð
t
þ
1
Þ
P
ð
t
þ
1
Þ
P
ð
0
Þ
e
r
¼
¼
:
(1-5)
P
ð
t
Þ
P
ð
0
Þ
e
rt
Thus, we can estimate r by:
P
ð
t
þ
1
Þ
r
¼
ln
¼
ln
ð
P
ð
t
þ
1
ÞÞ
ln
ð
P
ð
t
ÞÞ:
(1-6)
P
ð
t
Þ
Using that P(0)
7.2, and so forth, we give the estimated
values of r in column 3 of Table 1-3. If we average the values of r
(the method that gave the best estimate in the discrete case), we get
r
¼
5.3, P(1)
¼
¼
0.297. We can now estimate the population by using:
3 e
0
:
297t
P
ð
t
Þ¼
5
:
;
(1-7)
where t is the number of decades after 1800. The predicted U.S.
population appears in column 4 of Table 1-3.
