It is worth spending a bit of time considering the nature of the solution
of the difference Equation 4 and the differential Equation 14. First,
note that the solutions of both increase at the same "rate". If we sample
the population function p(t)p(t) at intervals of TT time units, a
geometric number sequence results. Let pnpn be the samples of p(t)p(t)
given by
p
n
=
p
(
n
T
)
n
=
0
,
1
,
2
,
.
.
.
p
n
=
p
(
n
T
)
n
=
0
,
1
,
2
,
.
.
.
(16)This give for Equation 15
p
n
=
p
(
n
T
)
=
p
o
e
r
n
T
=
p
o
(
e
r
T
)
n
p
n
=
p
(
n
T
)
=
p
o
e
r
n
T
=
p
o
(
e
r
T
)
n
(17)which is the same as (Reference) if
This means that one can calculate samples of the exponential solution of
differential equations exactly by solving the difference Equation 4 if
RR is chosen by Equation 18. Since difference equations are easily implemented
on a digital computer, this is an important result; unfortunately,
however, it is exact only if the equations are linear. Note that if the
time interval TT is small, then the first two terms of the Taylor's
series give
R
=
e
r
T
≈
1
+
r
T
R
=
e
r
T
≈
1
+
r
T
(19)which is somewhat similar to Equation 3.
Another view of the relation can be seen by approximating Equation 14 by
x
(
n
+
1
)
-
x
(
n
)
T
=
r
x
(
n
)
,
x
(
n
+
1
)
-
x
(
n
)
T
=
r
x
(
n
)
,
(20)which gives
x
(
n
+
1
)
=
x
(
n
)
+
r
T
x
(
n
)
x
(
n
+
1
)
=
x
(
n
)
+
r
T
x
(
n
)
(21)
=
(
1
+
r
T
)
x
(
n
)
=
(
1
+
r
T
)
x
(
n
)
(22)having a solution
x
(
n
)
=
x
(
0
)
(
1
+
r
T
)
n
x
(
n
)
=
x
(
0
)
(
1
+
r
T
)
n
(23)This implies Equation 21 also, and the method is known as Euler's method for
numerically solving a differential equation.
These approximations are used often in modeling. For population models a
differential evuation is often used, even though it is obvious that
births and deaths occur at random discrete times and populations can take
on only integer values. The approximation makes sense only if we use
large aggregates of individuals. We end up modeling a process that occurs
at random discrete points in time by a continuous time mode, which is then
approximated by a uniformly-spaced discrete time difference equation for
solution on a digital computer!
The rapidity of increase of an exponential is usually surprising and it is this fact that makes understanding it important. There are several ways to describe the rate of growth.
If
x
=
k
e
r
t
,
If
x
=
k
e
r
t
,
(24)
then
d
x
d
t
=
k
r
e
r
t
then
d
x
d
t
=
k
r
e
r
t
(25)
or
r
=
1
x
d
x
d
t
.
or
r
=
1
x
d
x
d
t
.
(26)This states that rr is the rate of growth per unit of xx. For
example, the growth rate for the U.S. is about 0.014 per year, or an
increase of 14 people per thousand people each year.
Another measure of the rate is the time for the variable to double in
value. This doubling time, TdTd, is constant and can easily be shown
to be given by
T
d
=
1
r
log
e
2
=
0
.
6931472
1
r
T
d
=
1
r
log
e
2
=
0
.
6931472
1
r
(27)For example, doubling times for several rates are given by
Table 1
| rr |
TdTd |
| .01 |
70 |
| .02 |
35 |
| .03 |
23 |
| .04 |
17 |
| .05 |
14 |
| .06 |
12 |
The present world population is about three billion, and the growth
rate is 2.1% per year. This gives
p
(
t
)
=
3
e
0
.
021
t
p
(
t
)
=
3
e
0
.
021
t
(28)with p(t)p(t) measured in billions of people and tt in years. This
gives a doubling time of 33 years. While it is easy to talk of growth
rate and doubling times, these have real predictive meaning only
if the growth is exponential.
