PROCESSES THAT CHANGE WITH TIME: INTRODUCTION TO DYNAMICAL SYSTEMS - An Invitation to Biomathematics - page 10

U.S.
Population
P(t)
(millions)

Predicted U.S.
Population
[millions] for
r ¼ 0.297

Relative Error [%] ¼
j Predicted Actual j
Actual

Time t
(decades)

r ¼ ln(P(t þ 1))
-ln(P(t))

100

0 5.3 0.306 5.300 0.000

1 7.2 0.288 7.133 0.931

2 9.6 0.295 9.599 0.010

3 12.9 0.282 12.919 0.147

4 17.1 0.305 17.387 1.678

5 23.2 0.303 23.399 0.858

6 31.4 0.306 31.491 0.290

TABLE 1-3.
Determination of r and evaluation of predicted population values.

As in the discrete case, our method of estimating the value of r was
rather primitive. The average value r

0.297 showed a good fit with the
census data, but we defer how to find the best value of r until Chapter 8.

¼

One purpose of a mathematical model may be to predict values that
cannot be measured directly. In our example, these may be values of the
U.S. population for past years for which no U.S. census data are
available, or values of the U.S. population for future years. In particular,
can we use the discrete and continuous models (1-1) and (1-2) (with
our best values of k 0.297) to predict the U.S. population

in the year 3000? Mathematically, this is not a problem. In the discrete
case, we rewrite our model P n - P n 1 ¼

¼ 0.345 and r ¼

(1.345) P n-1 .
Because time is measured in decades beginning with the year 1800, the
year 3000 will correspond to n

(0.345) P n 1 as P n ¼

120, and so we need to find the value of
P 120 . Knowing the U.S. population for n

¼

¼ 0 to be 5.3 million, we have

P 0 ¼ 5.3 and can compute P 1 ¼ (1.345) P 0 ¼ (1.345) (5.3) ¼ 7.1.

Having calculated P 1 , we can calculate P 2 ¼ 9.6,

and so on. We would therefore need to calculate 120 consecutive values
before we get P 120 . Alternatively, we could use a computer to get the
value of P 120 . In the continuous case, of course, we just substitute 120
for t into Eq. (1-7). Exercise 1-1 shows that a formula for direct
computation of P 120 can also be calculated for the discrete model.

1.5 P 1 ¼ (1.345) (7.1) ¼

E XERCISE 1-1

For the model P n - P n 1 ¼ kP n 1 , show that:

ð Þ ¼ð þ Þ

a P n 1 k P n 1

(1-8)

n P 0

ð b Þ P n ¼ð 1 þ k Þ :

k) n P 0 represents the analytical solution for
Eq. (1-1). Because we know the net per capita growth rate k

The expression P n ¼ (1 þ

¼ 0.345 and