Biology Reference
In-Depth Information
Change in
Population
P
n
P
n1
Time
(decades) n
U.S. Population
(millions) P
n
k ¼ (P
n
P
n1
)/ P
n1
0
5.3
—
—
1
7.2
1.9
0.358
2
9.6
2.4
0.333
3
12.9
3.3
0.344
4
17.1
4.2
0.326
5
23.2
6.1
0.357
6
31.4
8.2
0.353
TABLE 1-2.
Estimation of k from U.S. population data.
We next estimate the numerical value of k from the data. The
calculations are summarized in Table 1-2. Ideally, if all data points (P
n
1
,
P
n
- P
n
1
), n
, whose coordinates are given in the second and
third columns were perfectly lined up, the values of k calculated as k
¼
1, 2, 3,
...
¼
(P
n
P
n-1
)/P
n
1
in the third column would be exactly the same. In
reality, because of the noise and small inconsistencies that are always
present in the world of experimental data, the values of k vary slightly.
The numerical value chosen for k should be the value that provides the
best agreement between the actual population sizes and the values
predicted by the model. We could, in principle, test all of them and
visually determine the best fit of the predicted data with the actual data.
We did this for the smallest value of k (k
¼
0.326), the largest value of
k (k
0.345).
The results and corresponding graphs are presented in Figure 1-3.
¼
0.358), and the average of all calculated values for k (k
¼
Not surprisingly, the smallest k-value produces predictions that
systematically underestimate the population, while the largest value
generates overestimates. Using the average of the k-values in Table 1-2,
however, gave a very good overall fit. The question of what is meant
by ''best fit'' is certainly nontrivial and will be addressed later in
detail. For now, we shall note that the value of k
0.344 provides the
best fit with the data—just 0.001 below the average value of k we
calculated above.
¼
II. DISCRETE VERSUS CONTINUOUS MODELS
Our model is now P
n
-P
n
1
¼
(0.345)P
n-1
. One limitation of this model is
apparent almost immediately: Our model is discrete, that is, it can only be
used to describe changes that occur at specific time intervals. The
smallest unit it works with is a decade, and, thus, the model is
incomplete. For example, it does not allow us to compute the
