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In the figure below, we repeat from
Part 1 a plot of the actual U.S. census data
through 1940, together with a fitted logistic curve. (Recall that the data after
1940 did not appear to be logistic.)
In this part we will determine directly
from the differential equation
and, if so,
Our test case will be the U.S. Census
data, first up to 1940, then up to 1990. In Part 6 we will study the same questions,
but we will use the known form of the logistic solution from Part 4.
To determine whether a given set of
data can be modeled by the logistic differential equation,
we have to estimate values of the
derivative dP/dt from the data. We will do that by symmetric differences,
as shown in the following figure. The slope dP/dt at a given census year
t is approximated by the slope of the line joining the points 10 years
earlier and 10 years later.
For example, the growth rate dP/dt
in 1900 was approximately
Of course, for the period from 1790
through 1940, we can calculate these slope estimates only from 1800 through 1930,
because we need a data point before and after each point at which we are estimating
the slope.
We may rewrite the logistic equation
in the form
In this form the equation says that
the proportional growth rate (i.e., the ratio of dP/dt to P)
is a linear function of P. Thus, we have a test of logistic behavior:
The same graphical test tells us how
to estimate the parameters:
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