Differential Equations: The First Steps - Elements of Scientific Computing - page 63

Assume that we have an exponential growth model and let r 0 D 5:3: Here t D 0
corresponds to the year 1800: We want to estimate the population until 1900; i.e.,
we consider t between 0 and 100: Note that we count the population in millions,
i.e., in the year 1800 approximately 5.3 million people lived in the United States. 23

(a) Explain that an exponential growth model can be written in the form

r.t/ D 5:3e 0:03t

where t ranges from 0 to 100:
(b) We want to plot this solution and therefore we want t to go from 1800 to 1900.
Show that we can define

r.t/ D 5:3e 0:03.t 1800/

1900:
(c) Write a computer program to plot r.t/ for 1800

for 1800 6 t 6

1900: In the same plot
you should include the actual population given in the table below.
Year Population (in millions)
1800 5.3
1810 7.2
1820 9.6
1830 12.9
1840 17.0
1850 23.2
1860 31.4
1870 38.6
1880 50.2
1890 63.0
1900 76.2
(d) Use the model above to predict r.1980/: The correct 24

6 t 6

number is about 226.5

million.
(e) Use a logistic model with r 0 D 5:3 and a D 0:03: Plot the solutions for some
values of the carrying capacity. Can you find a value of R such that the logis-
tic model matches the data in the table better than the exponential model did?

23 That is the current population of Denmark and they probably have no idea of what is about to
happen. If you do not know much about Denmark, you should find a map and rethink the concept
of carrying capacity. Is it reasonable that in the next 100 years the population of Denmark will
develop as the U.S. population did from 1800 to 1900?
24 Unfortunately, it turns out that counting Americans is not much easier than counting rabbits, so
the numbers are not at all certain. In fact nobody knows the exact number of Americans at any
time. This is, however, not a typical American phenomenon. Nobody knows the exact number of
people in any reasonably big country.