1550078481-Ordinary_Differential_Equations__Roberts_

132 Ordinary Differential Equations

P(t)

t

Figure 3.9 P(t) = P 0 ek(t-to) for k > 0, k = 0, and k < 0.

For k > 0 the Malthusian population model indicates there is no limit

to the size of the population. This is totally unrealistic. Nonetheless, for

short intervals of time, exponential growth of a population is possible when

the population has enough room in which to expand and an abunda nce of

food and other natural resources to support its expansion. For example, the

population of the United States from 1800 through 1860 can be modelled

approximately by the function

(4) P(t) = Poe.D3(t-1soo)

where Po = 5.31 million is the population of the United States in 1800. Of
course, the function ( 4) is the solution of the Malthusian population model

dP = .03P·

dt '

P(1800) = 5.31 million.

A comparison of the predicted population and the actual population of the
United States from 1800 through 1900 is given in Table 3.1. As we can see
from this table, the function ( 4) does not model the population growth of the
United States from 1870 to 1900 very well. This is due to the fact that the
Malthusian population model ignores important factors such as the change
in the birth or death rate with time, wars, disease, immigration, emigration,
and changes in the age structure of the population. (What happened in the
United States between 1860 and 1870 which might account for the cessation
of exponential growth?)