1550078481-Ordinary_Differential_Equations__Roberts_

Applications of the Initial Value Problem y' = f(x, y); y(c) = d 133

Table 3.1 United States Population (in millions), 1800-1900

Year 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900

Actual 5.31 7.24 9.64 12.87 17.07 23.19 31.44 38.56 50.16 62.95 76.00

Predicted 5.31 7.17 9.68 13.06 17.63 23.80 32 .12 43.36 58.53 79.01 106.65

Error (%) 0.00 -. 97 .41 1.48 3.28 2 .63 2.16 12.44 16.69 25.51 40.33

For k < 0 the Malthusian population model predicts that the population
will decrease exponentially to zero. In 1946, the pesticide DDT began to be
used extensively throughout the United States. The following year a decrease
in the population of peregrine falcons was noted. By 1970, due to the use of
DDT and other similar pesticides, the peregrine falcon was nearly extinct in
the continental United States. This particular tragedy illustrates that it is
currently possible for humans to alter the death rate of a species so that it
becomes greater than the birth rate thereby producing a dramatic population
decrease. Similar events may occur in the future if the delicate balance of
environmental factors are not properly considered before various courses of
action are pursued.

In his text of 1835, Lambert Quetelet (1796-1874) criticized Malthus and
others who studied population growth for not establishing their results within
a more mathematical framework. Quetelet advanced the theory that popu-
lations tend to grow geometrically but the resistance to growth increases in
proportion to the square of the velocity with which the population tends to
increase. He drew an analogy between population growth and the motion of
a body through a resisting medium; however, he presented no mathematical
treatment of the problem. Pierre-Frarn;ois Verhulst (1804-1849) studied under
Quetelet in Ghent. In memoirs published in 1838, 1845 , and 1847, Verhulst
developed his "logistic growth" model for populations. He was frustrated in
his attempts to verify the model, because no accurate census information was
available at that time. Verhulst's population model lay dormant for approxi-
mately eighty years. It was independently rediscovered in the early 1920s by
two American scientists, Raymond Pearl (1879-1940) and Lowell Reed (1886-
1966). The logistic law model or Verhulst-Pearl model for population
growth is

dP = kP-EP^2

dt

where k and E are positive constants and E is small relative to k. When

the population P is small , the term EP^2 is very small compared to kP and

so the population will grow at nearly an exponential rate. However, as the

population becomes large, the term EP^2 will approach the term kP in size