Applications of the Initial Value Problem y' = f(x, y); y(c) = d 135Substituting the estimated value fork and the known 1960 values given above
for P and ( dP / dt) / P, we find that the vital coefficient E must satisfy.018 = .029 - c:(3 x 109 ).Solving this equation for E yields E = 3.667 x 10-^12. Using these vital coeffi-
cients, the logistic law model predicts a limiting value for the human popula-
tion of the earth of
K = k/c: = .029/(3.667 x 10-^12 ) = 7.91 x 109 people.
In 2000 the size of the population reached 6.08 x 109. The graph displayed
in Figure 3.10 is the logistic curve for the earth's human population ob-
tained from equation (6) by setting Po = 3 billion, to = 1960 , k = .029, and
E = 3.667 X 10-^12.
EXERCISES 3.3
l. Assume the human population of the earth obeys the Malthusian pop-
ulation model. In 1650 A.D. the earth's human population numbered
2.5 x 108. By 1950 A.D. the population had grown to 2.5 x 109. Write a
general equation for the population of the earth. (HINT: Use the infor-
mation given to determine the constants Po, k, and to in equation (2).
No computer is needed.) Suppose further that the earth can support at
most 2.5 x 1010 people. When will this limit be reached?- If a population is being harvested by a predator or dying due to a
disease at a constant rate H > 0, then a possible model for the popula-
tion based on a modification of the Malthusian model is
dP
- = kP-H·
dt '
P(O) =Po.
Suppose for a specific population k = .1 and the initial population size
is 500. Use SOLVEIVP or your computer software to compute and
graph the population on the interval [O, 10] for the following values of
the harvesting constant:a. H = 40 b. H =^50 c. H = 60
What do you think will happen to P(t) as t increases in each case?3. The logistic law model with k = .03134 and E = 1.589 x 10-^10 pro-
vides a model for the population of the United States. Based on this
model, what is the limiting population of the United States? Assume
that in 1800 the population of the United States was 5.31 million
(5.31 x 106 ). Use SOLVEIVP or your computer software to compute