Applications of Logistic Growth
(1) Some diseases spread through a (finite) population P at a rate proportional
to the number of people, N(t), infected by time t and the number, P − N(t), not
yet infected. Thus N′(t) = kN(P − N) and, for some positive c and k,
(2) A rumor (or fad or new religious cult) often spreads through a population
P according to the formula in (1), where N(t) is the number of people who have
heard the rumor (acquired the fad, converted to the cult), and P − N(t) is the
number who have not.
(3) Bacteria in a culture on a Petri dish grow at a rate proportional to the
product of the existing population and the difference between the maximum
sustainable population and the existing population. (Replace bacteria on a Petri
dish by fish in a small lake, ants confined to a small receptacle, fruit flies
supplied with only a limited amount of food, yeast cells, and so on.)
(4) Advertisers sometimes assume that sales of a particular product depend on
the number of TV commercials for the product and that the rate of increase in
sales is proportional both to the existing sales and to the additional sales
conjectured as possible.
(5) In an autocatalytic reaction a substance changes into a new one at a rate
proportional to the product of the amount of the new substance present and the
amount of the original substance still unchanged.
Example 20 __
Because of limited food and space, a squirrel population cannot exceed 1000. It
grows at a rate proportional both to the existing population and to the attainable
additional population. If there were 100 squirrels 2 years ago, and 1 year ago the
population was 400, about how many squirrels are there now?
SOLUTION:
Let P be the squirrel population at time t. It is given that
with P(0) = 100 and P(1) = 400. We seek P(2).