7.6. POPULATION GROWTH AND THE LOGISTIC EQUATION
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(a) Assume that the per capita growth rate
db/dt
b
is a linear function of b. Use the
measurements to find this function and write a logistic equation to describe
db
dt .
(b) What is the carrying capacity for the bacteria?
(c) At what population is the number of bacteria increasing most rapidly?
(d) If there are initially 1,000 bacteria, how long will it take to reach 80% of the
carrying capacity?
3. Suppose that the population of a species of fish is controlled by the logistic equation
dP
dt
= 0.1P(10 − P),
where P is measured in thousands of fish and t is measured in years.
(a) What is the carrying capacity of this population?
(b) Suppose that a long time has passed and that the fish population is stable at
the carrying capacity. At this time, humans begin harvesting 20% of the fish
every year. Modify the differential equation by adding a term to incorporate
the harvesting of fish.
(c) What is the new carrying capacity?
(d) What will the fish population be one year after the harvesting begins?
(e) How long will it take for the population to be within 10% of the carrying
capacity?