Suppose that rabbits are introduced onto an island where they have no natural enemies.
Because of natural conditions, the island can support a maximum of 1000 rabbits. Let P(t)
denote the number of rabbits at time t (measured in months), and suppose that the population
varies in size (due to births and deaths) at a rate proportional to both P(t) and 1000 − P(t).
That is, suppose that P(t) satisfies the differential equation dP/dt = kP(1000−P), where k is
a positive constant.
a) Find the value of P when the rate of change of the rabbit population is maximized.

Question

Suppose that rabbits are introduced onto an island where they have no natural enemies.
Because of natural conditions, the island can support a maximum of 1000 rabbits. Let P(t)
denote the number of rabbits at time t (measured in months), and suppose that the population
varies in size (due to births and deaths) at a rate proportional to both P(t) and 1000 − P(t).
That is, suppose that P(t) satisfies the differential equation dP/dt = kP(1000−P), where k is
a positive constant.
a) Find the value of P when the rate of change of the rabbit population is maximized.

perl · Answer

b) When is the rate of change of the rabbit population a minimum? Discuss your answers.