Name__________________________________
Period____
Calculus
Lesson 6.5 Logistic Growth and the Law of Limited Growth
Logistic Growth Models
dP
 kP  M  P  where M is the carrying capacity and P is the population.
dt
M
P
1  Ae Mk t
1. The growth rate of population P of bears in a newly established wildlife preserve is modeled by the differential
dP
 0.008P 100  P  , where t is measured in years.
equation
dt
(a) What is the carrying capacity for bears in this wildlife preserve?
(b) What is the bear population when the population is growing the fastest?
(c) What is the rate of change of the population when it is growing the fastest?
Logistic Growth Models
dP
 kP  M  P  where M is the carrying capacity and P is the population.
dt
P
M
1  Ae Mk t
2. In 1985 and 1987, the Michigan Department of Natural Resources airlifted 61 moose from Algonquin Park,
Ontario to Marquette County in the Upper Peninsula. It was originally hoped that the population P would reach
dP
 0.0003P 1000  P  .
carrying capacity in about 25 years with a growth rate of
dt
(c) Solve the differential equation with the initial condition P(0) = 61 and show that it conforms to the slope field.
The Law of Limited Growth
dP
 k  M  P  where M is the carrying capacity and P is the population.
dt
P  M  Ae kt , where A is a constant determined by an appropriate initial condition.
3. The population P of bears in a preserve at time t years, when t  0 , is increasing at a rate directly proportional
to 400 – P, where the constant of proportionality is k. There are initially 50 bears on the preserve and three years
later there are 120 bears,
(a) Define the population equation, P(t).
(b) Find the lim P  t 
t 