Logistic Growth
In exponential growth, we assume that the rate of increase (or decrease)
of a population at any time t is directly proportionalto the population P.
In other words, However in many situations population growth
levels off and approaches a limiting number M (the carrying capacity)
because of limited resources. In this situation the rate of increase (or
decrease) is directly proportional to both Thistype of growth is
called logistic growth. It is modeled by the differential equation .
If we find , wecan find out an important fact about the time when P is
growingthe fastest. We will do this in the example below.
Ex. 1The population of fish in a lake satisfies the logistic differential equation
, where t is measured in years, and .
(a) (b) What is the range of the solution curve?
(c) For what values of P is the solution curve increasing? Decreasing? Justify your answer.
(d) For what values of P is the solution curve concave up? Concave down? Justify your answer.
(e) Does the solution curve have an inflection point? Justify your answer.
(f) Use the information you found to sketch the graph of .
Ex. 2 The population of fish in a lake satisfies the logistic differential equation
, where t is measured in years, and .
(a) (b) What is the range of the solution curve?
(c) For what values of P is the solution curve increasing? Decreasing? Justify your answer.
(d) For what values of P is the solution curve concave up? Concave down? Justify your answer.
(e) Does the solution curve have an inflection point? Justify your answer.
(f) Use the information you found to sketch the graph of .
______
Ex. 3 The population of fish in a lake satisfies the logistic differential equation
, where t is measured in years, and .
(a) (b) What is the range of the solution curve?
(c) For what values of P is the solution curve increasing? Decreasing? Justify your answer.
(d) For what values of P is the solution curve concave up? Concave down? Justify your answer.
(e) Does the solution curve have an inflection point? Justify your answer.
(f) Use the information you found to sketch the graph of .
Logistic Growth
Ex4. The rate at which the flu spreads through a community is modeled by the logistic differential
equation , where t is measured in days.
(a) If solve for P as a function of t.
(b) Use your solution to (a) and your graphing calculator to find the size of the population
when t = 2 days.
(c) Use your solution to (a) and your graphing calculator to find the number of days that have
passed when 2400 people have contracted the flu.
Ex5: The number of students infected by measles in a certain school is given by the formula
where t is the number of days after students are first exposed to an infected student.
Show that P(t) is a solution of a logistic differential equation. Identify k and the carrying capacity.
Estimate P(0) and explain its meaning.
Answers to Examples
Day 1
Ex. 1
(a)
Since.
(b) The range of the solution curve is .
(c) is positive for 4000 < P < 18,000 so the solution curve isincreasing for 4000 < P < 18,000.
(d)
is positive for 4000 < P < 9000 and negative for 9000 < P < 18,000 so the solution
curve is concave up for 4000 < P < 9000 and concave down for 9000 < P < 18,000.
(e) The solution curve has an inflection point at P = 9000 because changes from positive
to negative there.
Ex. 2
(a)
If .
(b) The range of the solution curve is .
(c) is positive for 9000 < P < 18,000 so the solution curve is increasing for 9000 < P < 18,000.
(d)
is negative for 10,000 < P < 18,000 so the solution curve is concave down for
10,000 < P < 18,000.
(e) The solution curve does not have an inflection point because is negative for
10,000 < P < 18,000
Ex. 3
(a)
If .
(b) The range of the solution curve is .
(c) is negative for 18,000 < P < 20,000 so the solution curve is decreasing
for 18,000 < P < 20,000.
(d)
is positive for 18,000 < P < 20,000 so the solution curve is concave up
for 18,000 < P < 20,000.
(e) The solution curve does not have an inflection point because is positive for
18,000 < P < 20,000.
Day 2
Ex1:
(a)
(b) 2617.238 so approximately 2617 people
(c) P = 2400. At that time t = 1.821 days.
Ex2: p369 #32
k = 1; M=200; P(0) = 1…initially, 1 student had the measles