Download 3.7 Logistic and Gompertz Functions

Logistic and Gompertz
Functions
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Remarks
The sigmoid curve is
the s-shaped curve
Three functions of this type are the logistic growth
function, the logistic decay function, and the
Gompertz function
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Remarks
Logistic functions are good models of biological
population growth in species which have grown so
large that they are near to saturating their
ecosystems, or of the spread of information within
societies.
They are also common in marketing, where they
chart the sales of new products over time; in a
different context, they can also describe the decline
of demand for a product as a function of increasing
price
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Logistic Growth
Logistic functions combines characteristic of
exponential growth and decay
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Logistic Function
f  x 
c
1  aebx
where a, b, and c are constants
c is called the carrying capacity
If a > 0, a logistic function increases when b > 0
and decreases when b < 0
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Example – Cell Growth
www.seop.leeds.ac.uk/entries/ecology/
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2
Sales Volume of Archa Beer
Source: http://kelive.kimeng.co.th
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Logistic Growth Model
f  x 
c
1  aebx
If a > 0, b > 0
x-intercept – none
y-intercept – (0,c/(1+a))
Horizontal asymptotes y = 0, y = c
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Logistic Growth Model
y
yc
y0 
f  x 
c
1  aebx
b0
c
1 a
x
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3
Example
P t  
500
1  7e.25t
t = number of days
a) What was the initial amount of bacteria?
b) When will the amount of bacteria be 300
grams?
c) What is the carrying capacity?
d) Graph the function
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Example
P t  
500
1  7e.25t
a) What was the initial amount of bacteria?
P  0 
500
500

 62.5 grams
.25 0 
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1  7e
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Example
P t  
500
1  7e.25t
b) When will the amount of bacteria be 300 grams?
300 
500
1  7e.25t
9.4 days
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4
Example
P t  
500
1  7e.25t
c) What is the carrying capacity?
P t  
500
1  7e.25t
The carrying capacity is 500 grams
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Example
P t  
500
1  7e.25t
Carrying Capacity
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Example
Create logistic model
x
0
2
4
6
8
10
12
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y
4
10
24
46
70
86
94
98
99
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5
Example
Create logistic model
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Example
f  x 
Create logistic model
Logistic
c
d
1  aebx
Example
Logist83
f  x 
c
1  aebx
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Create logistic model
Regression: logistic
Allows for vertical translation
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Example
Create logistic model
Regression: logist83
Equivalent to TI-83 regression
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Example
A model for the number of
students at Palo Alto College
that have heard the latest
rumor might be
N d  
P
1  9e.25d
where P is the total number of students at Palo Alto
and d is the number of days that have elapsed since
the rumor began.?
If P = 8000 students, how
many students will know the
latest rumor in two days?
In ten days?
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Example
Rumor at Palo Alto College
Two days
N
8000
 1240 students
1  9e0.252
Ten days
N
8000
 4600 students
1  9e0.2510
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Example
Rumor at Palo Alto College
Graph of first 50 days
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Logistic Decay Model
f  x 
c
1  aebx
If a > 0, b < 0
x-intercept – none
y-intercept – (0,c/(1+a))
Horizontal asymptotes y = 0, y = c
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Logistic Growth Model
y
yc
c
y0 
1 a
f  x 
c
1  aebx
b0
x
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8
Gompertz Function
f  x   Ca R
t
0  R  1 is the expected rate of growth of a population
a is the proportion of the initial population
C is the carrying capacity
In Biology - The Gompertz growth law has been shown to
provide a good fit for the growth data of numerous
tumors
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Example
The number of micro DVD players sold by
the Palo Alto Manufacturing Company each
month is given by N  10, 000  0.2 
0.3t
(a) How many DVD players will be sold the first month?
(b) Graph the function for the first year
(c) What is the predicted upper limit on sales?
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Example
The number of micro DVD players sold by
the Palo Alto Manufacturing Company each
month is given by N  t   10, 000  0.2 
0.3t
(a) How many DVD players will be sold the first month?
N 1  10,000  0.2 
0.3
 6,170 units
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Example
The number of micro DVD players sold by
the Palo Alto Manufacturing Company each
month is given by N  10, 000  0.2 
0.3t
(b) Graph the function for the first year
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Example
The number of micro DVD players sold by
the Palo Alto Manufacturing Company each
month is given by N  10, 000  0.2 
0.3t
(b) Graph the function for the first year
To graph a Gompertz function on a TI-83/84 we must
use Y1 and Y2
Incorrectly
Correctly
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Example
The number of micro DVD players sold by
the Palo Alto Manufacturing Company each
month is given by N  10, 000  0.2 
(c) What is the predicted upper limit on sales?
Upper limit = 10,000 units
0.3t
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