6.5 day 2
Logistic
Growth
Columbian Ground Squirrel
Glacier National Park, Montana
Photo by Vickie Kelly, 2004
Greg Kelly, Hanford High School, Richland, Washington
kt
y

y
e
We have used the exponential growth equation
0
to represent population growth.
The exponential growth equation occurs when the rate of
growth is proportional to the amount present.
If we use P to represent the population, the differential
equation becomes: dP
dt
 kP
The constant k is called the relative growth rate.
dP / dt
k
P

kt
P

P
e
The population growth model becomes:
0
However, real-life populations do not increase forever.
There is some limiting factor such as food, living space or
waste disposal.
There is a maximum population, or carrying capacity, M.
A more realistic model is the logistic growth model where
growth rate is proportional to both the amount present (P)
and the carrying capacity that remains: (M-P)

The equation then becomes:
Logistics Differential Equation
dP
 kP  M  P 
dt
We can solve this differential equation to find the logistics
growth model.

Logistics Differential Equation
dP
 kP  M  P 
dt
1
dP  kdt
P M  P
1
A
B
 
P  M  P P M  P
1  A  M  P   BP Partial
Fractions
1
M
1 
1
 
 dP  Mkdt
P M P
ln P  ln  M  P   Mkt  C
P
ln
 Mkt  C
M P
1  AM  AP  BP
1  AM
1
A
M
0   AP  BP
AP  BP
AB
1
B
M

Logistics Differential Equation
dP
 kP  M  P 
dt
1
dP  kdt
P M  P
1
M
1 
1
 
 dP  Mkdt
P M P
ln P  ln  M  P   Mkt  C
P
ln
 Mkt  C
M P
P
 e Mkt C
M P
M P
 e  Mkt C
P
M
 1  e  Mkt C
P
M
 1  e  Mkt C
P

Logistics Differential Equation
P
 e Mkt C
M P
M
P
1  e  Mkt C
M P
 e  Mkt C
P
M
P
1  e C  e  Mkt
M
 1  e  Mkt C
P
M
 1  e  Mkt C
P
Let A  e C
M
P
1  Ae Mk t

Logistics Growth Model
M
P
 Mk t
1  Ae

Example:
Logistic Growth Model
Ten grizzly bears were introduced to a national park 10
years ago. There are 23 bears in the park at the present
time. The park can support a maximum of 100 bears.
Assuming a logistic growth model, when will the bear
population reach 50? 75? 100?

Ten grizzly bears were introduced to a national park 10
years ago. There are 23 bears in the park at the present
time. The park can support a maximum of 100 bears.
Assuming a logistic growth model, when will the bear
population reach 50? 75? 100?
M
P
1  Ae Mk t
M  100
P0  10
P10  23

M
P
 Mk t
1  Ae
100
10 
1  Ae0
M  100
P0  10
P10  23
At time zero, the population is 10.
100
10 
1 A
10 10 A  100
10 A  90
A9
100
P
 Mk t
1  9e

M
P
 Mk t
1  Ae
100
23 
1  9e100 k 10
1  9e
9e
e
1000 k
1000 k
1000 k
100

23
77

23
 0.371981
M  100
P0  10
P10  23
100
P
 Mk t
1  9e
After 10 years, the population is 23.
1000k  0.988913
k  0.00098891
100
P
1  9e 0.1t

100
P
1  9e 0.1t
We can graph
this equation
and use “trace”
to find the
solutions.
100
80
60
Bears
40
20
0
20
40
Years
60
80
100
y=50 at 22 years
y=75 at 33 years
y=100 at 75 years
p