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Photo by Vickie Kelly, 2004. Greg Kelly, Hanford High School, Richland, Washington. 6.5 day 2. Logistic Growth. Columbian Ground Squirrel Glacier National Park, Montana. We have used the exponential growth equation to represent population growth. - PowerPoint PPT Presentation
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6.5 day 2
Logistic Growth
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2004
Columbian Ground SquirrelGlacier National Park, Montana
We have used the exponential growth equationto represent population growth.
0kty y e
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
kPdt
The constant k is called the relative growth rate.
/dP dtk
P
The population growth model becomes: 0ktP Pe
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
dPkP M P
dt
We can solve this differential equation to find the logistics growth model.
PartialFractions
Logistics Differential Equation
dPkP M P
dt
1
dP kdtP M P
1 A B
P M P P M P
1 A M P BP
1 AM AP BP
1 AM
1A
M
0 AP BP AP BPA B1
BM
1 1 1 dP kdt
M P M P
ln lnP M P Mkt C
lnP
Mkt CM P
M
Logistics Differential Equation
Mkt CPe
M P
Mkt CM Pe
P
1 Mkt CMe
P
1 Mkt CMe
P
dPkP M P
dt
1
dP kdtP M P
1 1 1 dP kdt
M P M P
ln lnP M P Mkt C
lnP
Mkt CM P
M
Logistics Differential Equation
1 Mkt C
MP
e
1 C Mkt
MP
e e
CLet A e
1 Mk t
MP
Ae
Mkt CPe
M P
Mkt CM Pe
P
1 Mkt CMe
P
1 Mkt CMe
P
Logistics Growth Model
1 Mk t
MP
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?
1 Mk t
MP
Ae
100M 0 10P 10 23P
1 Mk t
MP
Ae
100M 0 10P 10 23P
0
10010
1 Ae
10010
1 A
10 10 100A
10 90A
9A
At time zero, the population is 10.
100
1 9 Mk tP
e
100M 0 10P 10 23P
After 10 years, the population is 23.
100
1 9 Mk tP
e
100 10
10023
1 9 ke
1000 1001 9
23ke
1000 779
23ke
1000 0.371981ke
1000 0.988913k
0.00098891k
0.1
100
1 9 tP
e
1 Mk t
MP
Ae
0.1
100
1 9 tP
e
Years
BearsWe can graph this equation and use “trace” to find the solutions.
y=50 at 22 years
y=75 at 33 years
y=100 at 75 years
