Unit 7 Test Tuesday Feb 11th
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Transcript of Unit 7 Test Tuesday Feb 11th
Unit 7 TestTuesday Feb 11th
AP #2
Friday Feb 7th
Computer Lab (room 253)
Monday Feb 10th
HW: p. 357 #23-26, 31, 38, 41, 42
Logistic growth is slowed by population-limiting factors
M = Carrying capacity is the maximum population size that an environment can support
Exponential growth is unlimited growth.
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 fraction of the carrying capacity that remains:M P
M
M
P1
The equation then becomes:
dP M PkP
dt M
Our book writes it this way:
Logistic Differential Equation
dP kP M P
dt M
We can solve this differential equation to find the logistic growth model.
PartialFractions
Logistic Differential Equation
dP kP M P
dt M
1 k
dP dtP M P M
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 kdP dt
M P M P M
ln lnP M P kt C
lnP
kt CM P
Logistic Differential Equation
kt CPe
M P
kt CM Pe
P
1 kt CMe
P
1 kt CMe
P
1 kt C
MP
e
1 C kt
MP
e e
CLet A e
1 kt
MP
Ae
Logistic Growth Model
1 kt
MP
Ae
Logistic Growth Model
Years
Bears
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 kt
MP
Ae 100M 0 10P 10 23P
1 kt
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 ktP
e
1 kt
MP
Ae 100M 0 10P 10 23P
After 10 years, the population is 23.
100
1 9 ktP
e
10
10023
1 9 ke
10 1001 9
23ke
10 779
23ke
10 0.371981ke
10 0.988913k
0.098891k
0.1
100
1 9 tP
e
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
Logistic Growth diff eq solution
PMM
kP
dt
dP
ktAe
MP
1
If you are told Logistic Growth you can go directly from diff eq to
ktAe
MP
1
Carrying Capacity
“Room to grow” constant
Population
rate
time