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We will discuss here the logistic growth model of an exponential function. Also, some examples related to the logistic growth model.

1.
Logistic Growth Model

Bears

Years

Greg Kelly, Hanford High School, Richland, Washington

Bears

Years

Greg Kelly, Hanford High School, Richland, Washington

2.
kt

We have used the exponential growth equation y y0 e

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

kP

dt

The constant k is called the relative growth rate.

dP / dt

k

P

We have used the exponential growth equation y y0 e

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

kP

dt

The constant k is called the relative growth rate.

dP / dt

k

P

3.
kt

The population growth model becomes: P P0 e

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

The population growth model becomes: P P0 e

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

4.
The equation then becomes:

dP M P

kP

dt M

Our book writes it this way:

Logistics Differential Equation

dP k

P M P

dt M

We can solve this differential equation to find the logistics

growth model.

dP M P

kP

dt M

Our book writes it this way:

Logistics Differential Equation

dP k

P M P

dt M

We can solve this differential equation to find the logistics

growth model.

5.
Logistics Differential Equation

dP k

P M P

dt M 1 A B

1 k P M P P M P

dP dt

P M P M 1 A M P BP Partial

Fractions

1 1 1 k 1 AM AP BP

dP dt

M P M P M

1 AM 0 AP BP

1 AP BP

ln P ln M P kt C A A B

M

P 1

ln kt C B

M P M

dP k

P M P

dt M 1 A B

1 k P M P P M P

dP dt

P M P M 1 A M P BP Partial

Fractions

1 1 1 k 1 AM AP BP

dP dt

M P M P M

1 AM 0 AP BP

1 AP BP

ln P ln M P kt C A A B

M

P 1

ln kt C B

M P M

6.
Logistics Differential Equation

dP k

P M P P

dt M e kt C

M P

1 k

dP dt

P M P M M P

e kt C

P

1 1 1 k

dP dt M

M P M P M 1 e kt C

P

ln P ln M P kt C M

1 e kt C

P

P

ln kt C

M P

dP k

P M P P

dt M e kt C

M P

1 k

dP dt

P M P M M P

e kt C

P

1 1 1 k

dP dt M

M P M P M 1 e kt C

P

ln P ln M P kt C M

1 e kt C

P

P

ln kt C

M P

7.
Logistics Differential Equation

P kt C M

e P

M P 1 e kt C

M P M

e kt C P

P 1 e C e kt

M C

1 e kt C Let A e

P

M M

1 e kt C P

P 1 Ae kt

P kt C M

e P

M P 1 e kt C

M P M

e kt C P

P 1 e C e kt

M C

1 e kt C Let A e

P

M M

1 e kt C P

P 1 Ae kt

8.
Logistics Growth Model

M

P

1 Ae kt

M

P

1 Ae kt

9.
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?

10.
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 M 100 P0 10 P10 23

1 Ae kt

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 M 100 P0 10 P10 23

1 Ae kt

11.
M

P M 100 P0 10 P10 23

1 Ae kt

100

10 At time zero, the population is 10.

1 Ae0

100

10

1 A

10 10 A 100

10 A 90

100

A 9 P

1 9e kt

P M 100 P0 10 P10 23

1 Ae kt

100

10 At time zero, the population is 10.

1 Ae0

100

10

1 A

10 10 A 100

10 A 90

100

A 9 P

1 9e kt

12.
M

P M 100 P0 10 P10 23

1 Ae kt

100

P

1 9e kt

100

23 After 10 years, the population is 23.

1 9e k 10

10 k 100 10k 0.988913

1 9e

23

k 0.098891

10 k 77

9e

23

10 k 100

e 0.371981 P

1 9e 0.1t

P M 100 P0 10 P10 23

1 Ae kt

100

P

1 9e kt

100

23 After 10 years, the population is 23.

1 9e k 10

10 k 100 10k 0.988913

1 9e

23

k 0.098891

10 k 77

9e

23

10 k 100

e 0.371981 P

1 9e 0.1t

13.
100

1 9e 0.1t

We can graph Bears

this equation

and use “trace”

to find the

Years

y=50 at 22 years

y=75 at 33 years

y=100 at 75 years

1 9e 0.1t

We can graph Bears

this equation

and use “trace”

to find the

Years

y=50 at 22 years

y=75 at 33 years

y=100 at 75 years