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


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


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


Years

Bears

Example:


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