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 P 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 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

p