Exponential Growth

Definition:

Growing without bound.

Ie. Nothing limits the growth.

What does it look like?

5

4

3

2

1

-1

-2

-4 -2 2 4

Population

Independent Axis

Dependent Axis

Population is a function of time:

P t( ) =P0ekt

Where:

P t( ) is the population after t years;P0 is the initial population;k is the growth rate constant;t is the time in years.

Population:

Write down the first thing you think of when you hear the word “population”.

Not a definition, just the first thing that pops into your head.

Population:

A population can be:

People

Rust

Bacteria

Rocks

Anything that can be grouped and categorized.

Example

A population of people increases from 2500 to 2700 in 25 years. What is the growth rate constant for this population?

Example

A population of bacteria doubles every 43 minutes. What is the growth rate constant?

Logistic Growthand

Exponential Decay

Logistic Growth

Logistic growth is where a population is growing toward a maximum number that is allowed for that population.

In other words, a population that is growing logistically has limitations.

What does it look like?

60

50

40

30

20

10

-10

-40 -20 20 40 60

Populations

Populations can’t grow exponentially forever.

What factors inhibit growth?

Factors that inhibit growth

Size of the area (overpopulation)

Disease

Predators

Natural Disasters

Man-made disasters

War

Limit of Growth

When a population reaches its maximum number, we say that the population has met its:

Carrying capacity

Logistic Modeling

The growth model for a population that grows logistically is:

P t( ) =N

1−ce−kt

Example

The number of fruit flies in a population after t days can be found by the equation:

P t( ) =230

1+ 56.5e−0.37t

Example (cont.)

What is the carrying capacity?

500

450

400

350

300

250

200

150

100

50

-50

5 10 15 20 25

f x( ) = 230

1+56.5 ⋅e-0.37 ⋅x

Example (cont.)

What is the initial population?

P t( ) =230

1+ 56.5e−0.37t

Example (cont.)

What will the population be after 15 days? After 30 days?

P t( ) =230

1+ 56.5e−0.37t

Decay

Exponential decay is normally discussed in terms of radioactive decay.We model decay with the equation:

P t( ) =P0ekt

Decay

Decay is normally discussed in reference to a half-life.

A half-life is the time it takes for half of the material to decay.

Example

A certain drug that a patient must receive has a half life of 13 days. If it is no longer potent once it has reached 40% of its original mass, how long after production does the doctor have to administer the drug?