Chapter 9: Population Growth Math 107

Sequence –

Terms –

Sequence Notation:

Ex 0: What comes next?

2, 4, 6, 8, …

3, 5, 7, …

It helps if the sequence has an explicit formula

Ex 1: Find the first 4 terms of the sequence 3 1NN

A = −

LC: Find the 8th term of the sequence.

Recursive Formula:

Fibonacci Sequence:

Population Sequence

Fibonacci’s Rabbits: A man put one pair of rabbits in a certain place entirely

surrounded by a wall. How many pairs of rabbits can be produced from that

pair in a year, if the nature of these rabbits is such that every month each pair

bears a new pair which from the second month on becomes productive?

Ex 2 (LC): How many pairs of rabbits will there be in the 12th month?

Linear Growth Model: Arithmetic Sequence

Linear Functions

F(x)=

Arithmetic Sequence

PN=

m = d

( ) = ( )

Recursive Formula for an Arithmetic Sequence:

Ex 3:

a) What is the common difference?

b) What is P0?

c) Write the Arithmetic Sequence. (Just write the right side, not PN.)

d) Predict the Unemployment Rate on January, 2013.

e) Predict when the United States would reach a zero unemployment rate.

Arithmetic Sum Formula

Let’s look at adding the first 200 terms of the sequence.

P0=

d=

Ex 4:

Ex 4 (LC):

Exponential Growth

Now we’re going to look at sequences that grow by a common __________,

not a common difference.

If a population has an initial value X (baseline) and a new value Y (end-value),

then we say the _______________ ______________ is the ratio

Y Xr

X

−=

By doing a little Algebra, we can calculate the value of Y if we’re given the

growth rate r.

A Population grows _______________________ if it grows by a constant

factor R

The explicit formula for the Nth term is

0

N

NP R P=

The recursive formula is 1N N

P R P −= ⋅

A numerical sequence that grows exponentially is called a geometric

sequence. If the growth rate is r, the sequence is written as

0 0

1 0 0 0

2

2 1 0

3

3 2 0

0

1

1 1

1 1

...

1N

N

P P

P P r P r P

P r P r P

P r P r P

P r P

=

= + ⋅ = +

= + = +

= + = +

= +

Another way to say this, is that the

common ratio = one + the growth rate

Ex 5:

Ex 6:

Ex 7: If you earned 1 cent today, 2 cents tomorrow, 4 cents the next day,

how much would you have after 31 days?

The Logistic Growth Model

Population Density –

A population’s growth rate is negatively impacted by the population’s density .

Habitat –

Growth Parameter –

The actual growth rate of a specific population doesn’t just depend on the growth

parameter, but also the amount of elbow room available as well.

Carrying Capacity –

If we call PN the current population, then “elbow room” ( )N

C P= −

The p-value is the percentage of the carrying capacity that is occupied. NN

Pp

C=

Now if we represent C as 100%, we can write “elbow room” as (1 )NP−

This finally leads us to the Logistic Equation (defined recursively).

( )11

N NNP r P P+ = −

Ex 8) In the following examples we are seeding a natural fish pond with rainbow trout,

which has a carrying capacity of 10,000C = fish, and growth parameter of 2.5r = .

a) Seed the pond with 2000 trout.

1

2

3

4

P

P

P

P

=

=

=

=

This is an example of a stable equilibrium .

b) Seed the pond with 3000 trout.

1

2

3

4

5

6

P

P

P

P

P

P

=

=

=

=

=

=

This is an example of an attracting point .

c) Seed the pond with the complementary seed of example b), 7000 trout, or 070%P = .

1

2

3

P

P

P

=

=

=

Ex 9) You decide to switch to goldfish which has a growth parameter 3.1r = .

Seed 20% of the tank’s carrying capacity.

This demonstrates the population settling into a two-cycle pattern.

Ex 10) A four-cycle pattern

Let’s look at a flour beetle (to feed those fish!) with a growth parameter of 3.5r = and

seed it with 00.44.P =

Ex 11) A Random Pattern

Ex 12 (LC!):