Chapter 9: Population Growth
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 AN = 3N −1
LC: Find the 8th term of the sequence.
Recursive Formula:
Fibonacci Sequence:
Math 107
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
P N=
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
r =Y − X
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
PN = R N P0
PN = R ⋅ PN −1
The recursive formula is
A numerical sequence that grows exponentially is called a geometric
sequence. If the growth rate is r, the sequence is written as
P0 = P0
P1 = P0 + r ⋅ P0 = 1+ r  P0


P2 = 1+ r  P1 = 1+ r 




2
P3 = 1+ r  P2


3
P0
= 1+ r 


P0
...
PN = 1+ r 


N
P0
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”
= (C − PN )
The p-value is the percentage of the carrying capacity that is occupied. pN =
Now if we represent C as 100%, we can write “elbow room” as
(1− PN )
This finally leads us to the Logistic Equation (defined recursively).
(
)
PN +1 = r 1− PN PN
PN
C
Ex 8) In the following examples we are seeding a natural fish pond with rainbow trout,
which has a carrying capacity of C = 10,000 fish, and growth parameter of r = 2.5 .
a) Seed the pond with 2000 trout.
P1 =
P2 =
P3 =
P4 =
This is an example of a stable
equilibrium .
b) Seed the pond with 3000 trout.
P1 =
P2 =
P3 =
P4 =
P5 =
P6 =
This is an example of an attracting
point .
c) Seed the pond with the complementary seed of example b), 7000 trout, or P0 = 70% .
P1 =
P2 =
P3 =
Ex 9) You decide to switch to goldfish which has a growth parameter
r = 3.1 .
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
seed it with P0 = 0.44.
Ex 11) A Random Pattern
r = 3.5 and
Ex 12 (LC!):