Section 10.1: Dynamics of Population Growth
POPULATION GROWTH: Creating mathematical models that demonstrate both positive (increase) and
negative (decay) growth in a population.
POPULATIONS:
Population growth is a ____________________________________ because it represents change over time.
1. CONTINUOUS GROWTH: the dynamics of change are in effect_______________________
2. DISCRETE GROWTH: ____________________________ (stop – and – go) where there is a
____________ change in the population after a certain amount of time with no change to
population

TRANSITION: _____________________________of time between changes to population.
Example:
Basic Questions of population growth:
 What will happen to a given population over time?
 What are the long-term patterns of growth (if any)?
TRANSITION RULES: rules that govern the transition
 WHEN does change happen
 HOW is the population changed
POPULATION SEQUENCE: a list of numbers that represents the changes to population over multiple
transitions or generations.
 Generic Population Sequence = P0, P1, P2, P3, …, PN, …

P=

N (subscript) =

Example: The number of cars produced is expected to increase by 25 every 2 months. If the
company started with 100 cars find the population sequence.
Transition =
Transition Rule =
Generation/
Transitions (N)
POPULATION
(PN)
N=0
P0 = Initial
Population
(SEED)
N=1
N=2
N=3
P1 = population of
1st generation
P2 = population of
2nd generation
P3 = population of
3rd generation
The transition rule is not sustainable over the long hall. Our goal is to best model the growth of
population, but no model is exact to nature or the real-life.
Section 10.2: Linear Growth – Arithmetic Sequences Part 1
Linear Growth Model: population changes (increases or decreases) by a constant amount each transition.
Example Problem: A city is considering a new law to restrict the monthly amount of garbage allowed
into the local landfill to a maximum of 120 tons a month. Although there is currently only 8000 tons of
garbage in the landfill, there is a concern that the landfill will reach its maximum capacity of 20,000
tons. Determine the transition and transition rule of this growth problem.
Transitions:
Transition Rule:
Write the population sequence:
Plot and connect the points of the sequence.
What type of graph is created?

How much garbage will there be in the landfill five years from now?
(Hint: How many transitions represent 5 years)

How long before the landfill reaches its 20,000-ton capacity?
Linear growth model produces an _____________________________ SEQUENCE.
 Definition: the DIFFERENCE between consecutive terms is constant (add or subtract constant)
The constant amount of change is called the ______________________________________, d.
The initial population is called the SEED of the arithmetic sequence, P0.
Which of the following is an arithmetic sequence?
a) 3, 7, 11, 15, 19, …
b) -7, 3, 8, 18, 23, 33, 38, …
c) -4, -11, -18, -25, - 32, …
RECURSIVE FORMULA for an Arithmetic Sequence:
The next population is the sum of the COMMON DIFFERENCE and the previous population.
** EXPLICIT FORMULA for an Arithmetic Sequence:
The term of the population sequence is the sum of the Initial population and the product of the
COMMON DIFFERENCE and the number of Transitions, N.
Find the EXPLICIT and RECURSIVE formulas.
a) 7, 13, 19, 25, …
b)-3, -1, 1, 3, 5, …
c) 2, -3, -8 , -13, -18, …
Pattern:
Pattern:
Pattern:
Recursive:
Recursive:
Recursive:
Explicit:
Explicit:
Explicit:
Find the nth of each arithmetic sequence described
a) P0 = 8, d = 3
Find P16 = ??
b) 5.3, 5.9, 6.5, 7.1, …
Find P39= ??
1
3
, P3  3
2
4
Find P36= ??
c) P2  1
A Frapucchino in Every Corner?: Cosmic Java opened in 2005 with 370 coffee shops and
plans to open 60 new coffee shops each year after that. Here are some questions of
relevance to the company’s long-term strategic plans:
(1) How many Cosmic Java coffee shops will there be in 2020?
(2) In what year will Cosmic Java reach the milestone of having 2000 coffee shops?
Nuclear Power: A nuclear power plant produces 12 lb of radioactive waste every month.
The radioactive waste is stored in a special storage tank. On January 1, 2000, there were
25 lb of radioactive waste in the tank. Let PN represent the amount of radioactive waste (in
pounds) in the storage tank after N months.
(1) Give an explicit description for PN.
(2) If the tank can only hold 217 lbs of nuclear waste safely, how soon should the power plant
purchase a new tank?
(3) Suppose the nuclear power plant intends to send all of its nuclear waste to YUCCA MOUNTAIN
in 2005. How many tanks will the power plant send to Nevada?
Write an equation for the nth term of each sequence both explicitly and recursively,
a) -3, -6, -9, -12, …
b) 2, 8, 14, 20, …
Linear models when the common difference or seed aren’t given directly.
Example #1: In a linear growth model, we only know that P9 = 1324 and P25 = 2684.
Write the model explicitly. How can we find the seed and common difference?
Find an explicit formula for the linear growth and find the 8th generation, P8.
a) P12 = 322 and P27 = 397
b) P8 = 408 and P32 = 24
c) P5 = 9107 and P18 = 11928
ADDITIONAL PRACTICE PROBLEMS
1) Consider a population that decays according to a linear model. The initial population is P0 = 238 and
the common difference is d = -17.
a. Find P24.
b. How many generations will it take for the population to reach 85?
c. How many generations will it take for the population to go extinct?
2) Consider a population that grows according to the recursive rule PN = PN-1 + 19, and P3 = 81.
a. Write the population growth explicitly.
b. Find P150.
HOMEWORK: pp. 364 – 365 #1 - 6
Section 10.2: Linear Growth – Arithmetic Sums Part 2
ARITHMETIC SUM FORMULA:
A small tractor manufacturer has a 12-week production schedule for its radically new 2008 model. The
plan is to manufacture 30 tractors each week for the next 12 weeks. After they are manufactured, the
tractors are stored in several warehouses until the release date. If the storage costs are $10 per tractor
per week, what is the total storage cost to the company over the entire 12-week period?
Write the population sequence for growth problem:
We want to know P0 + P1 + P2 + P3 + P4 + P5 + P6 + P7 + P8 + P9 + P10 + P11 + P12 = ???
Shortcut Method: Write the sum twice, once forward and once backward, line them up and add.
GENERAL FORMULA for the Sum of an Arithmetic Sequence:
The sum of N consecutive terms of an arithmetic sequence is one half the product of N (number of terms)
times the sum first and last term.
 A0  AN 1 N
A0 + A1 + A2 + A3 +…+ AN-2 + AN-1 =
2
Find the given sum for the arithmetic population sequence:
a) P0 = 25, P100= 1725; P0 + P1 + … + P99 + P100
c) P0 = 23 and d = 7; P100 + P101 + … + P999
b) P0 = 700 and P20 = 560; P0 + P1 + … + P20
Missing the Last Term? Consider the arithmetic sequence 5, 12, 19, …
What is the sum of the first 1000 terms? S = 5 + 12 + 19 + …
1000 terms
Find the given sum for the arithmetic population sequence:
a) For 200 terms: 5 + 16 + 27 + 38 + 49 + …
b) 2 + 5 + 5 + … + 5 and 2 + 7 + 12 + 17 + …
200 terms
100 terms
100 terms
Don’t know how many terms are being added? Consider 4 + 13 + 22 + 31 + … + 922
Find the sum for the arithmetic sequences:
a) 17 + 23 + 29 + … + 1511
b) 8012 + 7999 + 7986 + … + 6075
c) 6 + 27 + 48 + … + 363
Word Problem #1: An auditorium has 20 total rows of seats. In the first row there are 20 seats, second
row has 21, third row has 22 seats, and so on. What is the seating capacity of the auditorium? (how
many total seats)
Word Problem #2: A small business sells $10,000 worth of products during its first year of business.
The owner has set an annual goal for increased sales at $7500 for 19 years. Assuming the goal is met,
find the total sales during the first 20 years this business is in operation.
HOMEWORK: p. 365 #8 – 18 (even)
Section 10.3: Percent of Change - Part 1
PERCENT OF CHANGE: Increase or decrease expressed as a percent from original value

Percent of Change =
Calculate the Percent of Change:
1) Original: 25
New: 28
3) Original: 25
New: 18
2) Original: 30
New: 12
4) Original: 13.7
New: 40.2
Given Percent of Change Calculate:
 PERCENT INCREASE by x%: 100% + X%  Make a decimal  Multiply by C
 PERCENT DECREASE by x%: 100% - X%  Make a decimal  Multiply by C
5) A concert tickets costs $45. The sales tax is 6.25%, what is the total price of the ticket?
6) A sweater is on sale for 35% off the original price. If the original price of the sweater is $38, what is
the discounted price?
7) The population of 3000 students decreased by 20% with the creation of a new high school. How
many students will now populate the school?
8) A company predicts a 15% increase in production from the 405,500 units with a new marketing
campaign. How many units will the company now produce?
OVERALL PERCENT OF CHANGE: Multiple Stages
Example #1: - An item is manufactured at a cost of C dollars. The item is initially marked up 10% and
sold to a distributor. The distributor then marks the item up 20% (based on the price paid by the distributor
to make a profit) and sells it to a retailer. The retailer finally marks up the price 50% and sells the item to
the public.
a. Original Cost of the Item: ________
b. Cost to distributor: D = _______________
c. Cost item to Retailer: R = ________________ or in terms of C:________________________
d. Cost to the Public: P = ________________ or in terms of C:________________________
Example #2: A retailer buys an item for C dollars and marks it up 80%. To make room for new
merchandise, the item is put on sale for 40% off the marked place. After a while, an additional 10%
markdown is taken off the sale price.
a. Original Cost of the Item: ________
b. Cost after 80% Mark Up: P = _______________
c. Cost after 40% markdown: S = ________________ or in terms of C:________________
d. Cost after additional 10% markdown: F = ______________ or in terms of C:_______________
Practice Problems: HOMEWORK if not completed in class
1) The original price of a new computer is marked up 15% to the distributor, then marked up 30% to
the retailer and finally marked up 20% to the customer. What is the overall mark up of the original
price that a customer is paying?
2) A car company initially marks up the original price by 25%, then marked down the price by 30%,
and then marked up 10%. How much is the car valued at the final selling price based on its original
price?
3) A new car was marked up by 20% of its value. The car company is currently selling it for $50,000.
What was the approximate original price of the car?
4) The percent decrease is 22%. If the new number is 93.6, find the original number.
5) Ralph Lauren is having a sale on all polo shirts for 17% off the original price. If polo shirts are
now being sold for 29.05 dollars, then what is the approximate original price the polos were sold for?
6) A t-shirt original cost $10 and then was marked up the same amount twice at the store for a new
selling price of $12.10. What percent was the t-shirt marked up?
7) For two consecutive quarters the production of cell phones increases by the same percent. If the
original number of cell phones produced was 30,000 and after the 2 quarters is 41,772, then what was
the percent increase?
Section 10.3: Geometric Sequences/ Sums - Part 2
EXPONENTIAL GROWTH: is based on the idea of __________________________________________
– in each transition the population changes by a fixed factor called the common ratio.
Example Problem: The sum of $1000 is deposited in a retirement account that pays 10% annual interest.
How much money is there in the account after 25 years? How much money is in the account after N years?
Transitions:
Transition Rule:
Population Sequence:
KEY PROPERTIES OF EXPONENTIAL GROWTH:
 A CONSTANT GROWTH RATE

Each transition consists of multiplying the size of the population by a CONSTANT FACTOR
determined by the growth rate, but not equal to the growth rate.
CONNECTING TO SEQUENCES: A sequence defined by repeated multiplication by a fixed amount r is
called a _________________________ SEQUENCE.
The constant factor r is called the ____________________________________ of the geometric sequence.
Which of the following is a geometric sequence?
a) 12, 36, 108, 324, …
b) 1, 4, 9, 16, …
c) 6, -2, 2/3, -2/9, 2/ 27, …
Values of the Common Ratio for Population Growth: (r > 0)
 Increase:
 Decrease/ Decay:
RECURSIVE FORMULA for an Exponential Growth (r > 0):
Next term of the population sequence is product of the Common Ratio (r) and the previous term (PN-1)
EXPLICIT FORMULA for an Exponential Growth (r > 0):
A term of the population sequence is the product of the initial population (P0) and the Common Ratio (t)
to the power of the number of transitions, N.
Assume each of the following population sequences begins with P0. Find the explicit and recursive
formula for each population sequence.
a) 3, 6, 12, 24, 48, …
b) 20, 24, 28.8, 34.56, …
c) 400, 300, 225, 168.75, …
Recursive:
Recursive:
Recursive:
Explicit:
Explicit:
Explicit:
Example: The number reported cases of Gamma virus has been dropping by 70% a
year since 2004, when there were 1 million reported cases of the virus. If the present
rate continues, how many reported cases of the virus can we expect in 2010?
Find the nth of each exponential growth sequence described
a) Initial population is 20,000 with a 20% increase each transition. What is the 5th
generation?
c) 1000, 900, 810, 729, … P11= ??
b) P0 = 8, r = 1.7, P16 = ??
PRACTICE PROBLEMS:
1) A population grows according to an exponential growth model. The initial
population is P0 = 11 and the common ratio is r = 1.25.
a. Find P1.
b. Find P9.
c. Give an explicit formula for Find PN.
2) A population grows according to exponential growth with P1 = 9 and P2 = 12.
a. Find the common ratio.
b. Find P0.
c. Find P11.
3) Crime in Pleasantville is on the decline. Each year the number of crimes committed
increases by 20%. Assume that there were 400 crimes committed in 2000, and let PN
denote the number of crimes committed in the years after 2000.
a. Give recursive description of
PN.
b. Give explicit description of
PN.
c. If the trend continues,
approximately how many crimes
will be committed in 2006?
GENERAL FORMULA for a GEOMETRIC SUM:
The sum of N consecutive terms of a geometric
sequence a, a(r), a(r)2, a(r)3, a(r)4, …, a(r)N-1 is the
first term of the sequence times the fraction common
ratio to the Nth Power minus one over the common
ratio minus 1.
PROOF of SUM:
S = a + a(r) + a(r)2 + …+a(r)n-1
rS = a(r) + a(r)2 + …+a(r)n-1 + a(r)n
rS - S –= a(r)n - a
S (r - 1) = a (rn - 1)
S= a
( r N  1)
( r  1)
a + a(r) + a(r)2 + a(r)3 + a(r)4 + … + a(r)N-1 =
Find the following sums of geometric sequences
1) 4 + 4(.3) + 4(.3)2 + …+4(.3)11 2) Find the sum of the first 10 terms of a geometric
sequence with first term of 1 and common ratio of –2.
3) 10 + 10(2.7) + … + 10(2.7)8
4) Common Ratio = 0.75 with an initial population of
4500. What is the sum of the first 23 terms of this
geometric sequence?
Example #3: Consider an imaginary infectious disease called the X-virus, for which
there is no known vaccine. The first appearance of the X-virus occurred in 2005 (year
0) when a total of 5000 cases were reported. The virus will spread at a 40% annual rate
of growth. It is expected that it will take at least 10 years to create an effective vaccine.
How many estimated cases will there be of the X-virus over the 10-year period of 2005
– 2014?
HOMEWORK: P.366 # 20 - 26
Section 10.3: Compound Interest - Part 3
FINANCIAL APPLICATIONS OF EXPONENTIAL GROWTH
CASE #1: ANNUAL INTEREST
Interest is added to the account once a year and only at the end of the year. Assume the money
remains untouched in the account. The seed is given by the original investment or principal and the
common ratio r is derived from the interest rate, r = 1 + interest rate.
Annual Interest Formula:
P0 = principal or initial investment/ population
N = number of years i = annual interest rate (decimal)
Example #1: Suppose you invest $367.51 in account that pays an annual interest rate of 4.5% a year
and you let the money grow. How much money will there be in the account at the end of 7 years?
What if you leave the money in the account for 7.5 years?
Example #2: Suppose you invest $600 in account that pays an annual interest rate of 6.5% a year and
you let the money grow. After 10 years how much money have you accumulated in your savings
account? How much additional money would you gain by leaving it in the account for one more year?
Example #3: Suppose you invest $3420 in an account paying 7.5% annual interest. Suppose in 5 years
the annual interest rate will drop to 5.5 and you keep the money in the account for an additional 3
years. How much money do you have in the account at the end of the 8-year period? How much
money did you lose out on in the last 3 years of your account?
Example #4: Suppose you invest $4000 in an account paying 6% annual interest. Suppose in 6 years
the annual interest rate will drop to take out $1500 and leave the leftover money untouched for
another 4 years.
a) How much money do you have in the account at the end of the 10-year period?
b) How money would you have had if you didn’t need that $1500 after 6 years but after 10 years?
CASE #2: GENERIC COMPOUNDED INTEREST
Annual Interest Rate i is compounded k times in a year. The seed is given by the
original investment or principal and the common ratio r is derived from the periodic
interest rate, p 
i
.
k
General Compound Interest Formula:
P0 = principal or initial investment/ population
i = annual interest rate as decimal
N = number of years
k = number of times compounded in a year
QUARTERLY COMPOUNDED INTEREST:
A principal of $1000 in an account that pays 10% annual interest, and the interest is
compounded quarterly. If we don’t touch the money how much money will there be in the
account at the end of…
a) 1 Year
b) 5 years
c) 5.5 years
d) 25 years
MONTHLY COMPOUNDED INTEREST:
A principal of $2000 in an account that pays 15% annual interest compounded monthly. If we
don’t touch the money how much money will there be in the account at the end of …
a) 1 Year
b) 4 years
c) 4.5 years
d) 20 years
DAILY COMPOUNDED INTEREST:
A principal of $1500 in an account that pays 8% annual interest compounded daily. If we
don’t touch the money how much money will there be in the account at the end of …
a) 1 Year
b) 2 years
c) 8 years
d) 16 years
ANNUAL YIELD: TOTAL AVERAGE PERCENT INCREASE of an investment over a
one-year period.
 Annual Yield and Annual Interest Rate are not always the same percentage.
Example: Determine the annual yield if you began the year with $2400 and ended 1 year
later with $2784.
For example, you invest $1000 at an annual interest rate of 5%.
Compounded Interest:
Annual Yield:
Annually: 1000(1.05) = 1050
1050/ 1000 = 1.05  = 5%
Quarterly: 1000(1+.05/4)4 = 1050.94
1050.94/ 1000 = 1.05094  5.094%
Monthly: 1000(1+.05/12)12 = 1051.16
1051.16/ 1000 = 1.05116  5.116%
Daily: 1000(1+.05/365)365 = 1051.27
1051.27/1000 = 1.05127  5.127%
Not dependent on investment: If you know the annual interest and # of times compounded.
Example: Determine the annual interest on a savings account with a 16% annual interest
rate compounded monthly.
HOMEWORK: pp. 366 – 367 #27, 28, 30 – 33, 35 - 36
Section 10.4: Logistic Growth Model
Why are linear and exponential growth models inadequate at dealing with animals?
LINEAR GROWTH MODEL
EXPONENTIAL GROWTH MODEL
Population Biology: How do animal populations grow? What factors do we need to consider?
Rate of growth of an animal population is ___________________________________
Rate of growth depends on the _______________________________________

LOW population DENSITY

HIGH Population DENSITY
FACTORS for GROWTH:
HABITAT:
LOGISTIC GROWTH MODEL: Rate of Growth of the population is _________ ______________
proportional to the amount of “elbow room” (space available) in the population’s habitat
o Lots of elbow room =
o Little elbow room =
o No elbow room =
CARRYING CAPACITY, C: describes the total saturation point of a habitat
_______________________population that the habitat can sustain and support
“ELBOW ROOM”: Amount of POSSIBLE space that the population can increase

C – PN = Absolute amount of remaining space for population

C  PN
= RATIO represents amount of space available in proportion to the current population.
C
GROWTH PARAMETER, r: standard constant growth rate of the population
 From Exponential Growth Model: PN+1 = r (PN)
Combining Growth Parameter and Carrying Capacity: The main idea of Logistic
Growth Model is that the growth rate decreases in direct proportion to the amount of elbow room.
Example #1:
a) If the elbow room is 40% then the new growth rate = __________
b) If the elbow room is 100%, then the new growth rate = __________
c) If the elbow room is 60%, then the new growth rate = ___________
Example #2:
a) Carrying Capacity is 2,000 and current population is 1500.
b) Carrying Capacity is 340 and the current population is 204.
c) Carrying Capacity is 11,350 and the current population is 10,215.
For Logistic Growth Model, the growth rate for Nth population = _____________
Constant growth rate(r) times percentage of carry capacity remaining
TRANSITION RULE for logistic growth model as
EXAMPLE: Growth parameter of 2 and in an environment with carrying capacity of
100 and current population is 75.
a. How much growth will actually occur in the first transition? What will be the first
generation’s population?
b. How much growth will actually occur in the second transition? What will be the
second generation’s population?
p-values of population sequence (pN) represent the fraction or percentage of the carrying
capacity taken up by the Nth population.
 p-values range between 0 and 1
PN
o 0 = zero population
 pN 
C
o 1 = maximum population (carrying capacity)


***LOGISTIC EQUATION: pN  r 1  pN  1 pN  1 with p-values used
 r = growth parameter; 0 ≤ r ≤ 4
 p0 = seed = initial percentage of carrying capacity
 pn = p-value or percentage of carrying capacity that the current population represents
SHOW HOW TO DO RECURSION WITH GRAPHING CALCULATOR
1. Press the [Mode]. On the fourth line of the display, select Seq.
2. Press the [Y=]. Enter 0 for nMin. Enter your recursive equation for u(n).
3. To enter u(n-1), press [2nd] then [7] for u and [X] (to the right of Alpha) for n.
4. Enter your initial recursie value for u(nMin).
5. The recursive equation is now set up. Press[ 2nd] then [Mode] to quit to calculation mode.
6. Check your table ([2nd] [Graph]) for future values
EXAMPLE #1: For farm fishing, you have access to a large natural pond in which you
plan to raise rainbow trout which have an annual hatching season. The carrying
capacity of the pond is C = 10,000 fish, and the growth parameter is r = 2.5. You start by
seeding the pond with an initial population of 2000 rainbow trout.
p0 =
Trout Population at the beginning is ________ of carrying capacity
From the second year on, the hatchery is stuck at 60% of the pond capacity and nothing will
change unless external forces come into play.
This is called a __________________________________________ is the population that will
remain constant or stable between transitions.
EXAMPLE #2: For the same rainbow trout, suppose the pond is initially seeded with
3000 rainbow trout. Find the population sequence of p-values for the first 6 years. Use 4
decimal places.
The value __________ is called an ______________________________________________
of the population sequence .
EXAMPLE #3:
Raising a particular variety of goldfish have a growth parameter of r = 3.1. Suppose you
start by seeding a tank at 20% of its carrying capacity. What is the logistic equation for
this population of goldfish?
pN =
Look at first 15 years of goldfish populations: Do you notice any patterns or behavior?
EXAMPLE #4:
Raising a particular variety of flour-beetle have a growth parameter of r = 3.5. The seed
will be p0 = 0.44. What is the logistic equation for this population of beetle?
pN=
Look at first 28 years of the beetle population: Do you notice any patterns or behavior?
EXAMPLE #5:
Try p0 = 0.2 and r = 4. Graph the first 20 generations.
pN =
10.4 Logistic Model PRACTICE PROBLEMS: USE CALCULATOR AS NEEDED
1) A population grows according to a logistic growth model, with growth parameter r = 0.8.
Starting with an initial population given by p0 = 0.3,
i. Write the logistic equation for this model.
ii. Find p1 and p2
iii. What percent of the habitat’s carrying capacity is taken up by the third generation?
iv. What does the logistic growth model predict in the long term?
(Stable, Cycle, Attracting Point, Chaos)
2) A population grows according to the logistic growth model, with growth parameter
r = 2.2. The initial population was 65 in a habitat with a carrying capacity of 200.
i. Write the logistic equation for this model.
ii. Find p0, p1, and p2
iii. What percent of the habitat’s carrying capacity is taken up by the third generation?
iv. What does the logistic growth model predict in the long term for this population?
(Stable, Cycle, Attracting Point, Chaos)
3) A population grows according to the logistic growth model, with growth parameter
r = 1.5. Starting with an initial population given by p0 = 0.8,
i. Write the logistic equation for this model.
ii. Find p1 through p10
iii. What does the logistic growth model predict in the long term?
(Stable, Cycle, Attracting Point, Chaos)
4) A population grows according to the logistic growth model, with growth parameter
r = 2.5. Starting with an initial population given by p0 = 0.15,
i. Write the logistic equation for this model.
ii. Find p1 through p10
iii. What does the logistic growth model predict in the long term?
(Stable, Cycle, Attracting Point, Chaos)
5) A population grows according to the logistic growth model, with growth parameter r =
3.25. The initial population is 7000 in a habitat with carrying capacity of 35000.
i. Write the logistic equation for this model.
ii. Find p0 through p10
iii. What does the logistic growth model predict in the long term?
(Stable, Cycle, Attracting Point, Chaos)
6) A certain population of deer in a preserve, whose growth is described by the logistic
equation with a growth parameter of 4.0 for this type of deer, has a starting population of
300 deer and a carrying capacity of 8000 deer. Find the values of the first 5 breeding
seasons of the fish.
For #7 – 9: Identify long term behavior as STABLE, ATTRACTING, CYCLE, or CHAOS.
 Write the LOGISTIC EQUATION.
 If the long term behavior is STABLE, ATTRACTING, CYCLE, then provide the
value(s) of the long term prediction.
7) A population grows according to the logistic growth model, with r = 2.5 and p 0 = 0.3.
What does the logistic growth model predict in the long term for this population?
8) A population grows according to the logistic growth model, with r = 3.4 and p0 = 0.5.
What does the logistic growth model predict in the long term for this population?
9) A population grows according to the logistic growth model, with r = 4 and p0 = 0.4. What
does the logistic growth model predict in the long term for this population?