CHAPTER 10: Mathematics of Population Growth



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:

TYPES OF POPULATION GROWTH

1. Continuous Growth: the dynamics of change are in effect at _____________________

2. Discrete Growth: a __________________change in the population after a certain ____________________ of time with no change to population

TRANSITION: __________________________ of time between changes to population.

Discrete Growth 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 RULE: rule that governs the transition

▪ WHEN does the 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, …

PN =

P0 =

Example 1: 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) |N = 0 |N = 1 |N = 2 |N = 3 |

|Population |P0 = Initial Population |P1 = population of 1st |P2 = population of 2nd |P3 = population of 3rd |

| | |generation |generation |generation |

|(PN) | | | | |

1a) How many months have passed for the P9?

1b) What is the number of cars produced after 5 transitions?

1c) What is the number of cars produced after 1 year?

Example 2: Michael expects to double his profit every 6 months on his investments and has already earned $120 in profit.

Transition = Transition Rule =

2a) How much profit is Michael making in the 8th generation?

2b) How much profit has Michael made in the 3rd year of investing?

2c) How many years into the future does P36 represent for Michael?

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 1: A city is considering a new law to restrict the monthly amount of garbage allowed into the local landfill to a maximum of 12 tons a month. Although there is currently only 800 tons of garbage in the landfill, there is a concern that the landfill will reach its maximum capacity of 2,000 tons. Determine the transition and transition rule of this growth problem.

Transitions: 1 month

Transition Rule: PN = PN-1 + 12

1a) Write the population sequence:

1b) Plot and connect the points of the sequence.

1c) What type of graph is created?

1d) How much garbage will there be in the landfill five years from now?

1e) How long before the landfill reaches its 20,000-ton capacity?

Linear growth model produces an ARITHMETIC SEQUENCE.

▪ Definition: the difference between terms of a sequence is constant (add or subtract by constant)

The constant amount of change between consecutive terms is the COMMON DIFFERENCE, d.

The initial population is the seed of the arithmetic sequence, P0. (zero term)

SEED (Starting/Initial Value) = P0, P1, P2, P3, P4, …

Determine the common difference for each of the following arithmetic sequences?

a) 3, 7, 11, 15, 19, …

b) -4, -11, -18, -25, - 32, …

c) 123, 97, 71, 45 …

RECURSIVE FORMULA for an Arithmetic Sequence:

EXPLICIT FORMULA for an Arithmetic Sequence:

Seed, initial value of sequence, is zero term P0:

Seed, initial value of sequence, is first term a1:

Find the EXPLICIT and RECURSIVE formulas, assume each sequence begins with P0:

a) 7, 13, 19, 25, …

b) -3, -1, 1, 3, 5, …

c) 2, -3, -8 , -13, -18, …

d) -3, -6, -9, -12, …

e) 42, 61, 80, 99, …

Find the nth of each arithmetic sequence described using an explicit formula

a) P0 = 8, d = 3

P16 = ??

b) 5.3, 5.9, 6.5, 7.1, … P39= ??

c) [pic]

[pic]

Word Problem #1: Cosmic Java opened in 2005 with 370 coffee shops and plans to open 60 new coffee shops each year after that.

(1a) How many Cosmic Java coffee shops will there be in 2020?

(1b) In what year will Cosmic Java reach the milestone of having 2000 coffee shops?

Word Problem #2: 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.

(2a) Give an explicit description for PN.

(2b) If the tank can only hold 217 lbs of nuclear waste safely, how soon should the power plant purchase a new tank?

(2c) 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?

Linear models when the common difference or seed aren’t given directly.

1) Find the common difference like slope [pic]

2) Find the seed, P0, working backwards

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 (Series) Part 2

ARITHMETIC SUM FORMULA EXAMPLE PROBLEM

A small tractor manufacturer has a 12-week production schedule for its radically new 2008 model. The plan is to manufacture 3 additional tractors each week for the next 12 weeks from the original 10 they produced. These 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 (Series) 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 + A1 + A2 + A3 +…+ AN-2 + AN-1 = [pic]

Alternative Formula:

a1 + a2 + a3 +…+ an-1 + an = [pic]

Find the given sum for the arithmetic sequence:

a) P0 = 25, P100= 1725;

P0 + P1 + … + P99 + P100

b) P0 = 700 and P20 = 560;

P0 + P1 + … + P20

c) P0 = 23 and d = 7;

P100 + P101 + … + P999

Missing the Last Term? What is the sum of first 1000 terms of arithmetic sequence 5, 12, 19, …?

Find the given sum for the arithmetic sequence: Given number of terms and start of sequence

• Need to find last term of addition

a) For 200 terms: 5 + 16 + 27 + 38 + 49 + …

200 terms

b) 2 + 7 + 12 + 17 + …

100 terms

Don’t know how many terms are being added? Consider 4 + 13 + 22 + 31 + … + 922

Find the sum for the arithmetic sequences: Given first and last term of addition.

• Need to find the number of terms involved in the addition.

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. The first row has 20 seats, second row has 21, third row has 22 seats, and so on. What is the seating capacity of the auditorium? (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 of a number expressed as a percent from the original value

▪ [pic] as a percent statement (Positive = Increase and Negative = Decrease)

Calculate the Percent of Change:

1) Original: 25

New: 28

Exp: [pic]

0.12( 12% Increase

2) Original: 30

New: 12

3) New: 18

Original: 25

4) Original: 13.7

New: 40.2

5) Final: 94.86

Initial: 102

6) Start: 215

End: 464.4

Given Percent of Change Calculate:

PERCENT INCREASE by X%:

100% + X% ( Make a decimal

PERCENT DECREASE byX%:

100% - X% ( Make a decimal

Multiply: (Start Value)(Decimal) = (End Value) or Divide: (End)÷ (Decimal) = (Start)

7) The value of a computer will depreciate by 12% each year. If the computer was bought for $800, then what is the value of the computer next year?

Example: Depreciate = Subtract ( 100% - 12% = 88%

88% ( Decimal = 0.88

Multiply: 800(0.88) = $704

8) A concert tickets costs $45. The sales tax is 6.25%, what is the total price of the ticket?

9) 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?

10) The population of 3000 students decreased by 20% with the creation of a new high school. How many students will now populate the school?

11) 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?

12) A new car was marked up by 20% of it’s value. The car company is currently selling it for $50,000. What was the approximate original price of the car?

13) The percent decrease is 22%. If the new number is 93.6, find the original number.

14) 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?

OVERALL PERCENT OF CHANGE: Consecutive percent of change statements

• Find the decimal form of each percent of change

• Multiply all of those decimal forms together

• Determine the overall percent of change based on the product

Example #1: Percent Increase - 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. What is the overall percent of change from the original cost?

(Original Cost)(1.10)(1.20)(1.50) = Original Cost (1.98)

1.10 = 10% up 1.20 = 20% up 1.50 = 50% up [pic]

Example #2: Percent Increase and Decrease – 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 price. After a while, an additional 10% markdown is taken off the sale price.

Original Price (0.90)*(0.60)*(1.8) = original (0.972)

0.90 = 10% down 0.60 = 40% down 1.80 = 80% up [pic]

15) 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?

16) A car company initially marks up the original price by 25%, then marked down the price by 30%, and then marked up 10%. What is the overall percent of change of this car after all changes?

Section 10.3: Geometric Sequences - Part 2

EXPONENTIAL GROWTH: is based on the idea of a ____________________________________ in each transition the population changes by a fixed factor.

CONNECTING TO SEQUENCES: A sequence defined by repeated multiplication by a fixed (CONSTANT) amount is called a GEOMETRIC SEQUENCE. The constant factor is called the COMMON RATIO, r, of the geometric sequence.

For each of the following geometric sequences, calculate the common ratio:

a) [pic]

b) [pic]

c) [pic]

d) 16, 36.8, 84.64, 194.672, …

e) [pic]

f) [pic]

g) [pic]

h) [pic]

i) [pic]

Population Growth Description based on Values of the Common Ratio: (r > 0)

▪ Increase/Growth:

Decrease/ Decay:

Example Problem #1: A mom offers to double her daughter’s allowance every month for the rest of the year. The daughter currently only has a $5 monthly allowance.

1a) Transition:

1b) Transition Rule:

1c) Population Sequence:

1d) How much money will the daughter in the last month of the year receive for her allowance?

Example Problem #2: $1000 is deposited in a retirement account that pays 10% annual interest.

2a) Transitions:

2b) Transition Rule:

2c)Population Sequence:

2d) How much money is there in the account after 25 years, and in general after N years?

KEY PROPERTIES OF EXPONENTIAL GROWTH:

A constant growth rate: description that compares growth directly related to previous population

Example 1:

Example 2:

A constant factor: Actual value of multiplication used by each transition that is determined by the growth rate, but not equal to the growth rate.

Example 1:

Example 2:

RECURSIVE FORMULA for a Exponential Growth (r > 0): [pic]

.

EXPLICIT FORMULA for a Exponential Growth (r > 0):

Seed, initial value of sequence, is zero term P0:

Seed, initial value of sequence, is first term a1:

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, …

Word Problem: 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.

a. Transition?

b. Transition rule?

c. Initial population?

d.

e. What is the explicit formula of this population sequence?

f. 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

1) P0 = 8, r = 1.7, P16 = ??

2) 1000, 900, 810, 729, … P11= ??

3) Initial population is 20,000 with a 20% increase each transition.

What is the 5th generation?

4) Initial population is 85,000 with a 15% decrease each transition.

What is the 7th generation?

Exponential Growth 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.

|1a. Find P1. |1b. Find P9. |1c. Give an explicit formula for Find PN. |

| | | |

2) A population grows according to an exponential growth model with P1 = 9 and P2 = 12.

|2a. Find the common ratio. |2b. Find P0. |2c. Find P11. |

| | | |

3) Crime in Pleasantville is on the decline. Each year the number of crimes committed decreases 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.

|3a. Recursive description of PN. |3b. Explicit description of PN. |3c. How many crimes will be committed in 2006? |

Section 10.3: Geometric Sum (Series) – Part3

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 = [pic]

Alternative Formula:

a1 + a2 + a3 + a4 + … + an = [pic]

Find the following sums of geometric sequences

|1) 8 + 12 + 18 + 27 + 40.5 + 60.75 |2) Find the sum of the first 10 terms of a geometric sequence with first term of 1 and common ratio of 2.|

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|3) 4 + 4(.3) + 4(.3)2 + …+4(.3)11 |4) Find sum of the first 8 terms: 5 + 15 + 45 + … |

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|5) 10 + 10(2.7) + … + 10(2.7)8 |6) Common Ratio = 0.75 with an initial population of 4500. What is the sum of the first 23 terms of this |

| |geometric sequence? |

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Example #3: A imaginary infectious disease called the X-virus has no known cure. 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. 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 4

FINANCIAL APPLICATIONS OF EXPONENTIAL GROWTH

The initial population or seed, Po, is referred to as the original investment of money or principal.

Annual Interest Problems

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 common ratio r is derived from the interest rate, i, r = 1 + interest rate.

Annual Interest Formula: [pic]

P0 = principal or initial investment/ population

N = number of years (transitions since initial investment)

i = annual interest rate as 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.

a. How much money will there be in the account at the end of 7 years?

b. How much money is in the account after 7.5 years?

Example #2: Suppose you invest $600 in account that pays an annual interest rate of 6.5% a year.

a. After 10 years how much money have you accumulated in your savings account?

b. How much additional money would you gain by leaving it in the account for one more year?

Example #3: 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 for an additional 3 more years.

a. How much money do you have in the account at the end of the 8-year period?

b. How much money did you lose out on in the last 3 years of your account?

Example #4: 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 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. k represents the number of transitions in one year. The common ratio r is derived from the interest rate,[pic], which represents the multiplication during each transition from sharing the annual interest.

General Compound Interest Formula: [pic]

P0 = principal or initial investment/ population

N = number of years

i = annual interest rate as decimal

k = number of times interest compounded in year

kN = total number of transitions over the indicated number of years

QUARTERLY COMPOUNDED INTEREST: Transitions occur every _______________

A principal of $1000 in an account that pays 10% annual interest, and the interest is compounded quarterly.

a) Identify P0, i, k

b) Money after 1 year?

c) Money after 5 years?

d) Money after 5.5 yrs?

MONTHLY COMPOUNDED INTEREST: Transitions occur every _______________.

A principal of $2000 in an account that pays 15% annual interest, and the interest is compounded monthly.

a) Identify P0, i, k

b) Money after 2 Years?

c) Money after 4.75 years?

DAILY COMPOUNDED INTEREST: Transitions occur every _______________

A principal of $1500 in an account that pays 8% annual interest, and the interest is compounded daily.

a) Identify P0, i, k

b) Money after 3 years?

c) Money after 8 years

ANNUAL YIELD: PERCENT INCREASE of an investment over a one-year period.

For example, you invest $1000 at an annual interest rate of 5%.

|Compounded Interest: |Annual Yield: Use at least 3 decimal places for percent |

|Annually: |[pic]= 1.05 ( |

|[pic] = 1050 | |

|Quarterly: |[pic]= 1.05094 ( |

|[pic]= 1050.94 | |

|Monthly: |[pic]= 1.05116 ( |

|[pic]= 1051.16 | |

|Daily: |[pic]= 1.05127 ( |

|[pic]= 1051.27 | |

#1: Calculating Annual Yield when you know the start and end values for the year.

(Beginning of year)*r = (End of year) (Annual Yield = percent of change r represents

Example 1: Find the annual yield if the year began with $2400 and ended a year later with $2784.

Example 2: Determine the annual yield if the year ended with $3798 and $3600 was invested to start that year.

#2: Calculating Annual Yield when the initial investment is unknown (treat P0 as 100%)

[pic]

Example 3: Determine the annual yield on a savings account with a 16% annual interest rate compounded monthly.

Example 4: Find the annual yield on a savings account with a 9% annual interest rate compounded quarterly.

HOMEWORK: pp. 366 – 367 #27, 28, 30 – 33, 35 – 36

Section 10.4: Logistic Growth Model Part 1

Why are linear and exponential growth models inadequate at dealing with animals?

|LINEAR GROWTH MODEL |EXPONENTIAL GROWTH MODEL |

|Models situations where the _________________ of growth between transitions is |Model situations where the ___________________ of growth between transitions is |

|constant. |constant |

| | |

Both Models assume the same growth is occurring unrestrained forever with no changes.

What do we need to consider when trying to model animal populations?

Rate of growth of an animal population is __________________________________

Rate of growth depends on the population _________________

▪ Low density

▪ High density

Additional Factors that can limit a population’s growth besides its density:

HABITAT:

LOGISTIC GROWTH MODEL: Rate of Growth is ______________________________ proportional to the amount of available space in the habitat

o MORE Space

o LESS Space

o NO SPACE

CARRYING CAPACITY, C: describes the maximum population that the habitat can sustain and support

MEASUREMENTS OF SPACE IN HABITAT:

1) C – PN = Absolute amount of space

2) [pic]= ratio or proportion amount of space

Example #1: 32 desks in a classroom and there are 28 students enrolled in the class.

Example #2: There are 1230 fish in a lake that can hold 4100 fish.

GROWTH PARAMETER, r: standard constant growth rate of the population

Combining Growth Parameter and Carrying Capacity:

Amount of space affects the amount of growth parameter used at each transition.

Example #1: Calculate the growth factor for the next transition.

a) If the space available is 40% of the carry capacity and the growth the parameter of 3, then the growth factor for the next transition is

b) If the space available is 100% of the carry capacity and the growth the parameter of 2.6, then the growth factor for the next transition is

c) If the space available is 65% of the carry capacity and the growth the parameter of 1.8, then the growth factor for the next transition is

Example #2: Calculate what percent of the growth parameter, r, will be used as the growth factor

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.

GENERAL TRANSITION RULE for LOGISTIC GROWTH MODEL: [pic]

[pic] = growth factor for the transition, C = Carrying Capacity, and PN = population

EXAMPLE: Growth parameter of 2 and in an environment with carrying capacity of 100 and current population is 75.

a. Use the transition rule to find the first generation’s population?

b. Use the transition rule to find 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.

▪ [pic]

p-values range between 0 and 1

o 0 = zero population

o 1 = carrying capacity

***LOGISTIC EQUATION: [pic]

▪ 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

[pic]

SEQUENCE MODE: Press [pic], On the fourth line, Highlight “Seq”

Write Logistic Equation:

1. Press [pic]

2. Enter [pic]for nMin.

3. To write a “u” = press [pic]then [pic]for u

4. To write a “n” = press [pic]

5. u(nMin) = p0

WINDOW SETUP:

[pic]

nMin = 0

nMax = 30

XMin = 0

XMax = 30

YMin = 0

YMax = 1

Find p-values and Long Term Behaviors

Long Term Behaviors = STABLE, ATTRACTING POINT, CYCLE, CHAOS

TABLE:

[pic], [pic]: Scroll to find future p-values

GRAPH:

Press [pic], Scroll to the left of u(n) formula and select to have a solid diagonal line

Press [pic], Visual representation of possible long term behavior

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.

1a. What is p0?

1b. What is the logistic equation for this situation?

1c. Use the logistic equation to find p1?

1d. Use the logistic equation and your calculator to find p2, p3, and p4?

1d. Use the logistic equation and its graph to determine long term behavior?

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)

2a. What is p0?

2b. What is the logistic equation for this situation?

2c. Use the logistic equation and your calculator to find the p-values of the first 6 years?

p1 = ______________

p4 = ______________

p2 = ______________

p5 = ______________

p3 = ______________

p6 = ______________

2d. Use the logistic equation and its graph to determine long term behavior.

EXAMPLE #3: Raising a particular variety of goldfish has a growth parameter of 3.1. Suppose you start with 20% of its carrying capacity.

3a. What is the logistic equation for this population of goldfish?

3b. Use your calculator to find the first 15 years of goldfish population p-values

p0 = ________

p1 = ________

p2 = ________

p3 = ________

p4 = ________

p5 = ________

p6 = ________

p7 = ________

p8 = ________

p9 = ________

p10 = ________

p11 = ________

p12 = ________

p13 = ________

p14 = ________

p15 = ________

3c. Use the logistic equation and its graph to determine long term behavior.

EXAMPLE #4: Raising a particular variety of flour-beetle have r = 3.5. The seed will be p0 = 0.44.

4a.What is the logistic equation for this population of beetle?

4b. Use your calculator to find p20 through p27 (use three decimal places)

p20 =________

p24 = ________

p21 = ________

p25 = ________

p22 = ________

p26 = ________

p23 = ________

p27 = ________

4c. Use the logistic equation and its graph to determine long term behavior.

EXAMPLE #5: Try p0 = 0.2 and r = 4.

5a.What is the logistic equation for this population of beetle?

5b. Use the logistic equation and its graph to determine long term behavior.

10.4 Logistic Model PRACTICE PROBLEMS: Part 2

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 p0 = 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?

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MODE

Y=

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2ND

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X, T, θ, n

WINDOW

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GRAPH

Y=

GRAPH

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