Project 1: Exponentials, Logarithms and Vampires –Test Grade

In this project we will use exponential models to explore population growth and decline. We will also use

properties of logarithms to solve exponential problems. We will also be plotting our results and comparing the

different models and making conjectures about which model is the most effective.

Part 1: Introduction to Population Ecology and Exponential Models (50 Points)

On November 18, 2015, Count Dracula moved from Transylvania to Arlington, MA, in order to fully enjoy

everyone that Arlington has to offer.

In this section, we will model the growth of the vampire population (and the decline of the human population)

using different rates of vampire-human conversion.

As of 2010, the population of Arlington, MA is 42,844 humans, and let’s assume that Dracula converts to

vampire 3% of the humans per day. We can also assume that once he has sired a new vampire, that new vampire

is also going to convert humans at the same rate as Dracula. So we will have an increasing number of vampires in

Arlington (and a decreasing number of humans).

Let’s find the exponential equation that is going to describe the change in population of vampires…

a). What is the rate? _______________

b.) Initial number of vampires = _____________.

c.) Exponential Equation describing the growth of vampires in Arlington:

d.) Plotting the data

To get the number of humans that are left, we will subtract the number of new vampires from the original

population of Arlington. Plot the time on the x-axis. Plot the vampire population in blue, and the human

population in red.

e). In order to find out exactly when all of the humans in Arlington have been converted to vampires, we can

use the properties of logarithms to solve for x. (use the space below for showing work)

x = ____________

This is how much time is needed for the vampires to convert all of the humans. (don’t forget units)

Put a blue circle on the graph for this (x, y) point.

f.) Based on your graph, what is the time when the vampire population is equal to the human population (i.e.,

where do they intersect each other)?

g.) Based on your graph, what is the number of vampires/people when their populations are equal in number?

x (days) y

(# vampires)

# humans

remaining

0

30

60

90

120

180

210

240

270

300

330

360

? = 42,844 0

time (_____)

# v

am

pir

es (

blu

e)

# h

um

an

s (r

ed)

3.) After converting all of the humans in Arlington to vampire, Dracula and his Arlington-vampires move out to

the rest of the state of Massachusetts, with its coveted population of 6,547,629 humans. How long is it before he

is able to convert everyone in Massachusetts to being a vampire? (use the table below for showing work)

Initial Value :_______________

Final Value:______________

x = ____________

This is how much time is needed for the vampires to convert all of the humans in Massachusetts.

4.) After converting everyone in Massachusetts, Dracula and his blue-state vampires move on to the rest of the

United States (making their way across Alaska and Hawaii too). If the population of the US is estimated at

326,120,217 how long before they convert everyone here to vampire? (use the table below for showing work)

Initial Value :_______________

Final Value:______________

x = ____________

This is how much time is needed for the vampires to convert all of the humans in the United States.

5.) After converting everyone in the US, they move on to the rest of the Earth, with its population of

7,380,426,515 (approximately 7.3 Billion people including the USA as of 2015). How long before everyone is a

vampire? (use the table below for showing work)

Initial Value :_______________

Final Value:______________

x = ____________

This is how much time is needed for the vampires to convert all of the humans in Massachusetts.

6.)

7.) Why does the time to convert each new segment of the population get smaller than the previous time?

8.) Once everyone is converted to vampire, and there are no more humans left to eat, what happens to the

vampire population of Earth?

9.) Is this a sustainable ecological model? Why/Why not?

Time to convert all of Arlington =

Time to convert the rest of Massachusetts =

Time to convert the rest of the United States =

Time to convert the rest of Earth =

Total time to convert all humans on Earth =

Part 2: Population Decline (50 Points)

In Part 1, we had one vampire to start with, and he was able to convert all of the humans to vampire completely

unchecked. In this part, we introduce a way of reducing the vampire population, so that there is some means of

controlling their growth rate. (in Part 4, we will look at all factors combined).

So, back to our story: The werewolf, Wolfman, having heard about the increasing numbers of vampires in

Arlington, decides to follow Dracula from Transylvania to Massachusetts, as a means of keeping the vampire

population in check. Let’s also assume that after Dracula has converted all of Arlington, he and the rest of the

vampires take a break and hang out while Wolfman does his job (implausible, but...).

Let’s explore different models of vampire population decline:

1.) At first, Wolfman is only allowed to change into werewolf form at the full moon, which happens only once

every 28 days, and let’s assume that he is able to get rid of only one vampire a day.

b.) Calculating # vampires left:

Initial number of vampires =

# vampires culled each year =

time (years) =

# vampires left =

(what we are solving for)

___________ = b

__________ = m

_________ = x

= y

Putting it together: y b m x

Re-arranging to look familiar:

y m x b

a.) Wolfman’s yearly rate of change is:

Every 28 days he changes to a werewolf.

Which happens: 365

28 = __________times per year

If he is culling only one vampire per change, the

number of vampires he gets each year is:

_____________

time (years)

x

# vampires

y

0

42,844

1

2

10

50

0

Use this to

fill in the

table.

time (years)

# v

am

pir

es

c.) Plot the data here (in blue)

d.) At this rate, how long will it take

Wolfman to get rid of all of the vampires

in Arlington?

2.) In a different model, if Wolfman is can change into werewolf form every day, and he culls one vampire a day:

a.) Wolfman’s yearly rate of change is:

Everyday he changes to a werewolf.

Which happens: __________times per year

If he is culling one vampire per change, the number

of vampires he gets each year is:

_____________

c.) Plot the data on the graph on page 7 (in green).

d.) At this rate, how long will it take Wolfman to get rid of all of the vampires in Arlington?

3.) In a third model, if Wolfman changes into a werewolf every day, and he is able to cull 10% of the 42,844

vampires each day. Let’s see how long it takes Wolfman to get rid of all of the new vampires – leaving Dracula

alone (again, we are assuming all the vampires are hanging around waiting for Wolfman to finish them off).

a.) What is the Decay Factor for the decline in Vampire population:

a.) Initial number of Arlington Vampires = _____________.

d.) Exponential Equation:

e.) Put it together:

time (years)

x

# vampires

y

0

42,844

1

2

10

50

0

b.) Calculating # vampires left:

Initial number of vampires =

# vampires culled each year =

time (years) =

# vampires left =

(what we are solving for)

___________ = b

__________ = m

_________ = x

= y

Putting it together: y b m x

Re-arranging to look familiar:

y m x b

Use this to

fill in the

table.

d.) Plot this data on the graph on page 7 (in red).

e.) Why is a linear model used to describe the decline in vampires in the first two models?

f.) Why is an exponential model used to describe the decline in the third model?

g.) Which model is most effective (in terms of time) in decreasing the vampire population? Why?

time (days)

x

# vampires

y

0

42,844

1

2

50

100

1

Use the properties of logarithms

to determine how long it will take

for only one vampire to be left.

Part 3: Bounded Population Growth (Vampirism as a Disease Model)

In the previous sections, we used an exponential model that allowed for unbounded population growth (or spread

of disease, if you think of Vampirism as a disease model). Now we will take a look at a model of exponential

growth that takes into account a limit on the growth (i.e., there is a finite number of humans, so the disease of

vampirism cannot spread past the total number of people in a given area). This kind of model is called a logistic

function and it looks like this:

In our model of the spread of vampirism, we will be using the following equation:

𝑃(𝑡) = 𝑎

1 + 𝑏𝑒−𝑘𝑡

Where we define the terms as:

P(t) is the population of vampires at any point in time (t).

a = the mamximum number of vampires possible

P0 = initial number of vampires = a

1+b

k = rate of spread of disease

t = time

The time at which half of the population has been infected is an important point because it marks a change in the

curve from increasing growth to decreasing growth. The plot is still increasing, but the rate at which it increases

has changed.

Part A: Modeling Spread of Vampirism in Arlington. Assuming all parameters are the same from Part 1,

a = ____________

P0 = ________________

Equation:_________________________________

b = _______________

k = _______________

Plot the logistic function that models the spread of vampirism in Arlington from time = 0 until saturation.

Include a plot of the unbounded function from Part 1 (equation = _______________)

1. At what time does the logistic function start to level off? t = _____________________

2. What is P(0) = _______________

3. When do the two graphs pull apart? t = ___________________

4. What are the two horizontal asymptotes of the logistical function?________________________

5. At what time does half of the population become infected? t = ________________________ Mark this on your plot

time (_____)

# v

am

pir

es

Part B: Modeling Spread of Vampirism from Arlington to Massachusetts. Assuming all parameters are the

same from Part 1,

a = ____________

P0 = ________________

Equation:_________________________________

b = _______________

k = _______________

Plot the logistic function that models the spread of vampirism in Massachusetts from time = 0 until

saturation. Include a plot of the unbounded function from Part 1 (equation = _______________)

6. At what time does the logistic function start to level off? t = _____________________

7. What is P(0) = _______________

8. When do the two graphs pull apart? t = ___________________

9. What are the two horizontal asymptotes of the logistical function?________________________

10. At what time does half of the population become infected? t = ________________________ Mark this on your plot

time (_____)

# v

am

pir

es

Arlington, MA

Births Immigration

Deaths Emigration

BONUS (10 Points): Population Dynamics, Full Scale Model

In our previous models, we assumed that the populations were not changing while the various increases or

decreases were taking place (i.e., there were no human births while Dracula was converting everyone, and there

were not vampire deaths while Dracula was converting everyone). In this model, we will look at a more complex

model of population interaction and explore the meaning of population equilibrium.

Population changes can be described by the following figure:

The Change in population over time is best described by : Births – Deaths. But it is also governed by other

factors, such as the Carrying Capacity of an area (i.e., what is the maximum number of people an area can

support before running out of resources), the numbers of people moving in and out of the area. Additionally,

population change is a continuous function, so we are using the natural logarithm instead of the log10.

In our fictional model of vampires and werewolves, the Human-Vampire-Werewolf interaction can be described

by the following variables:

Human Vampire

H represents the human population of Arlington

(56, 173) V represents the vampire population

r growth rate of the human population.

this is due to births as well as people moving into

Arlington from elsewhere.

b the vampire birth rate

(Dracula won’t be able to convert people

at will anymore, since that was not a

sustainable ecological model).

K carrying capacity of Arlington. (in terms of people)

m the rate at which more vampires arrive

into Arlington from elsewhere

a # of human-vampire encounters that result in feedings

(these are not conversions)

w the rate at which the vampires are being

culled by the werewolves

(we’ll give Wolfman a break and allow

him to have help).

Setting our parameters in order to find the numbers of vampires that can make this population interaction stable

requires some assumptions:

1.) If the vampire encounters 100 people per day, and he feeds every 3 days,

the rate of feedings per encounter = 1

100 3 = ________ = a

2.) Vampires will interact with (i.e. feed on) humans every 3 days, resulting in

365

3________ feedings per year.

3.) Vampires sire a new vampire once in two years. In terms of feedings, the rate of vampire births is:

1

2feedings = 1

= _____________ = b

Human

H 56, 173

r 10% growth

ln(1.1) ________

K

100,000

100,000

a 1

300

= _____________

Human population change over time Vampire population Change over time

= ( )r H k H

k a H V

=b a H V m V w V

Plug these equations and values into Excel and plot your results with Vampire Population Number on the y-axis

and Human population Number on the x-axis.

Where is the equilibrium point? (i.e. how many vampires can Arlington sustain)?

Vampire

V

initial 1

b from #3

=________

m

10%

ln(1.1) ________

w 30% decline

= ln(1- 0.33) +1

= _______

Example: Initial vampire population is 50. Each row represents time. Use MS Excel to plot.

h k r a v b m w dhdt dvdt

56173 100000 0.0953 0.00333 50 0.00417 0.0953 0.6 2588.26 13.7661948

58761.26 100000 0.0953 0.00333 63.76619 0.00417 0.0953 0.6 2638.575 19.8481675

61399.83 100000 0.0953 0.00333 83.61436 0.00417 0.0953 0.6 2724.415 29.0897917

64124.25 100000 0.0953 0.00333 112.7042 0.00417 0.0953 0.6 2887.226 43.4740215

67011.48 100000 0.0953 0.00333 156.1782 0.00417 0.0953 0.6 3233.675 66.5050486

70245.15 100000 0.0953 0.00333 222.6832 0.00417 0.0953 0.6 4157.517 104.82394

74402.67 100000 0.0953 0.00333 327.5072 0.00417 0.0953 0.6 9625.299 173.075382

84027.97 100000 0.0953 0.00333 500.5825 0.00417 0.0953 0.6 -3192 331.446089

80835.97 100000 0.0953 0.00333 832.0286 0.00417 0.0953 0.6 -1190.89 514.024275

79645.08 100000 0.0953 0.00333 1346.053 0.00417 0.0953 0.6 -601.163 809.327127

79043.92 100000 0.0953 0.00333 2155.38 0.00417 0.0953 0.6 -337.79 1277.95007

78706.13 100000 0.0953 0.00333 3433.33 0.00417 0.0953 0.6 -199.687 2019.5573

78506.44 100000 0.0953 0.00333 5452.887 0.00417 0.0953 0.6 -121.316 3192.3834

78385.12 100000 0.0953 0.00333 8645.271 0.00417 0.0953 0.6 -74.87 5046.79483

78310.25 100000 0.0953 0.00333 13692.07 0.00417 0.0953 0.6 -46.6413 7978.69521

78263.61 100000 0.0953 0.00333 21670.76 0.00417 0.0953 0.6 -29.2227 12614.037

78234.39 100000 0.0953 0.00333 34284.8 0.00417 0.0953 0.6 -18.3742 19942.4569

78216.01 100000 0.0953 0.00333 54227.25 0.00417 0.0953 0.6 -11.5786 31528.5607

78204.44 100000 0.0953 0.00333 85755.82 0.00417 0.0953 0.6 -7.30638 49845.9635

78197.13 100000 0.0953 0.00333 135601.8 0.00417 0.0953 0.6 -4.61454 78805.402

78192.51 100000 0.0953 0.00333 214407.2 0.00417 0.0953 0.6 -2.91604 124589.671

78189.6 100000 0.0953 0.00333 338996.9 0.00417 0.0953 0.6 -1.84335 196973.642

78187.76 100000 0.0953 0.00333 535970.5 0.00417 0.0953 0.6 -1.16552 311411.179

78186.59 100000 0.0953 0.00333 847381.7 0.00417 0.0953 0.6 -0.73704 492334.519

78185.85 100000 0.0953 0.00333 1339716 0.00417 0.0953 0.6 -0.46612 778370.514

78185.39 100000 0.0953 0.00333 2118087 0.00417 0.0953 0.6 -0.2948 1230587.4

78185.09 100000 0.0953 0.00333 3348674 0.00417 0.0953 0.6 -0.18646 1945532.79

78184.91 100000 0.0953 0.00333 5294207 0.00417 0.0953 0.6 -0.11793 3075846.41

78184.79 100000 0.0953 0.00333 8370053 0.00417 0.0953 0.6 -0.07459 4862848.46

78184.71 100000 0.0953 0.00333 13232902 0.00417 0.0953 0.6 -0.04718 7688061.1

78184.67 100000 0.0953 0.00333 20920963 0.00417 0.0953 0.6 -0.02984 12154662.8

78184.64 100000 0.0953 0.00333 33075626 0.00417 0.0953 0.6 -0.01888 19216266.2

78184.62 100000 0.0953 0.00333 52291892 0.00417 0.0953 0.6 -0.01194 30380512.5

78184.61 100000 0.0953 0.00333 82672404 0.00417 0.0953 0.6 -0.00755 48030951.1

78184.6 100000 0.0953 0.00333 1.31E+08 0.00417 0.0953 0.6 -0.00478 75935923.1

dhdt=(C3*A3*(B3-A3))/(B3-D3*A3*E3)

dvdt = =F3*D3*A3*E3+G3*E3-H3*E3

Where the H value for the next year is equal to the H value of the previous year + dhdt

Where the V value for the next year is equal to the V value of the previous year + dvdt