Populations in Nature Graphing Activity:
Exponential vs Logistic Growth
NGSS PE: HS-LS2-1. Use mathematical and/or computational representations to support explanations of factors that
affect carrying capacity of ecosystems at different scales.
NGSS DCI: LS2.A: Interdependent Relationships in Ecosystems
Introduction:
If a population has ideal conditions for growth, such as unlimited food and no predators, then that population will grow
exponentially. To figure out if the population is growing or shrinking, one must add the births and immigration rates and
subtract them from the death and emigration rates. If the number is greater than zero, the population is growing. If the
number is less than zero, the population is decreasing. If the number equals zero, the population is stable. 'N' represents
the number of individuals in the population.
If a population is limited in growth, that population will follow the logistic growth curve. A
carrying capacity will be reached in a logistic growth model in which a maximum number of
individuals can be held in an environment healthfully. Predators, food, and territory can all put
limits on the growth of a population.
Part One: The Cottontail Rabbits
Cottontail rabbits are found in Southern California. Recently, the cottontail's predator, the
coyote, has been disappearing due to hunting by humans. Jim, a zoologist, documented and
recorded the number of cottontail rabbits in the field during this time of predator hunting. His
information is recorded below in the table.
Year
Number of Population (N)
Births and
Immigration
Deaths and
Emigration
Year 1
1,000
3,000
2,000
Year 2
Year 3
Year 4
Year 5
Year 6
2,500
4,500
8,500
16,500
32,500
4,000
8,000
9,000
17,000
33,000
2,000
4,000
1,000
1,000
1,000
Population Growing
(+) or Shrinking (-)?
Evidence?
Ex: 3,000-2,000 =
1,000: Growing
1) Calculate whether or not the population is decreasing or growing. Do this by subtracting births and immigration from
deaths and emigration. Place the number in the empty column and state whether the population is growing or shrinking.
The first data set is done for you.
2) Graph the data on the graph paper on the next page. Place the year on the x-axis and the population number (N) on
the y-axis. Make a line graph and connect the data points. Give the graph a title, use equal intervals, and label the axes.
3) Answer the following questions using complete sentences:
a) What type of growth is this (exponential/logistic)? How could you tell?
b) How could the rabbits possibly grow this way?
c) How would you describe the overall trend in the rabbit population?
d) What is the average population size of bunnies from Year 1 to Year 6? Show mathematical set up and include
units. Round answer to the nearest tenth.
e) Prediction: What will most likely happen to this population over time? What is your evidence for this
prediction?
Part Two: The Gray Fox
The gray fox is a native to the Santa Monica Mountains of California. An ecologist
Cammy has been studying the numbers of the gray fox for ten years. She is
amazed at how successful the gray fox is at living in this area. In the chart below
are the ecologist's records for the gray fox population of the Santa Monica
Mountains.
Year
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
Year 7
Year 8
Year 9
Year 10
Population Number
(N)
2,500
3,000
4,000
5,000
6,600
7,000
7,100
7,000
7,050
7,100
Births and
Immigration
700
1,100
1,200
1,700
500
200
100
100
150
200
Deaths and
Emigration
Population Growing (+)
or Shrinking (-)?
Evidence?
200
100
200
100
100
100
200
50
100
100
1) Calculate whether or not the population is decreasing or growing. Do this by subtracting births and immigration from
deaths and emigration. Place the number in the empty column and state whether the population is growing or shrinking.
2) Graph the data on the graph paper on the next page. Place the year on the x-axis and the population number (N) on
the y-axis. Make a line graph and connect the data points. Give the graph a title, use equal intervals, and label the axes.
3) Answer the following questions using complete sentences:
a) Describe the overall trend of the graph. What did you observe?
b) What type of growth is this (exponential/logistic)? How could you tell?
c) Define carrying capacity. Was a carrying capacity reached in this data?
d) Draw a line on the graph to show where the carrying capacity is. What is the carrying capacity?
e) What is the average population size of gray foxes from Year 1 to Year 5? Show mathematical set up and
include units.
f) What did you see happening in years 6 through 10? Why do you think this occurred?
g) Prediction: What do you think will happen to this population in the future? What is your evidence
for this prediction?
Helpful Diagrams:
Populations in Nature Graphing Activity:
Exponential vs Logistic Growth Teacher Key
NGSS PE: HS-LS2-1. Use mathematical and/or computational representations to support explanations of factors that
affect carrying capacity of ecosystems at different scales.
NGSS DCI: LS2.A: Interdependent Relationships in Ecosystems
Introduction:
If a population has ideal conditions for growth, such as unlimited food and no predators, then that population will grow
exponentially. To figure out if the population is growing or shrinking, one must add the births and immigration number
and subtract them from the death and emigration rates. If the number is greater than zero, the population is growing. If
the number is less than zero, the population is decreasing. If the number equals zero, the population is stable. 'N'
represents the number of individuals in the population.
If a population is limited in growth, that population will follow the logistic growth curve. A
carrying capacity will be reached in a logistic growth model in which a maximum number of
individuals can be held in an environment healthfully.
Part One: The Cottontail Rabbits
Cottontail rabbits are found in Southern California. Recently, the cottontail's predator, the
coyote, has been disappearing due to hunting by humans. Jim, a zoologist, documented and
recorded the number of cottontail rabbits in the field during this time of predator hunting. His
information is recorded below in the table.
Year
Number of Population (N)
Births and
Immigration
Deaths and
Emigration
Year 1
1,000
3,000
2,000
Year 2
Year 3
Year 4
Year 5
Year 6
2,500
4,500
8,500
16,500
32,500
4,000
8,000
9,000
17,000
33,000
2,000
4,000
1,000
1,000
1,000
Population Growing
(+) or Shrinking (-)?
Evidence?
Ex: 3,000-2,000 =
1,000: Growing
2,000; Growing
4,000: Growing
8,000: Growing
16,000: Growing
32,000: Growing
1) Calculate whether or not the population is decreasing or growing. Do this by subtracting births and immigration from
deaths and emigration. Place the number in the empty column and state whether the population is growing or shrinking.
The first data set is done for you.
2) Graph your results on the graph paper on the next page. Place the year on the x-axis and the population number (N)
on the y-axis. Make a line graph and connect the data points. Give the graph a title, use equal intervals, and label the
axes.
3) Answer the following questions using complete sentences:
a) What type of growth is this (exponential/logistic)? How could you tell?
This type of growth is exponential. I could tell because the population is doubling every year. The graph also
resembles a J, as in the exponential growth curve diagram. The population continues to grow.
b) How could the rabbits possibly grow this way?
The rabbits could grow this way by having an unlimited food supply, much territory, or lack of predators.
c) How would you describe the overall trend in the rabbit population?
The overall trend is that the rabbit population continues to increase year after year.
d) What is the average population size of bunnies from Year 1 to Year 6? Show mathematical set up and include
units. Round answer to the nearest tenth.
10,000 + 2,500 + 4,500 + 8,500 + 16,500 + 32,500 rabbits = 12,416.7 rabbits per year
6 years
e) Prediction: What will most likely happen to this population over time? What is your evidence for this
prediction?
If the rabbits continue to grow this way, I believe they will run out of natural resources. There are only some
many organisms that a habitat can hold. From the graph, you can see that the population continues to double.
This means that at some point, the environment will no longer be able to hold these organisms, and the
population will decline.
Part Two: The Gray Fox
The gray fox is a native to the Santa Monica Mountains of California. An ecologist
Cammy has been studying the numbers of the gray fox for ten years. She is
amazed at how successful the gray fox is at living in this area. In the chart below
are the ecologist's records for the gray fox population of the Santa Monica
Mountains.
Year
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
Year 7
Year 8
Year 9
Year 10
Population Number
(N)
2,500
3,000
4,000
5,000
6,600
7,000
7,100
7,000
7,050
7,100
Births and
Immigration
700
1,100
1,200
1,700
500
200
100
100
150
200
Deaths and
Emigration
200
100
200
100
100
100
200
50
100
100
Population Growing (+)
or Shrinking (-)?
Evidence?
500: Growing
1,000: Growing
1,000: Growing
1,600: Growing
400: Growing
100: Growing
-100: Decreasing
50: Growing
50: Growing
100: Growing
1) Calculate whether or not the population is decreasing or growing. Do this by subtracting births and immigration from
deaths and emigration. Place the number in the empty column and state whether the population is growing or shrinking.
2) Graph your results on the graph paper on the next page. Place the year on the x-axis and the number of the
population (N) on the y-axis. Make a line graph and connect the data points. Give the graph a title, use equal intervals,
and label the axes.
3) Answer the following questions using complete sentences:
a) Describe the overall trend of the graph. What did you observe?
The overall trend of the graph is that the population increases. However, the population begins to slow
down in its growth.
b) What type of growth is this (exponential/logistic)? How could you tell?
This is logistic growth. I observed a stabilizing or leveling off of the population growth on the graph.
c) Define carrying capacity. Was a carrying capacity reached?
Carrying capacity is the maximum number of individuals that a habitat or location can hold in a given
time period. Yes, a carrying capacity was reached in this population of foxes.
d) Draw a line on the graph to show where the carrying capacity is. What is the carrying capacity?
The carrying capacity is about 7,000-7,050 individuals.
e) What is the average population size of gray foxes from Year 1 to Year 5? Show mathematical set up and
include units.
2,500 + 3,000 + 4,000 + 5,000 + 6,600 foxes
5 years
= 4,200 foxes per year
f) What did you see happening in years 6 through 10? Why do you think this occurred?
In years six through ten, I observed a slowing down of the fox population growth. I think this occurred
because the foxes reached their carrying capacity in the area. The foxes were probably limited by food,
water sources, or the size of their territory.
g) Prediction: What do you think will happen to this population in the future? What is your evidence
for this prediction?
I predict the foxes will continue to hover at around 7,000 foxes, their carrying capacity. This holds true
only if their resources stay consistent. However, if the environment changes, then the fox carrying
capacity might also change. Thus, there may be more or less foxes in the area depending upon the
resources available.
Graph Keys
Carrying Capacity