INQUIRY &
I N V E S T I G AT I O N
Modeling Exponential Population
Growth
BONNIE MCCORMICK
The concept of population growth patterns is a key component
of understanding evolution by natural selection and population dynamics in ecosystems. The National Science Education
Standards (NSES) include standards related to population growth
in sections on biological evolution, interdependence of organisms,
and science in personal and social perspectives (NRC, 1996).
Organisms have the potential to achieve exponential growth under
ideal conditions, yet sustained exponential growth is not found in
nature. This observation is a cornerstone of the theory of evolution
through natural selection (Mayr, 1982). Factors that limit growth
can lead to evolutionary change in a population and can have “profound effects on the interactions between organisms” (NRC, 1996).
To promote understanding of the concept of exponential growth,
a set of activities was developed to engage students by integrating
mathematical principles with the science concepts.
1. The peppers distributed to the class represent the fruits
from one pepper plant.
Students explore the concepts of population growth by predicting the growth potential of a plant population. Understanding
exponential growth in a population includes knowledge of population dynamics, the mathematical principles used to calculate
growth over time, and the ability to interpret the graphical representations of population growth. To facilitate the calculation and
representation of growth patterns, students use a graphing calculator. Graphing calculators allow the students to predict population
growth trends when conditions change and to answer questions
about the future growth potential of a population. Students then
apply the concepts they have learned by predicting how human
population will grow in the future if current population trends
continue. Finally, students discuss limits to population growth in
nature and consequences of these limits to population phenotypic
structure.
7. There is only one generation of pepper plants per year.
Background
2. The number of seeds in the peppers is the offspring of one
pepper plant.
3. The number of seeds produced by every pepper plant in a
population will be equal to the total number of seeds in the
peppers counted by the class.
4. Conditions are ideal. There are unlimited space and unlimited nutrients to support growth. There is no plant predation or disease. Climate is stable and favorable for growth.
5. All the plants die at the end of the summer.
6. All seeds produced by the pepper plants will grow a plant
the following year.
8. The number of plants at the inception of the population
model is one.
These assumptions allow the students to calculate and graph a
simple population growth equation by eliminating survivorship
of plants in the next generation. The death rate is assumed to be
100%.
This activity has been used successfully in undergraduate biology courses for majors and non-majors, and graduate courses in
science teacher preparation. It requires one three-hour class period
or two shorter class periods. National Science Education Standards
(NRC, 1996) addressed in this inquiry activity are listed in Table 1.
Materials
• one bell pepper fruit per student group
• small tray or paper plate
The plant model used is the bell pepper plant (Capsicum
• small knife (a plastic knife works fine)
annuum). Bell peppers are annual plants (Crockett, 1972) and the
• graphing calculator
fruits are readily available in supermarkets. However, any annual
fruit with multiple seeds can be used. Before beginning the activProcedure
ity, it may be necessary to briefly review plant growth
Student groups are given a pepper to count the seeds
and development. Include the fact that the seeds are
(offspring) of the pepper plant. Each group reports its
the potential offspring and are found in fruits.
results so that a class total can be determined. The
Annual plants produce seeds that typically gerHuman population
class total is assumed to be the number of seeds
minate the next year (or later), and the parent
produced by one pepper plant. Since one bell
plant only survives one year. Some plants,
growth can have a propepper plant would be likely to produce more
including the bell pepper, can self-fertilize.
found effect on ecosystems
than the number of bell pepper fruits counted
Using an annual plant allows for simpliby the student groups, the total number of
fying assumptions to be made about the
as humans compete for
seeds obtained from adding the group results
potential for population growth. Students
resources and alter
produces a reasonable number of offspring from
can then develop their own equation to use
one pepper plant. The student protocol can be
in graphing the population potential over
the physical
found in Appendix 1. The results are recorded in
a number of years. In this model of plant
Table 2.
environment.
growth, the assumptions are:
THE AMERICAN BIOLOGY TEACHER
POPULATION GROWTH
291
The year of incepTable 1. National Science Education Standards addressed.
tion, Year 0, is the first
plant. Year 1 is the following year when all the
Standard (9-12)
Fundamental abilities and concepts
seeds counted grow and
produce a bell pepper
Content Standard A
Use technology and mathematics to improve
plant. The number of
Abilities necessary to do scientific inquiry
investigations and communications.
plants in the population
Biological evolution is a consequence of the interfor Year 1 equals the
Content Standard C
action of (1) the potential for a species to increase
total number of seeds
Biological Evolution
counted by the class. At
in numbers.
this point, the students
Living organisms have the capacity to produce
in each group work to
Content Standard C
populations of infinite size, but environment and
calculate the number
Interdependence of Organisms
of pepper plants in sucresources are finite.
ceeding generations if all
Populations grow or decline through combined
the seeds are viable and
effects of births and deaths.
produce plants with the
Concept Standard F
same number of seeds
Populations can increase through linear or expoScience in Personal and Social Perspectives
as the initial population.
nential growth with effects on resource use and
Population Growth
Encourage the students
environmental pollution.
to check with the instrucPopulations can reach limits to growth.
tor after the calculation for
the second year to make
sure they have correctly
Table 2. Projected growth of pepper plants (worked example).
calculated the population. For
example, the number of pepper
Total number of seeds in your bell pepper
43
plants in Year 0 is one. If the
class determines that one pepTotal number of pepper seeds of all bell pepper plant produces 1000 seeds
pers in the class. (This is how many seeds each
(the number of seeds found in
1,000
plant will produce.)
the peppers counted) and all
the seeds grow the following
1
Number of pepper plants at inception (Year 0)
year, then Year 1 is 1000 plants.
(Remember: Always assume
Number of pepper plants Year 1
1 x 1,000 = 1,000
that each plant produces the
Number of pepper plants Year 2
1000 x 1,000 = 1,000,000
same number of seeds and that
all the seeds grow into a plant
Number of pepper plants Year 3
1,000,000 x 1,000 =1,000,000,000
the next year.) Year 2 population is 1,000,000 plants from the
Number of pepper plants Year 4
1,000,000,000 x 1,000 = 1,000,000,000,000
1000 plants in Year 1 that each
produces 1000 seeds.
The next step in the activity is
to graph the population growth so that students can see the pattern of population growth. Population growth could be graphed
using graph paper, but this becomes difficult because of the rapidly (exponentially) increasing population. Using a graphing calculator to graph the calculated population growth potential allows
the students to make the graph, to predict population growth in
subsequent years, and to predict what will happen to the population curve if the number of seeds that are able to grow each year
is limited. To do this, the students must develop an equation that
represents their data so that they can enter it into the graphing calculator. Most students immediately protest that they cannot possibly do this because they are terrible at math, however with a little
encouragement, students can successfully develop an equation.
The students can begin by discussing and recording the
method they used to calculate the population for each year. It is
important that inception is the Year 0. The students will probably
need to be reminded that any number raised to the exponent zero
is one (the original population).
Given the assumptions of our population model, the equation
will always be Xn where X is equal to the number of seeds produced
292
THE AMERICAN BIOLOGY TEACHER
by one pepper plant and n is equal to the year. This exponential
equation is entered into the graphing calculator to produce the
graph. Entering data, setting the graph range, and using the graph
and table function to find information at specific points are demonstrated using overhead projection of the calculator screen. By
using the table function of the calculator, students can determine
the projected population in future years.
Once the students have entered the equation and produced
the graph on the graphing calculator, it is a good time to have a
class discussion about the shape of the graph. A J-shaped curve is
produced by growth that is exponential and describes unrestricted
population growth when conditions are ideal. If any of the assumptions in the model are violated, then growth would be restricted.
Sigmoid (S) shaped or logistic curves predict the carrying capacity
which is the population that is sustainable in an environment. The
steeply rising curve of the logistic model flattens as the growth rate
approaches 0. At this point, the population has reached carrying
capacity. Introducing the concept of carrying capacity facilitates
discussion of what factors limit growth in a population. Knowledge
of the graphical patterns of idealized population growth allows
students to recognize the presence of limiting factors in a popu-
VOLUME 71, NO. 5, MAY 2009
lation. Deviation from idealized
exponential or logistic growth
patterns indicate some change in
conditions that limit or promote
growth. Because these patterns
may be explained by changes
in environmental conditions and
by community interactions, they
are valuable in understanding
population ecology (Raven et
al., 2005).
Students are then asked
to use their calculator to determine how many pepper seeds
can grow each year without
producing exponential growth.
By changing the number of
seeds that grow each year (X in
the equation), students can rapidly evaluate their predictions.
They use the graph to determine
when the J-shape becomes flat.
In this model of the pepper plant,
the answer is one. Although it
seems that the answer would be
obvious, students typically start
with high numbers. This activity
helps them realize that in any population, each plant can only have one
viable offspring to prevent exponential
growth. Students extend these concepts by discussing the factors that
limit population growth in nature.
This sets the stage for future discussions that deal with the concepts of
change in populations through the
process of natural selection and the
dynamics of population, community,
and ecosystem ecology. Some suggested questions for discussion are listed
in Table 3.
Table 3. Topics for discussion.
Using your graph, predict what the population of the bell pepper plant will be in the sixth
year if the assumptions are valid. Would this growth of pepper plants be likely to occur?
What factors might influence pepper plant population growth?
What will the growth curve look like if only 10 seeds produce plants each generation? Is
this growth rate sustainable?
What factors might influence which individual plants survive? If there is a genetic trait that
favors survival, how might the characteristics found in the plant population change?
Use the calculator to determine how many seeds can survive to produce population
growth that is not exponential.
Using the graphing calculator, predict what the human population will be in the year
2020 if population follows the predicted curve. Predict the population for the year 2050.
Is the current growth rate of the human population sustainable? What factors might limit
or promote human population growth?
Identify and discuss the ethical and social justice issues associated with the growth of
human populations.
Table 4. Estimated world human population (U.S. Census Bureau, 2008).
If human population continues to
grow at the present rate, the population
is projected to be 13 billion by 2050
using the graphing calculator model. The
YEAR
WORLD POPULATION
Population Bureau (2008) estimates the
1930
2.07 x 109
human population will be between 7.5
billion and 13 billion in 2050, depend1940
2.30 x 109
ing on the fertility rate. The U.S. Census
Bureau (2008) projects the population
1950
2.52 x 109
to be 9.5 billion in 2050. The human
population growth rate slowed from 2.2
1960
3.02 x 109
in 1962 to 1.15 in 2005 so the growth
1970
3.70 x 109
rate is no longer exponential. However,
it is unclear what the carrying capac1980
4.40 x 109
ity population is for humans. There are
many factors other than birth rates that
1990
5.27 x 109
Application
may affect the sustainability of human
population growth. These include suit2000
6.06 x 109
Students can apply what they have
able space, energy, food, fresh water,
learned to human population growth
and disease. Humans are responsible
patterns in several ways. The U.S.
for changes in the physical environment
Census Bureau (2008) and the Population
including climate change, pollution, and acid rain that may increase
Reference Bureau (2008) are two excellent sources that examine
as population increases. Our future actions will affect not only us
human population growth and the factors that affect growth rate.
but also other organisms that share the planet.
Students can use these resources and questions in Table 3 as a
starting point to explore issues of human population dynamics.
An alternative way to evaluate future patterns and consequences of human population growth was suggested by students
in a non-majors biology class. After completing the pepper lab, the
students asked if we could use the graphing calculator to project
human population growth. Historic data for human population
estimates (Table 4) can be entered into the graphing calculator to
produce a graph. The equation of the line can then be determined,
and students can project the population into the future. These
population projections can be compared to Census Bureau (2008)
and Population Bureau (2008) resources that both project that the
human population will begin to show a logistic growth pattern
around 2050.
THE AMERICAN BIOLOGY TEACHER
Conclusions
This activity helps the student understand the key concept of
exponential growth. By using the graphing calculator to project
future populations, it becomes clear that exponential growth
cannot be sustained because resources are limited. In the pepper
population, the calculator demonstrates that unrestrained growth
quickly reaches a number that is incomprehensible. Students learn
to identify exponential growth by the shape of the curve. On their
lab assessment, nearly all of the students are able to draw a curve
that represents exponential growth.
Because students gain an understanding of idealized growth
potential when resources are unlimited and when predation
POPULATION GROWTH
293
References
Crockett, J. W. (1972). The Time-Life Encyclopedia of Gardening
Vegetables and Fruits. New York: Time-Life Books.
Mayr, E. (1982). The Growth of Biological Thought. Cambridge, MA:
Belknap Press.
National Research Council. (1996). National Science Education
Standards. Washington: National Academy Press.
Population Reference Bureau. (2008). World Population Highlights
2007: Overview of World Population. Available online at:
http://www.prb.org/Articles/2007/623WorldPop.aspx
Raven, P.H., Johnson, G. B., Losos, J. B. & Singer, S. R. (2005). Biology,
Seventh Edition. Boston: McGraw Hill.
U.S. Census Bureau. (2008). Total Midyear Population of the
World: 1950-2050. Available online at: http://www.census.
gov/ipc/www/idb/worldpop.html
Vitousek, P. M., Monney, H. A., Lubchenco, J. M., and Melillo, J. M.
(1997). Human dominated ecosystems. Science, 277(5325),
494-499.
Appendix 1. Student Protocol
1. Cut open the pepper and count the number of
seeds inside. Record the total in Data Table 2 and
on the board so that a class total can be obtained.
Assume that the number of bell peppers in the
class represents the number of peppers produced
by one bell pepper plant in a growing season.
2. Record the total number of seeds counted for the
class.
and disease are not a factor, they are able to recognize that when the
assumption of idealized conditions is violated, population growth patterns change. Limits to space, food, and nutrients and the presence
of competition, predation, and disease determine carrying capacity.
Because of these limiting factors, logistic growth provides an estimate
of population growth potential when carrying capacity is considered.
Growth rates tend to decrease, and death rates tend to increase as populations approach or exceed carrying capacity.
3. To complete the Data Table, assume that:
(a) Only one bell pepper plant existed the first
year and that it produced the number of
seeds represented by the class total.
(b) Each seed always grows into a new bell pepper plant the next year.
(c) Each new plant always produces the class
total number of seeds.
This knowledge can be applied to evolution by natural selection
because factors that limit population growth are the selective pressures
on populations. Traits that confer survival advantage and are inherited
by offspring can be the cause of evolutionary change. Human population growth can have a profound effect on ecosystems as humans
compete for resources and alter the physical environment. These
effects have consequences for the entire planet because of the relatively
large amount of resources humans currently require to sustain their
population growth (Vitousek et al., 1997). There are serious social,
economic, and political issues related to population growth as we begin
to approach carrying capacity. By using a mathematical model, students
are able to answer questions about patterns found in nature and practice inquiry skills by collecting data, making predictions, and discussing
implications of their findings.
(d) All pepper plants die at the end of the year.
4. Find the number of bell pepper plants that grow
in the second year and record this number in
Data Table 2. (Check with other groups and the
teacher to make sure that this step is performed
correctly before continuing.)
5. Calculate the number of bell pepper plants that
will grow in the second, third, and fourth years.
Describe the method you used to calculate the
number of plants for each generation. Write the
method as a mathematical equation.
6. Graph your results using the graphing calculator.
Sketch your graph.
Julie Barker, former Instructor of Mathematics at the University of the
Incarnate Word, provided assistance with the integration of mathematics and graphing calculators for this activity. This project was supported
in part by NASA Opportunities for Visionary Academics (NOVA), a
program funded by the National Aeronautics and Space Administration,
although the views expressed here are those of the author only.
294
THE AMERICAN BIOLOGY TEACHER
BIO
Acknowledgments
BONNIE MCCORMICK, Ph.D., is Associate Professor of Biology,
University of the Incarnate Word, San Antonio, TX 78209; e-mail:
mccormic@uiwtx.edu.
VOLUME 71, NO. 5, MAY 2009