A Fibonacci Simple Ecosystem: Prey and Predator
Author(s): Jorge Garcia
Source: Teaching Children Mathematics, Vol. 20, No. 6 (February 2014), pp. 392-396
Published by: National Council of Teachers of Mathematics
Stable URL: http://www.jstor.org/stable/10.5951/teacchilmath.20.6.0392
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iSTEM
J o rg e G a rc i a
◗
Ecological balance is the state of
equilibrium whereby different
species live in balance with
one another; species come
and go at similar rates, and
a steadiness exists among
members. However, sometimes this delicate balance
is disrupted when some
species become extinct or
a new species is introduced.
If this happens, then the ecosystem might become chaotic
and unbalanced. This happened in
Yellowstone National Park in 1940, where
wolves had helped control the elk population
(Alston 1987). When wolves and other predators were eradicated by humans as well as by
fire, the equilibrium in the ecosystem became
unbalanced.
The branch of biology that studies ecosystems, habitats, their developments, and the
equilibrium of species is called conservation
biology (Soule 1986). It is important to study
this balance of species within an ecosystem,
where species might live for a short period of
time (such as mosquitoes, the main source of
food for frogs) or live for centuries (such as the
Galapagos turtles). Studying ecosystems helps
scientists predict issues that may be of great
importance to the survival of species.
By using mathematics, we will analyze the
behavior of a simple ecosystem that has two
species: rabbits (prey) and coyotes (predator). In this model that we are studying, the
predators eat their prey at a constant rate. For
example, a coyote will eat two rabbits each day.
The rabbits reproduce according to a pattern.
The number of live rabbits after the second day
follows a Fibonacci sequence: The number of
live rabbits on a given day is the sum of the surviving rabbits of the previous two days.
Let’s say that on day 4, seven rabbits survive
and that on day 5, ten survive. Then on day 6
392
in the morning, seventeen live rabbits
will exist before the predator eats
its prey.
The problem scenario in
our ecosystem consists of
determining what happens
with the population of rabbits when variables change.
We will examine different
variables: (1) the number of
predators, (2) the number of
rabbits on days 1 and 2, and (3)
the number of rabbits on the first
day the predators come in.
Various Standards are addressed in
grades 6–8. A Data Analysis and Probability
Standard states that “students should formulate
questions, design studies, and collect data about
a characteristic shared by two populations or
different characteristics within one population”
(NCTM 2000, p. 249), and a Representation Standard states that students should “use representations to model and interpret physical, social,
and mathematical phenomena” (NCTM 2000,
p. 280). In addition to these Standards, “constructing explanations and designing solutions”
is an important practice of the Next Generation
Science Standards (NGSS) at the intermediate
elementary level. Within this standard, students
are encouraged to “use evidence (e.g., measurements, observations, patterns) to construct a
scientific explanation or design a solution to a
problem” (NGSS 2013, p. 11).
Preparation
Give students the initial values of the variables.
On day 1 (let’s say it is a Monday), there is one
rabbit; on day 2 (Tuesday), one new rabbit
arrives, hence there are a total of two rabbits
on the night of day 2. Only one predator, the
coyote, will start hunting on the third day, eating two rabbits each day. Give students a copy
of table 1 (see p. 395) to do their computations. For clarity, some of the cells in the table
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MOCOO/THINKSTOCK
A Fibonacci simple ecosystem:
Prey and predator
have been filled. To help students complete
the table, explain that on the morning of day
3 (Wednesday), three rabbits exist (3 = 1 + 2,
according to Fibonacci’s rule) but that after
the predator comes in during the day, only
one rabbit survives at the end of day 3. The
important rule is that the number of rabbits in
the morning is the sum of the surviving rabbits
from the previous two nights. Ask, “How many
rabbits will there be on the morning of day 4?”
It is important to state that the Fibonacci rule
applies to the number of surviving rabbits from
the previous two nights. Here the answer is that
three rabbits will have survived on the morning
of day 4 (3 = 2 + 1).
Activities
After students understand the problem, they
finish filling in the first table and compute the
number of surviving rabbits during the first
nine days. During this first activity, students
must count how many rabbits exist in the
morning and how many survive (at night) after
the predator comes in. After students complete
table 1, ask them to display their data on a bar
graph. (A copy of a graph template is provided
in the online appendix.)
For a second activity, we modify our
variables. We can change the day the predator
comes in by saying, for example, that the coyote
comes on day 4 instead of day 3. This simple
modification produces completely different
results on the population of surviving rabbits.
Students will use table 2 (see p. 395), and it is
also convenient that they represent their data
from table 2 in a bar graph. For a third activity,
table 3 (see p. 396) has different values; this
time two coyotes come for the first time on
day 5. In these four activities, the number of
predators remains constant. Finally, table 4
(see p. 396) is available for students to create
their own initial values and complete their
own table and graph. This will be the student’s
model to address a problem in an ecosystem.
If necessary, print more empty bar graphs for
students to display their data as well as more
empty tables (both available online) to create
more models of an ecosystem.
Guiding questions
To support these activities, ask some questions
regarding our variables:
www.nctm.org
• In the first activity, what will happen at the
end of day 8? Will some rabbits survive?
Explain.
• In the first activity, what do you think would
happen if we started with one more rabbit
on day 1? Did you represent your data on a
bar chart and observe the difference? Why is
there such a big difference?
• Would it make any difference to bring in an
extra predator if we want to eradicate the
rabbits? How do you know?
• What will happen if the rabbits are eradicated? What would happen to the coyotes if
they could eat only rabbits?
• Sometimes a predator becomes the prey
of a bigger predator (for example, a human
hunter). Let’s say that a hunter tracks one
coyote each day and that we start with ten
coyotes. Could you think of a different model
to count the number of surviving rabbits and
surviving coyotes each day?
• Would it be possible to keep the number
of surviving rabbits within some range by
sometimes hunting some coyotes? Why
would it be important to do so?
The number of rabbits indeed follows a
Fibonacci sequence (Knott 2008). Rabbits are
hunted by coyotes. These are examples in a
real ecosystem where math plays an important
role. Name others species that are prey and
predators. Do you think this model could also be
applied to plants? Elaborate.
Visualizing concepts
Some of the main concepts presented here can
enhance the perception of our environment and
the application of mathematics to other contexts.
By analyzing the models for prey and predators,
students are constructing mathematical explanations and potentially designing solutions regarding ecological balance. The representation of the
data in a graph immediately suggests the contrasting solutions when the original variables are
modified. The STEM concepts shown here allow
students to visualize and apply mathematical
concepts to populations in ecosystems, emphasizing the survival of a species. Through model
designs, variations of the initial constraints, and
mathematical computations, students are able to
address a simple environmental problem as well
as propose a solution to the given model.
Vol. 20, No. 6 | teaching children mathematics • February 2014
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393
iSTEM
RE F E RE N C E S
Alston, Chase. 1987. Playing God in Yellowstone:
The Destruction of America’s First National
Park. Orlando, FL: Harcourt Brace and Co.
Knott, Ron. “Fibonacci’s Rabbits.” 2008.
Guildford, U.K.: University of Surrey Faculty
of Engineering and Physical Sciences.
National Council of Teachers of Mathematics
(NCTM). 2000. Principles and Standards for
School Mathematics. Reston: NCTM.
Next Generation Science Standards. 2013.
http://www.nextgenscience.org
/next-generation-science-standards
Soule, Michael E. 1986. “What Is Conservation
Biology?” BioScience 35 (11): 727–34.
Jorge Garcia, jorge.garcia@csuci.edu, teaches at
California State University Channel Islands. He is interested in problem solving and using new techniques to
teach elementary school mathematics. Edited by Terri
Kurz, terri.kurz@asu.edu, who teaches math and
mathematics methodology at Arizona State University,
Polytechnic in Mesa. TCM’s iSTEM (Integrating Science,
Technology, and Engineering in Mathematics) department is the venue for ideas and activities that stimulate
student interest in the integrated fields of science,
technology, engineering, and mathematics (STEM) in
K–grade 6 classrooms. Send submissions of no more
than 1500 words to this department by accessing tcm.
msubmit.net. For detailed submission guidelines for all
departments, go to www.nctm.org/tcmdepartments.
Download one of the free
apps for your smartphone.
Then scan this tag, or go to
www.nctm.org/tcm051,
to access the empty table
and graph templates.
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➺ activity sheet 1
Name___________________________________
A Fibonacci Simple Ecosystem:
Prey and Predator
TABLE 1
One coyote eats two rabbits starting on day 3. There are one and two rabbits on day 1 and
day 2, respectively.
TABLE 2
One coyote eats two rabbits starting on day 4. There are one and two rabbits on day 1 and
day 2, respectively.
From the February 2014 issue of
This content downloaded from 132.203.227.62 on Fri, 20 Nov 2015 15:48:28 UTC
All use subject to JSTOR Terms and Conditions
➺ activity sheet (page 2 of 4)
Name___________________________________
A Fibonacci Simple Ecosystem:
Prey and Predator
TABLE 3
Two coyotes each eat two rabbits starting on day 5. There are one and two rabbits on day 1
and day 2, respectively.
TABLE 4
Model your own ecosystem.
From the February 2014 issue of
This content downloaded from 132.203.227.62 on Fri, 20 Nov 2015 15:48:28 UTC
All use subject to JSTOR Terms and Conditions