Title
A Food Web (Jo Ann Ellerbrock)
Goals
(1) Students will explore how digraphs can be used to model the food web.
(2) Students will explore using associated matrices to represent the digraph.
(3) Students will formulate and defend a position on an environmental issue.
(4) Students will discuss how the environment is altered in unexpected ways by human actions.
Abstract
This activity asks students to use matrices to evaluate the impact of introducing an insecticide on the food web for the area. Students are asked to evaluate the impact and then report their findings. This activity provides motivation for the discussion of matrix arithmetic associated with digraphs.
Problem Statement
Seemingly small changes in the local ecosystem can cause a chain reaction which affects a larger environment. This activity will allow students to explore how a change in the food web can impact seemingly unaffected animals. (Popular literature usually discusses food "chains," whereas the concept of food web is more relevant to real-world situations.)
Research has verified the relationships in the food web.
Instructor Suggestions
(1) It would be helpful to have a biology teacher give a brief introduction to food web concepts before the students work on the details.
(2) It is suggested that the following three-animal food web be discussed with the entire class before the students do the work in small groups.
The matrix associated with the digraph is
| C | R | M | |
| C | 0 | 1 | 1 |
| R | 0 | 1 | 1 |
| M | 0 | 0 | 0 |
(3) Students need to have been exposed to matrix arithmetic (matrix multiplication and addition) prior to using this application.
(4) Following the discussion of the three-animal food web, its associated matrix, and the matrix operations used to model changes in the food web, break students into small groups to complete the activity sheet.
(5) Students may present and justify their positions either orally or in writing as time permits.
(6) This problem acts as a natural precursor for a problem in Markov chains. An example of such a problem would be:
A vigorous television campaign is conducted to educate residents on the impact of use of the insecticide in the mountain park and urging the use of natural biological predators instead. For each of several weeks, a survey is made and it is found that each week 80% of those favoring using the insecticide continue to favor it and 20% switch to favoring biological predators. It is also found that of those favoring the use of biological predators, 20% change their minds and want the insecticide used instead.
(a) Write the transition matrix, assuming the transition percentages continue to hold for succeeding weeks.
(b) If 20% of the people want the insecticide at the start of the television campaign, what percentage will favor using the insecticide 1 week later? 2 weeks later?
(c) What portion of the people will want insecticide used at the end of the summer. (Find the steady state matrix.)
Materials
"A Food Web" Activity Sheet, Pencils, Paper, Calculators
Time
This activity takes one to three class periods depending on the amount of discussion regarding food webs and the amount of time allocated to student presentations.
Mathematics Concepts
Discrete Mathematics Concepts
Graphs, transition matrices, Extension - Markov Chains
Related Mathematics Concepts
Multiplying and adding matrices, mathematical modeling
NCTM Standards Addressed
Problem Solving, Reasoning, Connections, Algebra, Discrete Mathematics.
Colorado Model Content Standards Addressed
Algebraic Techniques (2), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into a traditional Algebra II class or any other class where matrix arithmetic is being used. With modifications to accommodate calculations done with matrices, it could also be used to introduce using matrices in an introductory class such as Algebra I.
Further Investigation
Analyze the environmental effect of a pollutant that kills the trout, frogs, and crayfish, but does not directly affect the insects.
Variations/Comments
This activity is based on a modeling exercise presented in Mathematical Modeling in the Secondary School Curriculum.
References/Resources
Swetz, Frank, & Hartzler, J. S. (1991). Mathematical Modeling in the Secondary School Curriculum. Reston, VA: National Council of Teachers of Mathematics
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete mathematics Through Applications. New York: W.H. Freeman and Company.