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Teaching Mathematics to Biologists and Biology to Mathematicians

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### Teaching Mathematics to Biologists and Biology to Mathematicians

Gretchen A. Koch

Goucher College

MathFest 2007

Introduction

- Who: Undergraduate students and faculty
- What: Improving quantitative skills of students through combination of biology and mathematics
- When: Any biology or mathematics course
- Simple examples interspersed throughout semester
- Common example as theme for entire semester

How??

- Communication is key
- Talk with colleagues in natural sciences
- Use the same language
- Make the connections obvious
- Example: Why is Calculus I required for many biology and chemistry majors??
- Case studies, ESTEEM, and the BioQUEST way
- Have an open mind and be creative

What is a case study?

- Imaginative story to introduce idea
- Self-discovery with focus
- Ask meaningful questions
- Build on students’ previous knowledge
- Students expand knowledge through research and discussion.
- Assessment

Case Studies – Beware of…

- Clear objectives = easier assessment
- Clear rubric = easier assessment
- Focused questions = easier assessment
- Too much focus = students look for the “right” answer
- Provide some starting resources
- Continue building your database
- Have clear expectations (Communication!)
- Be flexible

Where do I start???

http://bioquest.org/icbl/

C3: Cal, Crabs, and the Chesapeake

- Cal, a Chesapeake crabber, was sitting at the end of the dock, looking forlorn. I approached him and asked, “What’s the matter, Cal?” He replied, “Hon – it’s just not the same anymore. There are fewer and fewer blue crabs in the traps each day. I just don’t know how much longer I can keep the business going. You’re a mathematician – and you always say math is everywhere…where’s the math in this???”

Case Analysis – Use for Discussion

- What is this case about?
- What could be causing the blue crab population to decrease?
- Can we predict what the blue crab population will do?
- Can we find data to show historic trends in the blue crab population?
- How will you answer these questions?

Learning Objectives - Mathematics

- Use different mathematical models to explore the population dynamics
- Linear, exponential, and logistic growth models
- Precalculus level
- Continuous Growth ESTEEM Module
- Predator-prey model
- Calculus, Differential Equations, Numerical Methods
- Two Species ESTEEM Module
- SIR Model
- Calculus, Differential Equations, Numerical Methods
- SIR ESTEEM Module

Learning Objectives - Biology

- Explore the reasons for the decrease in the crab population
- Habitat
- Predators
- Food Sources
- Parasites
- Invasive species
- In field experiments
- Journal reviews of ongoing experiments

Assessment and Evaluation Plan

- Homework questions to demonstrate understanding of use of ESTEEM modules
- Homework questions to demonstrate comprehension of topics presented in ESTEEM modules
- Group presentations of background information
- Exam questions to demonstrate synthesis of mathematical concepts using different examples

Sources

- Blue Crab
- Chesapeake Blue Crab Assessment 2005
- Maryland Sea Grant The Living Chesapeake Coast, Bay & Watershed Issues Blue Crabs

But – what’s the answer??

- Assessments and objectives vary
- Knowledge of tools and structure
- Adopt and adapt

First Growth Model

- Suppose you ask Cal to keep track of the number of crabs he catches for 10 days. He gives you the following:
- Do you see a pattern?

Linear Growth Model

- Simplest model:

where C is the number of crabs on day t, and D is some constant number.

- Questions to ask:
- What is D? Can you describe it in your own words?
- What’s another form for this model?
- Describe what this model means in terms of the crabs.
- Does this model fit the data? Why or why not?
- Is this model realistic?

ESTEEM Time!

- Continuous Growth Module

Summary of Manipulations

- Entered data in yellow areas
- Clicked on “Plots-Size” tab
- Manipulated parameters using sliders until fit looked “right”
- Asked questions about what makes it right

Exponential Growth Model

- Simplest model:

where C is the number of crabs on day t, and r is some constant number.

- Questions to ask:
- What is r? Can you describe it in your own words?
- What’s another form for this model?
- Describe what this model means in terms of the crabs.
- Does this model fit the data? Why or why not?
- Is this model realistic?

ESTEEM Time!

- Documentation
- Continuous Growth Module

Compare the two models…

- Why can the initial population be zero in the linear growth model, but not in the exponential growth model?
- Why do such small changes in r make such a big difference, but it takes large changes in D to show a difference?
- What do these models predict will happen to the number of crabs that Cal catches in the future?

Logistic Growth Model

- Canonical model:

where C is the number of crabs on day t, and r and K are constants.

- Questions to ask:
- What are r and K? Can you describe them in your own words?
- Describe what this model means in terms of the crabs.
- Does this model fit the data? Why or why not?
- Is this model realistic?

Further Analysis

- What does the initial population need to be for each of the three models to fit the data well?
- Why is the logistic model more realistic?
- How did the parameters (D, r, K) affect the models?
- What does each model say about the total capacity of Cal’s traps?
- Do these models give an accurate prediction of the future of the crab population?

Let’s kick it up a notch!

- How do we model the entire crab population?
- According to http://www.chesapeakebay.net/blue_crab.htm, blue crabs are predators of bivalves.
- Cannibalism is correlated to the bivalve population.

Predator-Prey Equations

- Canonical example (Edelstein-Keshet):
- Assumptions (pg 218):
- Unlimited prey growth without predation
- Predators only food source is prey.
- Predator and prey will encounter each other.

Why does multiplication give likelihood of an encounter ??

- Law of Mass Action (Neuhauser)
- Given the following chemical reaction

the rate at which the product AB is produced by colliding molecules of A and B is proportional to the concentrations of the reactants.

- Translation to mathematics
- Rates = derivatives, k is a number
- What about [A] and [B]?

Another version

- Cushing:
- What are the variables? Put them into context.
- What’s the extra term?
- Did the assumptions change?

ESTEEM Two-Species Model

- Isolation (discrete time):
- What kind of growth?
- What are the terms and variables?

ESTEEM Two-Species Model

- Discussion Questions
- What do the terms mean?
- Which species is the predator, which is the prey?
- What other situations could these equations describe?
- Why discrete time?
- For what values of the rate constants does one species inhibit the other? Have no effect? Have a positive effect?
- Can we derive the continuous analogs?

ESTEEM Time!

- Documentation
- Two-Species Module

Summary of Manipulations

- Use sliders to change values of parameters.
- Examine all graphs.
- Columns B and C have formulas for numerical method.

Discussion Questions

- How did one species affect the other?
- What did the different graphs represent?
- Did one species become extinct?
- How can you have 1.25 crabs?
- What would happen if there was a third species? Write a general set of equations (cases as relevant).
- Can you determine the numerical method used?

Simple SIR Model

- Yeargers:
- Susceptible, Infected, Recovered
- Given the above equations, explain the assumptions, variables, and terms.

Connections to Case Study and Beyond

- Possible ideas for research projects
- Parasites and crabs
- Is there a disease affecting the crab population?
- Pick an epidemic, research it, and model it.
- Analytical or numerical solutions
- Make teams of biology majors and math majors.
- ESTEEM module…

SIR ESTEEM Module - Equations

Hosts (S, I, R) are infected by vectors (U, V) that can carry one

of three strains of the virus (i=1, 2, 3).

ESTEEM Time!

- Documentation
- SIR ESTEEM Module
- Red boxes are for user entry.

SIR Module Discussion

- Can you draw a diagram representing the SIR model?
- What are all of the variables and parameters in the SIR model?
- Can you find the continuous analog for the system?
- Can you rewrite the system in matrix form?
- What numerical method was used?
- Why did some values of the parameters work, while others did not?

Conclusion

- Many, many ways to bring biology into the classroom
- Build on students’ intuition and knowledge
- Make obvious connections between ideas
- Don’t be afraid to try something new.
- Experiment and experiment some more!
- Have fun!

Works Consulted/Cited

Texts:

Cushing, J.M. (2004) Differential Equations: An Applied Approach. Pearson Prentice Hall.

Edelstein-Keshet, L. (1988) Mathematical Models in Biology. Birkhäuser.

Neuhauser, C. (2004) Calculus for Biology and Medicine. 2 ed. Pearson Education.

Yeargers, E.K., Shonkwiler, R.W., and J.V. Herod. (1996) An Introduction to the Mathematics of Biology with Computer Algebra Models. Birkhäuser.

Online Sources:

BioQUEST Sources

BioQUEST: http://bioquest.org

ICBL: Investigative Case Based Learning: http://bioquest.org/icbl/

ESTEEM Module Documentation

Continuous Growth Models Documentation (John R. Jungck, Tia Johnson, Anton E. Weisstein, and Joshua Tusin): http://www.bioquest.org/products/files/197_Growth_models.pdf

SIR Model Documentation (Tony Weisstein): http://www.bioquest.org/products/files/196_sirmodel.doc

Two Species Documentation (Tony Weisstein, Rene Salinas, John Jungck): http://www.bioquest.org/products/files/203_TwoSpecies_Model.doc

Blue Crab Resources

Blue Crab: http://www.chesapeakebay.net/blue_crab.htm

Chesapeake Blue Crab Assessment 2005: http://hjort.cbl.umces.edu/crabs/Assessment05.html

Maryland Sea Grant The Living Chesapeake Coast, Bay & Watershed Issues Blue Crabs: http://www.mdsg.umd.edu/issues/chesapeake/blue_crabs/index.html

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