Teaching Mathematics to Biologists and Biology to Mathematicians

1. Teaching Mathematics to Biologists and Biology to Mathematicians Gretchen A. Koch Goucher College MathFest 2007

2. 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

3. 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

5. 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

6. 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

7. Where do I start??? http://bioquest.org/icbl/

9. 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???”

10. 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?

11. A Good Starting Place for Students

12. 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

13. 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

14. 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

15. Sources • Blue Crab • Chesapeake Blue Crab Assessment 2005 • Maryland Sea Grant The Living Chesapeake Coast, Bay & Watershed Issues Blue Crabs

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

26. But – what’s the answer?? • Assessments and objectives vary • Knowledge of tools and structure • Adopt and adapt

27. Continuous Growth Models Module

28. 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?

29. 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?

30. ESTEEM Time! • Continuous Growth Module

31. 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

32. 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?

33. ESTEEM Time! • Documentation • Continuous Growth Module

34. 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?

35. 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?

36. 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?

37. 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.

38. 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.

39. Put it into context!

40. 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]?

41. Another version • Cushing: • What are the variables? Put them into context. • What’s the extra term? • Did the assumptions change?

42. ESTEEM Two-Species Model • Isolation (discrete time): • What kind of growth? • What are the terms and variables?

43. 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?

44. ESTEEM Time! • Documentation • Two-Species Module

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

46. 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?

47. Simple SIR Model • Yeargers: • Susceptible, Infected, Recovered • Given the above equations, explain the assumptions, variables, and terms.

48. 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…

49. 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).

50. ESTEEM Time! • Documentation • SIR ESTEEM Module • Red boxes are for user entry.