1. GMST 570 Modeling Change in Mathematics and Science Day Two: Monday, July 15, 2002 Discrete Models

2. Overview of Class  Discussion and sharing of homework  The modeling process as a cycle  Classes of models  Discrete models: Hands-on example  Discrete models: Spreadsheets  Discrete models: Mathematics involved  Wrap-up

3. Basic modeling process Real World Model World Model Occam’s Razor Interpreting and Testing Formulating Model World Problem Model Results Mathematical Analysis Modeling Diagram (Taken from Mooney and Swift, page 4)

4. More examples of models  The Solar System and Gravity From circles and four elements… From circles and four elements… To Ptolemy’s epicycles… To Ptolemy’s epicycles… To Kepler’s three laws… To Kepler’s three laws… To Newton’s calculus… To Newton’s calculus… To Einstein’s relativity… To Einstein’s relativity…

5. Some classes of models Either this…  Time dependent  Deterministic  Empirical  Continuous  Extrapolation  Qualitative Or this…  Steady state  Stochastic  Theoretical  Discrete  Interpolation  Quantitative

6. Chapter 1 Overview  Topic: Discrete Models of Population Growth  What to model? Single “population” Single “population” Systems of interacting populations Systems of interacting populations Population could be any quantity that changes Population could be any quantity that changes  Ways to model this? Compartmental diagrams Compartmental diagrams Spreadsheet solutions Spreadsheet solutions Equations Equations Graphs Graphs  Equations Difference equations Recurrence relations  Things to study Changes in assumptions Graphs of results Closed-form solutions Fixed points & stability  Population changes Hacking Births Deaths

7. Hands on squirrels  Groups of 3-4  Every group needs squirrels in a box  Start with 10 squirrels at time 0  For every 3 entire squirrels in the population, add one new squirrel  Repeat for about 10 time steps  Refinements? (Age, death, etc.)

8. Sandhill Crane Population Model  Compartmental diagram  Spreadsheet model 100 cranes 100 cranes Three growth rates: 0.0194, -0.0324, -0.3820 Three growth rates: 0.0194, -0.0324, -0.3820  Equations for this  Check qualitative behavior  Modify the model diagram

9. Discussion  What is change?  What do your students study that will change over time? Is it a discrete process? Is it a discrete process? What are the variables/assumptions to consider? What are the variables/assumptions to consider? What are the constraints/limitations to consider? What are the constraints/limitations to consider?

10. Interacting Populations  In what ways can two populations interact?  How would the compartmental diagrams look for these cases?  How will that affect the spreadsheet model?  What are the long-term possibilities for two interacting populations?  Are we limited to just two populations?

11. Cobweb Diagrams  Tools for studying Graph paper Graph paper ODE Architect Discrete Tool ODE Architect Discrete Tool  Why look at the line y=x?  How does this help us study stability of a solution? How can we locate fixed points?  Try sketching a graph that has two fixed points. Try: two attracting and one repelling fixed points. What would these tell you about the populations?

12. Modeling Practice  Work in pairs  You will work on one of the projects at the end of the chapter (pp. 38-46)  I’ll give you an hour to get somewhere, while I come around and help you  We’ll spend the last hour discussing and sharing our work