GMST 570 Modeling Change in Mathematics and Science

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
Occams Razor
Model World
Interpreting and Testing
Real World
Formulating Model World Problem
Model Results
Model
Modeling Diagram (Taken from Mooney and Swift,
page 4)
Mathematical Analysis
4
More examples of models

• The Solar System and Gravity
• From circles and four elements
• To Ptolemys epicycles
• To Keplers three laws
• To Newtons calculus
• To Einsteins 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
• Systems of interacting populations
• Population could be any quantity that changes
• Ways to model this?
• Compartmental diagrams
• Spreadsheet solutions
• Equations
• 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
• 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?
• What are the variables/assumptions 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
• ODE Architect Discrete Tool
• Why look at the line yx?
• 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)
• Ill give you an hour to get somewhere, while I
  come around and help you
• Well spend the last hour discussing and sharing
  our work