Day 3 in math modeling
- Assignment #3: Exercises p. 38, #3a, 3c, 4; Begin working on either
Project 1.2 or 1.3 (your choice), pp. 39-42.
- Discuss standards for grading (5 minutes):
- Discuss homework problem (15 minutes):
- Review:
- four axes
- UPCE
- Stella - for creating simple or complex modeling equations
- Qualitative modeling
- Recurrence/Recursion relations (45 minutes):
- Definition, and order
- Simplest Examples: order 1, linear and affine
- Qualitative modeling: linear response shapes, and
Proposition 1.1, p. 21
(Linear model, exponential solutions)
- Fixed points in the linear case
- Discuss homework problems (15 minutes):
- Exercises p. 38, #1 - induction
- Exercises p. 38, #2 - induction, too
- fixed points in affine case (x = Rx+a)
- linearizing about the fixed point
(or steady state)
- Qualitative modeling: affine response shapes
- Complex Examples: order 1
- Qualitative modeling: Shapes of f(x)
- Fixed points (x=f(x)) more generally
- Stability of fixed points (cobwebbing); Exercise 3d,
p. 38 (10 minutes)
- Qualitative modeling: Response shapes
- Example, order 2: Linear homogeneous with constant coefficients
- Eigenvalues for the linear problem - ultimately "the
bully" (largest in absolute value) dominates.
- Fibonacci as a modeling problem: a pair of rabbits are
put in a field and, if rabbits take a month to become mature and then produce a
new pair every month after that, how many pairs will there be in twelve months
time? (from this
site)
- Plant compartmental model
- Stella, and the hunt for wild shapes
- Start on Project 1.2 or 1.3:
- Formulate compartmental diagram and Stella model
- Begin analysis
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