1. After Calculus I… Glenn Ledder University of Nebraska-Lincoln gledder@math.unl.edu Funded by the National Science Foundation

2. The Status Quo Biology majors Calculus I (5 credits) Baby Stats (3 credits) Biochemistry majors Calculus I (5 credits) Calculus II (5 credits) No statistics No partial derivatives

3. Design Requirements Calculus I + a second course Five credits each Biologists want – Probability distributions – Dynamical systems Biochemists want – Statistics – Chemical Kinetics

4. My “Brilliant” Insight The second course should NOT be Calculus II.

5. My “Brilliant” Insight The second course should NOT be Calculus II. Instead: Mathematical Methods for Biology and Medicine

6. Overview 1.Calculus (≈5%) 2.Models and Data (≈25%) 3.Probability (≈30%) 4.Dynamical Systems (≈40%)

7. CALCULUS the derivative Slope of y=f(x) is f´(x) Rate of increase of f(t) is Gradient of f(x) with respect to x is

8. CALCULUS the definite integral Area under y=f(x) is Accumulation of F over time is Aggregation of F in space is

9. CALCULUS the partial derivative For fixed y, let F(x)=f(x;y). Gradient of f(x,y) with respect to x is

10. MODELS AND DATA mathematical models Independent Variable(s) Dependent Variable(s) Equations Narrow View

11. MODELS AND DATA mathematical models Independent Variable(s) Dependent Variable(s) Equations Narrow View Parameters Behavior Broad View (see Ledder, PRIMUS, Feb 2008)

12. MODELS AND DATA descriptive statistics Histograms Population mean Population standard deviation Standard deviation for samples of size n

13. MODELS AND DATA fitting parameters to data Linear least squares – For y=b+mx, set X=x-x̄, Y=y-ȳ – Minimize Nonlinear least squares – Minimize – Solve numerically

14. MODELS AND DATA constructing models Empirical modeling Statistical modeling – Trade-off between accuracy and complexity mediated by AICc

15. MODELS AND DATA constructing models Empirical modeling Statistical modeling – Trade-off between accuracy and complexity mediated by AICc Mechanistic modeling – Absolute and relative rates of change – Dimensional reasoning

16. Example: resource consumption

17. Time is split between searching and feeding S – food availability R(S) – overall feeding rate a – search speed C – feeding rate while eating

18. Example: resource consumption Time is split between searching and feeding S – food availability R(S) – overall feeding rate a – search speed C – feeding rate while eating ------- = --------- · --------- · ------- food total t search t total t space search t food space

19. Example: resource consumption Time is split between searching and feeding S – food availability R(S) – overall feeding rate a – search speed C – feeding rate while eating ------- = --------- · --------- · ------- food total t search t total t space search t food space search t total t feed t total t --------- = 1 – -------

20. MODELS AND DATA characterizing models What does each parameter mean? What behaviors are possible? How does the parameter space map to the behavior space?

21. MODELS AND DATA nondimensionalization and scaling

22. PROBABILITY distributions Discrete distributions – Distribution functions – Mean and variance – Emphasis on computer experiments (see Lock and Lock, PRIMUS, Feb 2008)

23. PROBABILITY distributions Discrete distributions – Distribution functions – Mean and variance – Emphasis on computer experiments (see Lock and Lock, PRIMUS, Feb 2008) Continuous distributions – Visualize with histograms – Probability = Area

24. PROBABILITY distributions frequency width --------------- frequency width --------------- y = frequency/width means area stays fixed at 1.

25. PROBABILITY independence Identically-distributed – 1 expt: mean μ, variance σ 2, any type – n expts: mean nμ, variance nσ 2, →normal

26. PROBABILITY independence Identically-distributed – 1 expt: mean μ, variance σ 2, any type – n expts: mean nμ, variance nσ 2, →normal Not identically-distributed –

27. PROBABILITY conditional

28. DYNAMICAL SYSTEMS 1-variable Discrete – Simulations – Cobweb diagrams – Stability Continuous – Simulations – Phase line – Stability

29. Simulations Matrix form Linear algebra primer – Dominant eigenvalue – Eigenvector for dominant eigenvalue Long-term behavior (linear) – Stable growth rate – Stable age distribution DYNAMICAL SYSTEMS discrete multivariable

30. Phase plane Nullclines Linear stability Nonlinear stability Limit cycles DYNAMICAL SYSTEMS continuous multivariable

31. For more information: gledder@math.unl.edu