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### After Calculus I…

Glenn Ledder

University of Nebraska-Lincoln

Funded by the National Science Foundation

The Status Quo

Biology majors

Biochemistry majors

Calculus I (5 credits)

Calculus II (5 credits)

No statistics

No partial derivatives

- Calculus I (5 credits)
- Baby Stats (3 credits)

Design Requirements

- Calculus I + a second course
- Five credits each
- Biologists want
- Probability distributions
- Dynamical systems
- Biochemists want
- Statistics
- Chemical Kinetics

My “Brilliant” Insight

- The second course should NOT be Calculus II.

My “Brilliant” Insight

- The second course should NOT be Calculus II.
- Instead: Mathematical Methods for Biology and Medicine

Overview

- Calculus (≈5%)
- Models and Data (≈25%)
- Probability (≈30%)
- Dynamical Systems (≈40%)

CALCULUSthe 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

CALCULUSthe definite integral

- Area under y=f(x) is
- Accumulation of F over time is
- Aggregation of F in space is

CALCULUSthe partial derivative

- For fixed y, let F(x)=f(x;y).
- Gradient of f(x,y) with respect to x is

MODELS AND DATAmathematical models

Equations

Independent

Variable(s)

Dependent

Variable(s)

Parameters

Behavior

Narrow View

Broad View

(see Ledder, PRIMUS, Feb 2008)

MODELS AND DATAdescriptive statistics

- Histograms
- Population mean
- Population standard deviation
- Standard deviation for samples of size n

MODELS AND DATAfitting parameters to data

- Linear least squares
- For y=b+mx, set X=x-x̄, Y=y-ȳ
- Minimize
- Nonlinear least squares
- Minimize
- Solve numerically

MODELS AND DATAconstructing models

- Empirical modeling
- Statistical modeling
- Trade-off between accuracy and complexity mediated by AICc

MODELS AND DATAconstructing models

- Empirical modeling
- Statistical modeling
- Trade-off between accuracy and complexity mediated by AICc
- Mechanistic modeling
- Absolute and relative rates of change
- Dimensional reasoning

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

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

------- = --------- · --------- · -------

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 – -------

MODELS AND DATAcharacterizing models

- What does each parameter mean?
- What behaviors are possible?
- How does the parameter space map to the behavior space?

PROBABILITYdistributions

- Discrete distributions
- Distribution functions
- Mean and variance
- Emphasis on computer experiments
- (see Lock and Lock, PRIMUS, Feb 2008)

PROBABILITYdistributions

- 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

PROBABILITYdistributions

frequency

width

frequency

width

---------------

---------------

y = frequency/width means area stays fixed at 1.

PROBABILITYindependence

- Identically-distributed
- 1 expt: mean μ, variance σ2, any type
- n expts: mean nμ, variance nσ2, →normal

PROBABILITYindependence

- Identically-distributed
- 1 expt: mean μ, variance σ2, any type
- n expts: mean nμ, variance nσ2, →normal
- Not identically-distributed

DynamicalSystems1-variable

- Discrete
- Simulations
- Cobweb diagrams
- Stability

- Continuous
- Simulations
- Phase line
- Stability

DynamicalSystemsdiscrete multivariable

- Simulations
- Matrix form
- Linear algebra primer
- Dominant eigenvalue
- Eigenvector for dominant eigenvalue
- Long-term behavior (linear)
- Stable growth rate
- Stable age distribution

DynamicalSystemscontinuous multivariable

- Phase plane
- Nullclines
- Linear stability
- Nonlinear stability
- Limit cycles

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