After Calculus I…

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After Calculus I…. Glenn Ledder University of Nebraska-Lincoln [email protected] 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.

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### After Calculus I…

Glenn Ledder

[email protected]

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)

Narrow View

MODELS AND DATAmathematical models

Equations

Independent

Variable(s)

Dependent

Variable(s)

Parameters

Behavior

Narrow View

(see Ledder, PRIMUS, Feb 2008)

• 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