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After Calculus I…. Glenn Ledder University of Nebraska-Lincoln gledder@math.unl.edu. 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

University of Nebraska-Lincoln

gledder@math.unl.edu

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

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


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?


MODELS AND DATAnondimensionalization and scaling


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


PROBABILITYconditional


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


For more information:

gledder@math.unl.edu


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