After Calculus I…

Glenn LedderUniversity of Nebraska-Lincoln

gledder@math.unl.edu

Funded by the National Science Foundation

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

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

1. Calculus (≈5%)2. Models and Data (≈25%)3. Probability (≈30%)4. 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 dt

df

dx

df

CALCULUSthe definite integral

• Area under y=f(x) is

• Accumulation of F over time is

• Aggregation of F in space is

b

adxxf )(

b

adttF )('

b

adxxF )('

CALCULUSthe partial derivative

• For fixed y, let F(x)=f(x;y).

• Gradient of f(x,y) with respect to x is

dx

dF

x

f

x

f

MODELS AND DATAmathematical models

IndependentVariable(s)

DependentVariable(s)

Equations

Narrow View

MODELS AND DATAmathematical models

IndependentVariable(s)

DependentVariable(s)

Equations

Narrow View

Parameters Behavior

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-x, Y=y-y– Minimize

• Nonlinear least squares– Minimize – Solve numerically

n

iii mXYmF

1

2)()(

n

iii mxfymF

1

2)];([)(

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

0

5

10

15

20

25

30

35

0 50 100 150

food available

cons

umpti

on ra

te

Example: resource consumption

• Time is split between searching and feeding

S – food availability R(S) – overall feeding ratea – search speed C – feeding rate while eating

Example: resource consumption

• Time is split between searching and feeding

S – food availability R(S) – overall feeding ratea – search speed C – feeding rate while eating

------- = --------- · --------- · ------- foodtotal t

search ttotal t

spacesearch t

foodspace

SaCRfSR ),()(

Example: resource consumption

• Time is split between searching and feeding

S – food availability R(S) – overall feeding ratea – search speed C – feeding rate while eating

------- = --------- · --------- · ------- foodtotal t

search ttotal t

spacesearch t

foodspace

SaCRfSR ),()( search ttotal t

feed ttotal t--------- = 1 – -------

C

RCRf 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

frequencywidth

--------------- frequencywidth

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

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– )()(),( BPAPBAP

PROBABILITYconditional

1)()(

)()()(

)()()(

C

CCCC

C

BPBP

APBAPBAP

APBAPBAP

CA

A

CBB

)|()()( BAPBPBAP

DYNAMICAL SYSTEMS1-variable

• Discrete– Simulations–Cobweb diagrams– Stability

• Continuous– Simulations–Phase line– Stability

• Simulations• Matrix form• Linear algebra primer– Dominant eigenvalue– Eigenvector for dominant

eigenvalue

• Long-term behavior (linear)– Stable growth rate– Stable age distribution

DYNAMICAL SYSTEMSdiscrete multivariable

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

DYNAMICAL SYSTEMScontinuous multivariable

For more information:gledder@math.unl.edu