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 I (5 credits)
• Baby Stats (3 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%)
CALCULUS
the derivative
• Slope of y=f(x) is f´(x)
• Rate of increase of f(t) is
df
dt
• Gradient of f(x) with respect to x is
df
dx
CALCULUS
the definite integral
b
• Area under y=f(x) is a
f ( x) dx
b
• Accumulation of F over time is a F ' (t ) dt
b
• Aggregation of F in space is a F ' ( x) dx
CALCULUS
the partial derivative
• For fixed y, let F(x)=f(x;y).
f dF

x dx
• Gradient of f(x,y) with respect to x is
f
x
MODELS AND DATA
mathematical models
Independent
Variable(s)
Equations
Narrow View
Dependent
Variable(s)
MODELS AND DATA
mathematical models
Parameters
Independent
Variable(s)
Equations
Dependent
Variable(s)
Narrow View
Broad View
(see Ledder, PRIMUS, Feb 2008)
Behavior
MODELS AND DATA
descriptive statistics
•
•
•
•
Histograms
Population mean
Population standard deviation
Standard deviation for samples of size n
MODELS AND DATA
fitting parameters to data
• Linear least squares
– For y=b+mx, set X=x-x̄, Y=y-ȳ
n
– Minimize F (m)   (Yi  mX i ) 2
i 1
• Nonlinear least squares
n
– Minimize F (m)   [ yi  f ( xi ; m)]2
i 1
– Solve numerically
MODELS AND DATA
constructing models
• Empirical modeling
• Statistical modeling
– Trade-off between accuracy and
complexity mediated by AICc
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
Example: resource consumption
consumption rate
35
30
25
20
15
10
5
0
0
50
100
food available
150
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 search t space food
------= --------- · --------- · ------total t total t search t space
R ( S )  f ( R, C )  a  S
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 search t space food
------= --------- · --------- · ------total t total t search t space
R ( S )  f ( R, C )  a  S
search t
feed t
--------=
1
–
------total t
total t
R
f ( R, C )  1 
C
MODELS AND DATA
characterizing models
• What does each parameter mean?
• What behaviors are possible?
• How does the parameter space map to
the behavior space?
MODELS AND DATA
nondimensionalization and scaling
PROBABILITY
distributions
• Discrete distributions
– Distribution functions
– Mean and variance
– Emphasis on computer experiments
• (see Lock and Lock, PRIMUS, Feb 2008)
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
PROBABILITY
distributions
frequency
width
---------------
frequency
width
---------------
y = frequency/width means area stays fixed at 1.
PROBABILITY
independence
• Identically-distributed
– 1 expt: mean μ, variance σ2, any type
– n expts: mean nμ, variance nσ2, →normal
PROBABILITY
independence
• Identically-distributed
– 1 expt: mean μ, variance σ2, any type
– n expts: mean nμ, variance nσ2, →normal
• Not identically-distributed
–
P( A, B)  P( A)  P( B)
PROBABILITY
conditional
B
A
C
A
B
C
C
P( A  B)
P( A  B )
P( A)
C
C
C
C
P( A  B) P( A  B ) P( A )
C
P( B)
P( B )
1
P( A  B)  P( B)  P( A | B)
DYNAMICAL SYSTEMS
1-variable
• Discrete
– Simulations
– Cobweb diagrams
– Stability
• Continuous
– Simulations
– Phase line
– Stability
DYNAMICAL SYSTEMS
discrete multivariable
• Simulations
• Matrix form
• Linear algebra primer
– Dominant eigenvalue
– Eigenvector for dominant eigenvalue
• Long-term behavior (linear)
– Stable growth rate
– Stable age distribution
DYNAMICAL SYSTEMS
continuous multivariable
•
•
•
•
•
Phase plane
Nullclines
Linear stability
Nonlinear stability
Limit cycles
For more information:
gledder@math.unl.edu