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A Terminal Post-Calculus-I
Mathematics Course for
Biology Students
Glenn Ledder
Department of Mathematics
University of Nebraska-Lincoln
[email protected]
funded by NSF grant DUE 0536508
My Students
• From Calculus I:
– Biochemistry majors
– Pre-medicine majors
– Biology majors
• From Business Calculus:
– Natural Resources majors
• Took Calculus I in a past life:
– Biology and Agronomy graduate students
My Course Format
• 15 weeks
• 5 x 50-minute periods each week
• Computer lab access as needed
– We use the lab an average of 2 x per week
– I use R, which is popular among biologists
Formatting Note
The rest of the talk is lists of topics, with
comments and examples as needed:
Topics in blue are elaborated on 1 or
more additional slides.
Topics in black aren’t. (I have little to add to
what is readily available elsewhere.)
Outline of Topics
1. Mathematical Modeling
2. “Review” of Calculus
(2-3 weeks)
(1 week)
3. Probability
(4-5 weeks)
4. Dynamical Systems
(5 weeks)
5. Student Presentations
(1 week)
Unexpected Difficulties
(1 week)
1. MATHEMATICAL MODELING
• Functions with Parameters
• Concepts of Modeling
• Fitting Models to Data
• Empirical/Statistical Modeling
• Mechanistic Modeling
1. MATHEMATICAL MODELING
Functions with Parameters
• Parameter: a quantity in a mathematical
model that can vary over some range, but takes
a specific value in any instance of the model
• Perform algebraic manipulations on functions
with parameters.
• Identify the mathematical significance of a
parameter.
• Graph functions with parameters.
Functions with Parameters
y = e-kt
y = x3 −2x2 +bx
The half-life is
½ = e-kT,
or kT=ln2
Parameters can
change the
qualitative behavior.
Concepts of Modeling
• The best models are valid or useful, not
correct or true.
• Mathematics can determine the properties
of models, but not the validity. (data)
• Models can be analyzed in general;
simulations illustrate instances of a model.
• The same model can take different
symbolic forms (ex: dimensionless forms).
1. MATHEMATICAL MODELING
Fitting Models to Data
• Fit the models
Y = mX, y = b + mx, z = Ae-kt
using linear least squares.
• In what sense are the results “best”?
Fitting Models to Data
• The least squares fit for m in Y = mX is the
vertex of the quadratic function
F(m) = (∑X2) m2 − 2 (∑XY) m +(∑Y2) .
• The least squares fit for b and m in
y = b + mx comes from fitting Y = mX to
X = x – x̄, Y = y - ȳ
(We assume the best line goes through the mean
of the data.)
1. MATHEMATICAL MODELING
Empirical/Statistical Modeling
• Explain where empirical models come
from. (looking at graphs of data)
• Use AICc (corrected Akaike Information
Criterion) to compare statistical validity of
models.
Empirical/Statistical Modeling
The odd-numbered points were used to fit a line
and a quartic polynomial (with 0 error). But the
even-numbered points don’t fit the quartic at all.
• Measured data comprise only 0% of the points on
a curve. Complex models are unforgiving
of small measuring errors.
1. MATHEMATICAL MODELING
Mechanistic Modeling
• Discuss the relationship between real
biology, a conceptual model, and a
mathematical model. (Ledder, PRIMUS 2008)
• Derive the Monod growth function (Holling II).
• Use linear least squares to approximately fit
models of form y = mf ( x; p) to data from
BUGBOX-predator.
Mechanistic Modeling
Fitting y = mf ( x; p):
1. Let ti = f (xi; p) for any given p.
2. Then y = mt with data for t and y.
3. Define G(p) by
G( p)  min
m
 ( y  mt ) 
2
i
4. Best p is the minimum of G.
i
2. “REVIEW” OF CALCULUS
•
•
•
•
The derivative as the slope of the graph.
The definite integral as accumulation in
time, space, or “structure.”
Calculating derivatives.
Calculating elementary definite integrals by
the fundamental theorem (and substitution).
Approximating definite integrals.
Finding local and global extrema.
•
Everything with parameters!
•
•
Demographics / Population Growth
Let l(x) be the probability of survival to age x.
Let m(x) be the rate of production of offspring for
parents of age x.
Let r be the population growth rate.
Let B(t) be the total birth rate.
How do l and m determine B (and r)?
1. The birth rate should increase exponentially with
rate r. (it has to grow like the population)
2. The birth rate can be computed by adding up the
births to parents of different ages.
Demographics / Population Growth
B(t  x) dx
Population of age x if no deaths:
B(t  x)l ( x) dx
Actual population of age x:
Birth rate for parents of age x: B(t  x)l ( x)m( x) dx
Total birth rate at time t:

B(t )   B(t  x)l ( x)m( x) dx
0
Total birth rate at time t:
Euler equation:
B(t )  B(0) ert

1  e
0
 rx
l ( x)m( x) dx
3. PROBABILITY
•
•
•
•
•
•
•
Characterizing Data
Basic Concepts
Discrete Distributions
Continuous Distributions
Distributions of Sample Means
Estimating Parameters
Conditional Probability
Distributions of Sample Means
Frequency histograms for sample
means from a geometric distribution
(p=0.25), with n = 4, 16, 64, and ∞
4. DYNAMICAL VARIABLES
• Discrete Population Models
Example: Genetics and Evolution
• Continuous Population Models
Example: Resource Management
• Cobweb Plots
• The Phase Line
• Stability Analysis
Genetics and Evolution
Sickle cell anemia biology:
• Everyone has a pair of genes (each either
A or a) at the sickle cell locus:
– AA: vulnerable to malaria
– Aa: protected from malaria
– aa: sickle cell anemia
• Babies get A from an AA parent and
either A or a from an Aa parent.
Let p by the prevalence of A.
Let q=1-p be the prevalence of a.
Let m be the malaria mortality.
Genotype
AA
Aa
aa
Frequency
p2
2pq
q2
1-m
1
0
(1-m) p2
2pq
0
Fitness
Next Generation
The next generation has 2 pq of a and
2(1-m) p2 + 2 pq of A:
2 pt qt
qt
qt 1 

2
2(1  m) pt  4 pt qt (1  m)(1  qt )  2qt
Resource Management
Let X be the biomass of resources.
Let K be the environmental capacity.
Let C be the number of consumers.
Let G(X) be the consumption per consumer.
dX
 X
 R X 1    C G ( X )
dT
 K
• Holling type 3 consumption
– Saturation and alternative resource
2
QX
G( X )  2
A X2
Q
0.75Q
G 0.5Q
0.25Q
0
0
A
2A
X
3A
4A
Dimensionless Version
t
K
CQ
X  Ax, T  , k  , c 
R
A
RA
1  x 
dx
x 
 cx  1   
2
dt
c
k
1

x

 

k represents the environmental capacity.
c represents the number of consumers.
4. DISCRETE DYNAMICAL
SYSTEMS
• Discrete Linear Models
Example: Structured Population
Dynamics
• Matrix Algebra Primer
• Eigenvalues and Eigenvectors
• Theoretical Results
Presenting Bugbox-population, a real biology
lab for a virtual world.
http://www.math.unl.edu/~gledder1/BUGBOX/
Boxbugs are simpler than real insects:
– They don’t move.
– Development rate is chosen by the experimenter.
– Each life stage has a distinctive appearance.
larva
pupa
adult
• Boxbugs progress from larva to pupa to adult.
• All boxbugs are female.
• Larva are born adjacent to their mother.
Structured Population Dynamics
The final “bugbox” model:
Let Lt be the number of larvae at time t.
Let Pt be the number of juveniles at time t.
Let At be the number of adults at time t.
Lt+1 = sLt
+ fAt
Pt+1 = pLt
At+1 =
Pt + aAt
Computer Simulation Results
A plot of Xt/Xt-1 shows
that all variables tend to
a constant growth rate λ
The ratios Lt:At
and Pt:At tend to
constant values.
4. CONTINUOUS DYNAMICAL
SYSTEMS
• Continuous Models
Example: Pharmacokinetics
Example: Michaelis-Menten Kinetics
• The Phase Plane
• Stability for Linear Systems
• Stability for Nonlinear Systems
Pharmacokinetics
Q(t)
blood
x(t)
k1 x
k2 y
tissues
y(t)
rx
x′ = Q(t) – (k1+r) x + k2 y
y′ = k1 x – k2 y
References
• PRIMUS 18(1), 2008
– R.H. Lock and P.F. Lock, Introducing statistical inference to
biology students through bootstrapping and randomization
• Teaching statistics through discovery
– T.D. Comar, The integration of biology into calculus courses
• Demographics, genetics
– L.J. Heyer, A mathematical optimization problem in
bioinformatics
• Excellent introductory problem in sequence alignment
– G. Ledder, An experimental approach to mathematical
modeling in biology
• Modeling, theory and pedagogy
• Britton (Springer)
• Cobweb plots
• Brauer and Castillo-Chavez (Springer)
• Resource management