1. A Terminal Post-Calculus-I Mathematics Course for Biology Students
Glenn Ledder
Department of Mathematics
University of Nebraska-Lincoln
funded by NSF grant DUE

2. My Students From Calculus I: From Business Calculus:
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

3. 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

4. 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.)

5. Outline of Topics Mathematical Modeling (2-3 weeks)
“Review” of Calculus (1 week)
Probability (4-5 weeks)
Dynamical Systems (5 weeks)
Student Presentations (1 week)
Unexpected Difficulties (1 week)

6. 1. Mathematical Modeling
Functions with Parameters
Concepts of Modeling
Fitting Models to Data
Empirical/Statistical Modeling
Mechanistic Modeling

7. 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.

8. Functions with Parameters
y = e-kt
y = x3 − 2x2 + bx
The half-life is
½ = e-kT,
or kT = ln 2
Parameters can change the qualitative behavior.

9. 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).

10. 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”?

11. F(m) = (∑X2) m2 − 2 (∑XY) m + (∑Y2) .
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.)

12. 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.

13. 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.

14. 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 = m f ( x; p) to data from BUGBOX-predator.

15. Mechanistic Modeling Fitting y = m f ( x; p):
Let ti = f (xi; p) for any given p.
Then y = mt with data for t and y.
Define G(p) by
Best p is the minimum of G.

16. 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!

17. 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)?
The birth rate should increase exponentially with rate r. (it has to grow like the population)
The birth rate can be computed by adding up the births to parents of different ages.

18. Demographics / Population Growth
Population of age x if no deaths: Actual population of age x: Birth rate for parents of age x: Total birth rate at time t: Euler equation:

19. 3. Probability Characterizing Data Basic Concepts
Discrete Distributions
Continuous Distributions
Distributions of Sample Means
Estimating Parameters
Conditional Probability

20. Distributions of Sample Means
Frequency histograms for sample means from a geometric distribution (p=0.25), with n = 4, 16, 64, and ∞

21. 4. Dynamical Variables Discrete Population Models
Example: Genetics and Evolution
Continuous Population Models
Example: Resource Management
Cobweb Plots
The Phase Line
Stability Analysis

22. 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.

23. The next generation has 2 pq of a and 2(1-m) p2 + 2 pq of A:
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
Fitness
1-m 1
Next Generation
(1-m) p2
The next generation has 2 pq of a and
2(1-m) p2 + 2 pq of A:

24. 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.

25. Holling type 3 consumption
Saturation and alternative resource

26. Dimensionless Version
k represents the environmental capacity. c represents the number of consumers.

27. 4. Discrete Dynamical Systems
Discrete Linear Models
Example: Structured Population Dynamics
Matrix Algebra Primer
Eigenvalues and Eigenvectors
Theoretical Results

28. Presenting Bugbox-population, a real biology lab for a virtual world.
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.

29. 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 = s Lt f At
Pt+1 = p Lt
At+1 = Pt + a At

30. 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.

31. 4. Continuous Dynamical Systems
Continuous Models
Example: Pharmacokinetics
Example: Michaelis-Menten Kinetics
The Phase Plane
Stability for Linear Systems
Stability for Nonlinear Systems

32. Pharmacokinetics Q(t) k1 x blood tissues k2 y x(t) y(t) r x
x′ = Q(t) – (k1+r) x + k2 y
y′ = k1 x – k2 y

33. References PRIMUS 18(1), 2008 Britton (Springer)
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