Instructor: Mette S Olufsen

Office: SAS 3216

Office Hours: By appointment via email.

Phone Number: 515-2678

Email address: msolufse@ncsu.edu

This
course provides an introduction to modeling and analysis of
dynamic biological systems. We focus on analysis of models that
employ ordinary differential equations (ODEs). The aim is to
develop an appreciation for the formulation, analysis,
interpretation and criticism of dynamic mathematical models in
biology.

The course
will develop the theory of ordinary differential equations,
worrying less about the more esoteric mathematical aspects of
the theory than would be the case in an applied math course on
ODEs, and instead focusing on biological motivation and
interpretation.

By the end of the course you should be able to construct, analyze, interpret and criticize ordinary differential equation models of biological systems. You should be able to use appropriate analytic, geometric and numerical techniques to study a particular model. You should be able to use computer packages (such as Maple or Matlab) to facilitate goal. You should also have an appreciation of the broad range of behaviors that these models can exhibit and how model behavior can change (for instance as a parameter is varied).

Although
this is the first course in a two semester sequence, BMA 771 can
be taken as a stand-alone course. BMA 772 (Biomathematics II)
goes on to look at stochastic models (ones that incorporate
randomness). Beyond this, BMA 773 also looks at stochastic
models and BMA 774 considers partial differential equation
models (ones where there is more than one independent variable,
such as space and time).

**Text: ***Nonlinear Dynamics and
Chaos* by Steven Strogatz. Focus will cover the first
8 chapters, this book focuses mainly on the analysis of models,
rather than their formulation and criticism. Therefore, our
emphasis will be slightly different in places. Supplemental
material will be given in the form of handouts or reserve
reading. Computer tools will be used, e.g. Matlab, Maple,
Mathematica, and *XXPAUT***.**

F**ormat:**
During the course a number of smaller and larger homework
projects will be assigned. Students should be prepared to
discuss results of homework problems on the board or using
electronic media in class. While homework problems can be worked
out in a group, everybody should be able to discuss the solution
to a given homework. In addition to homework problems, one
in-class (mid-term) and one take-home (final) test will be
given. Results of the take-home test should be typed up and
handed in electronically.

**Evaluation criteria:** The course will be
graded based on the work performed in homework (50% of the final
grade) and in the one-two tests (midterm 20% and final 30%).

**Prerequisites:
**This course is suitable for grad students and undergrads,
depending on your math background. You are expected to have
knowledge of calculus corresponding to two semesters
courses (e.g. MA 141/241 or MA 131/231). In addition you should
have basic knowledge of arithmetic involving complex numbers,
and some knowledge of linear algebra would be useful (we will
need to work with eigenvalues and eigenvectors of matrices), but
we shall review the material we need. Note: A course in
differential equations is not required; BMA 771 is an
introductory differential equations course.

Homework: I encourage you to work with others on the
homework, but remember - if you work with someone else, put
all names on the paper for a common grade. If computers are
used for solving aspects of an homework problem (e.g., Maple,
Matlab, XPP, or Mathematica) you must indicate so on the
homework. Nicely commented computer code should be attached as
part of the homework. Points will be deducted for homework
handed in late - graded homework can be reworked for some
extra credit. Some homework problems will be discussed in
class, and I will call on you to go through your solution on
the board. For homework, anyone in the group can be called to
discuss the solution of any problem.

2. Course policies

3. Modeling

4. Introduction to eigenvalues and eigenvectors

5. Notes: Introduction to eigenvalues and eigenvectors

6. Phase portraits pplane: http://math.rice.edu/~dfield/dfpp.html

7. Numerical solutions of ODEs: Excel example, Matlab ex 1, Matlab ex 2

8. Runge Kutta methods

9. Bifurcation plots in matlab: Ex dx = r - x^2, Ex dx = r - x^2, Ex dx = r -x - exp(-x)

10. Maple code for sketching solutions and implicit 3d plots (spruce budworm)

11. Matlab for spruce budworm

12. Syncrhonous fireflies Smokey Mountains, video 1, video 2, video 3

13. Notes from Edelstein-Keshet 2D system

14. Notes from Edelstein-Keshet Chemostat model

15. Chemostat code

16. Van der Pol oscillator code

17. Test limit cycle code

18. Population models

19. Oscillating population models

20. Neural modeling

21. FitzHugh Nagumo Lecture

22. FtizHugh Nagumo Matlab

23. Hopf bifurcation

24. Neural network

X. Selkov glycolysis paper

X. Glycolysis: http://www.emc.maricopa.edu/faculty/farabee/biobk/biobookglyc.html#Glycolysis

X. JSIM: http://www.physiome.org/jsim/models/webmodel/NSR/Selkov/

X. Glycolysis process

X. Bifurcations

X. Matlab supercritical Hopf bifurcation

Project report due 12/12 before 5pm

1: Identify 4 biological models and classify them according to definitions given in the pdf discussing models.

2: Give 2 examples of mathematical models that have made an impact in biological sciences.

3: Discuss early contributions to mathematical biology involving dynamics.

4: For the population problem with juveniles and adults, calculate both eigenvalues and eigenvectors. Give the complete

solution for the problem and compare results for the complete solution with those obtained by "time-stepping" in excel.

Also graph the solution, the age distribution (the ratio of adults to juveniles), and the change of population in successive

time periods (i.e. x(t+1)/x(t) and y(t+1)/y(t)). See specific questions in the pdf file (Note 4).

Homework 2 (Due Thur 9/5/2013)

1: Solve problems 2.1.1, 2.1.2, 2.1.3, 2.1.5 from Strogatz.

2: Solve problems 2.2.3 and 2.2.4 from Strogatz.

3: Solve problems 2.3.1, 2.3.3, and 2.3.4 from Strogatz. Also solve this equation numerically using Eulers method, the 2-step Runge Kutta

method, and Matlab's build in ODE solver ode45. Calculate the error between analytical and numerical solutions. What do you observe?

4: Solve problem 2.4.8. Solve this equation using Eulers method, the 2-step Runge Kutta method, and using Matlab's build in ODE solver ode45.

Calculate the error between analytical and numerical solutions. What do you observe?

Set parameter values and initial values such that the model equations represent real biological populations. Include a paragraph (with a literature reference)

describing the populations chosen.

2: Solve one of problems (3.7.3 + 3.7.4) or 3.7.5 or 3.7.6.

Use a computational tool to sketch the fix-points [plot (x vs dx)], directional fields, and solve equations numerically.

3: Solve problem 4.5.1 from Strogatz.

2: Solve the Chemostat model numerically find appropriate parameter values and use code given above. Discuss your choice for parameter values.

2: In class we had dv/dt = v ( v- a ) ( 1 - v ) - w, now assume that the neuron is stimulated by I(t)

A) assume I(t) take the form of a step function I = 0 for t < t0; I = I0 for t > t0.

B) assume I(t) take the form of a pulse I = I0, for t0 <= t <= t1; and I = 0 otherwise

C) assume I(t) is constant I(t) = I0

Explore dynamics of the system for each of these simulations.

3: For the Fitz-Hugh Nagumo model, show how parameter values of a affect stability of the limit cycles, use the Hopf Bifurcation Theorem and the stability criterion (from notes)

1: For examples 1 (page 261), 2 (page 262), and 3 (page 263)

A: Rewrite equation in cartesian coordinates

B: Use pplane 8 to plot phase portraits for mu < muc, mu = muc, and mu > muc, note mu is always positive

C: Plot solutions as functions of time (in cartesian coordinates) [ plot x and y, what is more interesting to study? ]

For example 3 try to recreate figures shown in 8.4.3 note your figures will have more lines on them, which is fine, the book have only displayed important lines.

2: Solve problems 8.4.1 to 8.4.4

1: Strogatz problem 8.1.6

2: Strogatz problem8.2.8

3: Strogatz problem 8.6.2 questions a, c, d

NOTE: IF YOU DO NOT HAVE MATLAB ON YOUR PERSONAL COMPUTERS, THE COMPUTERS IN SAS HALL LABS ALL HAVE MATLAB!

Homework 50%

Test 20% (in-class)

Project 30% (Presentation (15%) and report
(15%))