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 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. Numerical solutions of ODEs: excel example, matlab ex 1, matlab ex 2

7. Runge Kutta methods

8. Bifurcation plots in matlab: examplebifur, examplebifur2, examplebifurSIR

9. Maple code for sketching solutions and implicit 3d plots

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

11. Notes from Edelstein-Keshet 2D system

12. Notes from Edelstein-Keshet Chemostat model

13. Notes from Edelstein-Keshet Predator prey model

14. Van der Pol oscillator code

15. Chemostat code

16. Test limit cycle code

17. Selkov Glycolysis paper

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

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

20. Glycolysis process

21. Neural modeling

22. FitzHugh Nagumo Lecture

23. FtizHugh Nagumo matlab

24. Bifurcations

25. Matlab supercritical Hopf bifurcation

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)).

Homework 2 (Due Thursday 9/6/2012)

1: Solve problems 2.1.1 - 2.14 from Strogatz. For this ODE also solve it using Euler's method, the 2nd order Runkge Kutta

Method, and using Matlab's ode-solver ODE45.

Calculate the error between numerical and the analytical solution.

2: Solve problems 2.2.1 and 2.2.5 from Strogatz

3: Solve problems 2.3.3 and 2.4.8 from Strogatz. For this example also solve the equations numerically in Matlab using your favorite method

Homework 3 (Due Thursday 9/20/2012 - hand in via email ONLY)

1: Solve problems 3.1.1, 3.1.4, 3.2.2, and 3.2.5 from the book.

Use Matlab to a) plot f(r,x) also use graphical methods for assessing the fix-points.

Use Matlab to make bifurcation plots and discuss plots obtained against results from analysis.

Note you need to use your analysis results to set the range for the parameter and for x.

2: Find one differential equations (autonomous) describing a biological phenomenon. Does this equation

have any fix-points? If so, can you make a bifurcation diagram? Remember, we are working on flow on a line,

so you should find a problem on the form dx/dt = f(r,x)

Homework 4 (Due Tuesday 10/2/2012 - Assigned by Sharon Lubkin)

We nondimensionalized the harvesting model 3 different ways. We used Maple to examine the graphs of dx/dt

against x as the parameters varied for 2 of the models.

Study the bifurcation diagram for the third version.

1: Is it the same kind of bifurcation?

2: What is the critical value of gamma?

3: What is the x value at which the bifurcation happens?

4: Take all 3 different bifurcation analyses and give an interpretation in dimensional terms.

That is, how much can we safely harvest?

5: For the equation dx/dt = rx - x^3 + k plot the vector field for all possible combinations of r and k (r < 0, r = 0, r > 0 and k < 0, k = 0. k > 0)

Maple notes:

A: I don't require you to do this in Maple, but you can. Alternatively, a graphing calculator should be fine.

B: I started you on the wrong path. Don't use "gamma" which is a reserved word in Maple. Call it "gam" or some other Greek letter.

Homework 5 (Due Tuesday 10/9/2012)

1: Solve one of the problems: 3.7.3, 3.7.4, 3.7.5, or 3.7.6

2: Problem 4.2.1 (on the church-bells)

3: Problems 4.3.3 and 4.3.7

4: Problem 4.5.3 (We have not yet gone over material for this problem)

1: For the nondimensionalized chemostat model

dN/dt = F(N,C) = a1 CN/(1+C) - N

dC/dt = G(N,C) = - CN/(1+C) - C + a2

a) Calculate Jacobian A = [FN FC; GN GC] where F and G are the right hand side

b) Show that this has two fix-points (0,a2) and (a1 (a2 - 1/(a1-1)) , 1/(a1-1))

c) Determine stability of the two fix-points

d) Find and plot the solution for appropriate values of the parameters and initial conditions (you can work with the non-dimensionalized equations)

e) Interpret the solution in terms of the dimensional quantities

2: Problems 7 and 10 (see notes link 12 above)

3: Define all words marked in bold in sections 5.1 and 5.2

4: Problems 5.2.3, 5.2.5, 5.2.7, 5.2 9 (In Strogatz)

1: Solve problems 6.4.4 - 6.4.6

2: For the rabbits vs. sheep and predator prey models do the following;

A: Find the fix points and use linear stability analysis to determine their kind

B: Plot phase portraits

C: Plot solutions as functions of time - you may use numerical methods to determine these, i.e., x(t) and y(t)

D: Compare the two models

2: For the FitzHugh Nagumo model (see notes) or a model of your choice show that as the critical parameter varies a super critical Hopf bifurcation occur.

Try to make the bifurcation diagram as well as representative phase portraits.

Homework 50%

Midterm 20% (in-class)

Final 30% (take-home)