BMA 771 Biomathematics I

Tue/Thur 1.30-2.45

    Class-room: 2229 SAS Hall

    Instructor: Mette S Olufsen
    Office: SAS 3216
    Office Hours: By appointment via email.
    Phone Number: 515-2678
    Email address:

Course Information

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.

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. If you are at all worried about having the necessary mathematical background (particularly with regard to linear algebra), please come and see me. It will be possible to fill in some gaps during the semester, and I will do what I can to help.

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.


    1.    Syllabus
    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:
  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:
  19.    JSIM:
  20.    Glycolysis process
  21.    Neural modeling
  22.    FitzHugh Nagumo Lecture
  23.    FtizHugh Nagumo matlab
  24.    Bifurcations
  25.    Matlab supercritical Hopf bifurcation

TEST - Thursday October 11th on Part 1 Flows on the line - Chapters 2 to 4.


Homework 1 (Due Tuesday 8/28/2012)
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)

Homework 6 (Due Thursday 11/1/2012)

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)

Homework 7 (Due Tuesday 11/5/2012)
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

Homework 8 (Due Wednesday 12/5/2012) FINAL (30%)
1: Solve problems 7.1.8, 7.3.5, 7.5.1, 7.6.7, 8.1.10, 8.2.3, 8.2.8, 8.2.9
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)

This page was last modified August 30h 2012.