Instructor: Mette S Olufsen
Office: SAS 3216
Office Hours: By appointment via email.
Phone Number: 515-2678
Email address: firstname.lastname@example.org
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
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
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).
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.
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%).
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.
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.
Project presentations 12/3 and 12/5
Project report due 12/12 before 5pm
Homework 1 (Due Thur 8/29/2013) 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
4: For the population problem with juveniles and
adults, calculate both eigenvalues and eigenvectors. Give
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.
3 (Due Thur 9/19/2013) 1: Solve
problems 3.1.1, 3.1.4, 3.2.2, 3.2.5, 3.4.2, 3.4.3,
3.5.7 from Strogatz.
Solve one of problems (3.7.3 + 3.7.4) or 3.7.5
computational tool to sketch the fix-points
[plot (x vs dx)], directional fields, and solve
3: Solve problem 4.5.1 from
(Due Tue 10/8/2013) 1: In the notes for the
Chemostat model solve problems 6-10.
Solve the Chemostat model numerically find appropriate
parameter values and use code given above. Discuss your
choice for parameter values.
5 (Due Tue 11/5/2013) 1:
In class we discussed impact of changing the
parameter alpha, investigate dynamics further by
exploring changes in the other parameter values
(epsilon and gamma).
2: In class we had dv/dt = v (
v- a ) ( 1 - v ) - w, now assume that the neuron
is stimulated by I(t)
assume I(t) take the form of a step function I = 0
for t < t0; I = I0 for t > t0.
assume I(t) take the form of a pulse I = I0, for
t0 <= t <= t1; and I = 0 otherwise
assume I(t) is constant I(t) = I0
dynamics of the system for each of these
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). Homework 6 (Due
Tue 11/20/2013) This homework is
associated with section
8.4 in Strogatz
examples 1 (page 261), 2 (page
262), and 3 (page 263)
equation in cartesian
B: Use pplane
8 to plot phase portraits for mu
< muc, mu = muc, and mu >
muc, note mu is always positive
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
Solve problems 8.4.1 to 8.4.4
7 (Due Tue
- INDIVIDUAL 1:
Strogatz problem 8.1.6
2: Strogatz problem8.2.8
3: Strogatz problem 8.6.2 questions a,
NOTE: IF YOU DO NOT HAVE MATLAB ON YOUR PERSONAL
COMPUTERS, THE COMPUTERS IN SAS HALL LABS ALL HAVE MATLAB!
Test 20% (in-class)
Project 30% (Presentation (15%) and report