Instructor: Mette S Olufsen
Office: SAS 3216
Office Hours: By appointment via email.
Phone Number: 515-2678
Email address: email@example.com
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 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.
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
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)). 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
Method, and using Matlab's
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
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
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)
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
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) -
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
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
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
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.
NOTE: IF YOU DO NOT HAVE MATLAB ON YOUR
PERSONAL COMPUTERS, THE COMPUTERS IN SAS HALL LABS ALL
Midterm 20% (in-class)
Final 30% (take-home)