Math 1360: Modeling in Applied Math I Fall 2011
Instructor: David
Swigon
Office: Thackeray 511, 4126244689, swigon@pitt.edu
Lectures:
MWF 10:0010:50am, OEH 316
Office
Hours: W 11:302:00pm, Thackeray 511,
or by appointment.
CourseWeb Page: check frequently for changes and updates
Course
Description
This course
is an introduction to mathematical modeling. We shall discuss different types
of models and learn about all stages of the modeling process, including model
construction, model analysis, parameter fitting, and model verification.
Examples are taken from biological sciences, but the techniques and model types
presented are universally applicable.
Prerequisites
Singlevariable
calculus (Math 0220, 0230, or equivalent) including
Textbook
G. de Vries et al., A Course in Mathematical Biology, SIAM,
2006. ISBN 0898716128
Grading
Scheme
Homework
Assignments: 25%
Term Project: 25%
Exams: 50% (25% each)
Schedule
The precise
schedule and a list of homework problems will be given out in advance and
posted on the web. Homework is due at the beginning of a class one week
after it was assigned. You may work with other students on homework but do not
submit solutions that are identical copies of each other, as such will be
discarded. You may use a computer program (e.g., MATLAB, Maple, Mathematica) in solving the homework unless I specifically
say otherwise. Each such use should be explicitly documented (e.g.,’eigenvalues computed with MATLAB’).
Term Project
Term
projects will be assigned about one month into the semester and will be due one
week before the final exam. The goal of the project is to design a mathematical
model of a biological, chemical, mechanical, or an economical system, fit the
model using real or simulated data, and write a project report. Detailed
information is included in the PROJECT
HANDOUT
Policy
Attendance is expected. No late homework will be
accepted and there will be no make up exams.
Disability Resource Services
If
you have a disability for which you are or may be requesting an accommodation,
you are encouraged to contact both your instructor and Disability Resources and
Services, 140 William Pitt Union, 4126487890 or 4123837355 (TTY) as early
as possible in the term. DRS will verify your disability and determine
reasonable accommodations for this course.
Academic Integrity
Cheating/plagiarism will not be tolerated. Students
suspected of violating the University of Pittsburgh Policy on Academic
Integrity will incur a minimum sanction of a zero score for the quiz, exam or
paper in question. Additional sanctions may be imposed, depending on the
severity of the infraction. On homework,
you may work with other students or use library resources, but each student
must write up his or her solutions independently. Copying solutions from other
students will be considered cheating, and handled accordingly.
Online resources
Brief
online MATLAB tutorial can be found here: http://www.cyclismo.org/tutorial/matlab/
XPP/XPPAUT is a free
software developed by my colleague Dr. Ermentrout that allows you to simulate
numerically a wide range of dynamical systems, including discrete, continuous
ODE, PDE systems and stochastic systems, systems with delays, etc. This
essential tool of any applied mathematician is available for Windows or Mac
OS. There is also a free iPad
app of the program available, with complete functionality of the full
computer version.
1D discrete systems
Here are
some applets for the logistic map:
http://math.bu.edu/DYSYS/applets/nonlinearweb.html
http://math.la.asu.edu/~chaos/logistic.html
http://www.ibiblio.org/enotes/MSet/Logistic.htm
(allows you
to plot higher iterates)
http://users.dickinson.edu/~richesod/math271/Bifurcations.html
(includes
other maps)
For tent
map you can use
http://www.enm.bris.ac.uk/staff/berndk/chaosweb/mapa.html
Cobweb plot
for userspecified function
http://www.emporia.edu/mathcs/yanikjoe/Chaos/CobwebPlot.htm
2D discrete systems
http://www.dean.usma.edu/math/research/mathtech/java/DDSSystemNH/ddssystempage.html
(allows iterations of linear systems  nice for the
love relations dynamics
1D and 2D dynamical systems
Applets for numerical simulation of dynamical systems (use DFIELD for 1D systems,
PPLANE for 2D systems)
http://math.rice.edu/~dfield/dfpp.html
Matlab versions of these applets
exist and are available for download. The benefit  can be used offline.
Parameter Estimation
An example of MATLAB code for parameter estimation
is provided by the following files:
paramfit1D.m and
Sfun1D.m for a 1dimensional ODE model
paramfit2D.m and
Sfun2D.m for a 2dimensional ODE model
paramfit2Ddiscrete.m
and Sfun2Ddiscrete.m for a
2dimensional discrete model
Read the accompanying handout for explanation.
WEEK 
CHAPTERS 
TOPICS 
HOMEWORK 
TERM
PROJECT 
Aug
29  Sep 2 
1
& 2.12.2 
Introduction Discrete time models Cobwebbing 


Sep
7  Sep 9 
2.32.5 
Discrete
logistic equation Stability
analysis 
HW#1:
2.4.1,5,7,10 due Sep 16 

Sep
12  Sep 16 
2.3 
Systems
of discrete equations 
HW#2: 2.4.13, 16, 19a,b
(19c is optional) due Sept 23 

Sep
19  Sep 23 
3.13.3,
3.7 
Ordinary differential equation models 1D
models Bifurcations 


Sep
26  Sep 30 
3.4 
2D
systems 
HW#3: 3.9.2, 3, 6 ((e) and
(f) go together) due Oct 5 
Group selection 
Oct
3  Oct 7 
3.53.6 
3D
systems 
HW#4: 3.9.7, 13, 16 due Oct 12 
Project proposal due Oct 7 
Oct
11  Oct 12 

Review Practice Problems and their results 


Oct 14 

EXAM I 


Oct
17  Oct 21 
7.17.3 
Parameter estimation Likelihood
function Model
comparison 
Requires MATLAB code for
parameter estimation (see above the syllabus) due Nov 4 

Oct
24  Oct 28 
7.4 
Optimization
algorithms Model
sensitivity 
Preliminary analysis, first report (2 page)
due Oct 28 

Oct
31  Nov 4 
4.14.2 
Partial differential equation models Age
structured model 


Nov
7  Nov 11 
4.3 
Reactiondiffusion
systems Traveling
waves 
HW#6: 4.5.2, 5, 7 due Nov 21 

Nov
14  Nov 18 
5.15.4 
Stochastic models Markov
chains Diffusion 

Term project draft report (complete)
due Nov 18 
Nov
21 
5.65.7 
Birth
& death process 
due Nov 30 

Nov
28  Nov 30 
Birth
& death process 



Dec
2 
Review Practice problems and their solutions 

Term project report due Dec 4 at 5:00pm (electronic version) 

Dec 5 

EXAM II 


Dec 7, 9 

PANEL REVIEW OF PROJECT REPORTS 

