Math 1360:      Modeling in Applied Math I        Fall 2011

Instructor: David Swigon

Office: Thackeray 511, 412-624-4689, swigon@pitt.edu

Lectures:  MWF 10:00-10:50am, OEH 316

Office Hours: W 11:30-2:00pm, 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

Single-variable calculus (Math 0220, 0230, or equivalent) including Taylor series, multivariable calculus (Math 0240 or equivalent), linear algebra and familiarity with separable differential equations is recommended.  Other techniques and theory will be reviewed and developed where needed

Textbook

G. de Vries et al., A Course in Mathematical Biology, SIAM, 2006. ISBN 0-89871-612-8

Textbook typos

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, 412-648-7890 or 412-383-7355 (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/nonlinear-web.html

http://math.la.asu.edu/~chaos/logistic.html

http://www.ibiblio.org/e-notes/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 user-specified function

http://www.emporia.edu/math-cs/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.

http://math.rice.edu/~dfield/

 

Parameter Estimation

An example of MATLAB code for parameter estimation is provided by the following files:

paramfit1D.m and Sfun1D.m for a 1-dimensional ODE model

paramfit2D.m and Sfun2D.m for a 2-dimensional ODE model

paramfit2Ddiscrete.m and Sfun2Ddiscrete.m for a 2-dimensional discrete model

Read the accompanying handout for explanation.


 

 

 

WEEK

CHAPTERS

TOPICS

HOMEWORK

TERM PROJECT

Aug 29 - Sep 2

1 & 2.1-2.2

Introduction

Discrete time models

Cobwebbing

 

 

Sep 7 - Sep 9

2.3-2.5

Discrete logistic equation

Stability analysis

HW#1: 2.4.1,5,7,10

due Sep 16

Solutions

 

Sep 12 - Sep 16

2.3

Systems of discrete equations

HW#2: 2.4.13, 16, 19a,b (19c is optional)

due Sept 23

Solutions

 

Sep 19 - Sep 23

3.1-3.3, 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

Solutions

Group selection

Oct 3 - Oct 7

3.5-3.6

3D systems

HW#4: 3.9.7, 13, 16

due Oct 12

Solutions

Project proposal due Oct 7

Oct 11 - Oct 12

 

 

Review

Review Topics

Practice Problems and their results

 

 

Oct 14

 

EXAM I

 

 

Oct 17 - Oct 21

7.1-7.3

Parameter estimation

Likelihood function

Model comparison

Homework #5

Requires MATLAB code for parameter estimation (see above the syllabus)

due Nov 4

Solutions

 

Oct 24 - Oct 28

7.4

Optimization algorithms

Model sensitivity

Preliminary analysis, first report (2 page) due Oct 28
Checklist

Oct 31 - Nov 4

4.1-4.2

Partial differential equation models

Age structured model

 

 

Nov 7 - Nov 11

4.3

Reaction-diffusion systems

Traveling waves

HW#6: 4.5.2, 5, 7 due Nov 21

Solutions

 

Nov 14 - Nov 18

5.1-5.4

Stochastic models

Markov chains

Diffusion

 

Term project draft report (complete) due Nov 18

Nov 21

5.6-5.7

Birth & death process

Homework #7

due Nov 30

Solutions

Nov 28 - Nov 30

Birth & death process

 

 

Dec 2

Review

Review topics

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

Panel review handout

(with review sheet)