Math 336, Fall 2008: Syllabus

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The chapter and section numbers labeled EK are those in Mathematical Models in Biology by L. Edelstein-Keshet. Reading assignments are also given for the on-line notes by Professor Sontag. Additional material may also be assigned and links are given.

Outlines of lectures

Date Topics TextAssignments
1 9/2 Dynamical models, introduction
Single species population models
Logistic model.
EK, 4.2 and 6.1
Review of elementary ode
Sontag, 1.1-1.5
Assignments due 9/4 and 9/9
Solutions
2 9/4 Qualitative analysis of scalar ode;
Logistic equation, continued;
Spruce budworm
EK, 4.2 and 6.1
Supplementary notes, qualititative ode analysis
 
3 9/9 Spruce budworm model analysis
The chemostat model
EK, 4.3--4.5
Sontag, 1.6--1.10
Spruce budworm project, due 9/18
Solutions
4 9/11 The chemostat model, scaling, equilibria and stability EK, 4.3--4.10
Sontag, 1.6--1.10, Chapter 2
Problems due on 9/18
Solutions
5 9/16 Stability of the origin for non-linear systems;
Linearization and stability for equilibria of nonlinear systems
EK, 4.3--4.10
Sontag, Chapter 2
Class notes
 
6 9/18 Stability of the chemostat
Solving linear systems in dimension 2 Notes
Drug infusion model
EK, 4.11
Sontag, 1.6--1.10, Chapter 2
Problems due on 9/25
Solutions
7 9/23 Phase plane analysis: null clines and phase portraits EK, Chapter 5
Sontag, Chapter 4
 
8 9/25
Phase portraits near equilibria
for linear and nonlinear systems
EK, Chapter 5
Sontag, Chapter 4
Problems due on 10/2
Solutions, part I
Solutions, part II
9 9/30 Multi-species models EK, 6.3, 6.4  
10 10/2 Phase plane analysis of a model with competition
Lecture 10 Notes
EK, 6.3, 6.4 Problems due on 10/9
Solutions
11 10/7 Chemical kinetics and quasi-steady state approximation EK, 7.1-7.2, Sontag, Chapter 6.  
12 10/9 Michaelis-Menten kinetics via quasi-steady states EK, 7.2, Sontag, Chapter 6  
13 10/14 First midterm Open book; open notes.  
14 10/16 Chemical kinetics equations: general approach
Quasi-steady state analysis of a model with inhibition
EK, 7.4, Sontag, Chapter 6 Problems due on October 23
Solutions
15 10/21 Quasi-steady state approximations, continued; Limit cycles EK, 8.1, Sontag, Chapter 8  
16 10/23 Limit cycles, Poincare-Bendixon theorem EK, 8.3, Sontag, Chapter 8 Problems due on October 30
Solutions
Solutions, part II
Van der Pol Applet
JODE manual
17 10/28 Applyling Poincare-Bendixon and the
Bendixson negative criterion;
The Van der Pol oscillator
EK, 8.4, Sontag Chapter 8  
18 10/30 The Van der Pol oscillator EK, 8.4, Sontag Chapter 8
Class Notes
Problems due on November 6.
Solutions
19 11/4 The Van der Pol oscillator, continued    
20 11/6 Relaxation Oscillations, Fitzhugh-Nagumo EK, 8.5; Sontag, 8.7
Class Notes on Fitzhugh-Nagumo
Problems due November 13 Problem A; Solution
Problem B Solution to B
21 11/11 Kolmogorov theory of predator prey EK, 8.7
Class Notes on predator prey.
Problems due November 13 Problem A
Problem B
22 11/13 Introduction to pde modeling EK, 9.1-9.5
Problems due November 18 Solutions
23 11/18 Introduction to pde modeling EK, 9.1-9.5
Problems For Dec. 4
Solutions
24 11/20 Midterm Exam    
25 11/25 Diffusion equations; boundary conditions, separation of variables EK, Appendix to Chapter 9
 
26 12/2 Diffusion equations; steady states
transit times, diffusive motion
EK, 9.5, 9.6
 
27 12/4 Transport equations and traveling wave solutions
Equations with attraction/repulsion, chemotaxis
Sontag, Chapter 9 Problems For Dec. 9
Solutions
28 1/29 Review    
2912/15 FINAL EXAM, DEC. 15; SEC-211, 8AM-11AM PDE material on final will be selected from:
from EK, sections 9.2, 9.3, 9.4, 9.5 and appendices, (to extent
covered in homework problems); stationary solutions;
Sontag, 9.4-9.6, and homework problems/class discussion.
Please note: New version to solutions to problem set 11
(was due Dec 4.) with typographical corrections to flux equalities
in solution of problem 9, chapter 9.