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 Text Assignments
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
29 12/15 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.

	Date	Topics	Text	Assignments
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
29	12/15	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.