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Homework sets will be uploaded here, together with due dates (typically one week after uploading). Solutions will appear on Sakai . Exam dates will also be posted.
ES denotes Prof. Sontag's notes.
| Lecture | Date | Topics | Reading and homework assignments |
|---|---|---|---|
| Week 1 | |||
| 1 | Tue 09/06 | Introduction to course, single difference equations, advantages/disadvantages of framework, exponential (Malthusian) equation | Section 1.1 in ES Review homework (due 09/13) |
| 2 | Thu 09/08 | Derivation of logisitic difference equation, cobwebbing, steady states, stability, and linearization about steady states | Sections 1.2 and 1.3 (beginning) in ES |
| Week 2 | |||
| 3 | Tue 09/13 | Linearized stability of steady states, systems of difference equations | Finished Section 1.3 in ES, Systems example (please read) Homework 1 (due 09/20) |
| 4 | Thu 09/15 | Studied dynamical behavior of (non-dimensionalized) logistic equation as parameter "r" varied. Periodic orbits (calculating and stability), bifurcation diagrams, period doubling, chaos. | Section 1.4 in ES |
| Week 3 | |||
| 5 | Tue 09/20 | Finished analysis of logistic difference equation (including chaotic behavior), introduction to ODEs, began exponential growth derivation | Finished Section 1.4 in ES, Sections 2.1.1-2.1.2 in ES Homework 2 (due 09/27) |
| 6 | Thu 09/22 | Finished exponential growth derivation, logistic growth derivation, alternate logistic derivation via nutrient consumption | Sections 2.1.3-2.1.4 in ES |
| Week 4 | |||
| 7 | Tue 09/27 | Non-dimensionalization of logistic equation, derivation of chemostat model | Sections 2.1.5-2.1.7 in ES Homework 3 (due 10/04) |
| 8 | Thu 09/29 | Finished chemostat derivation, Michaelis-Menten kinetics, Lineweaver-Burk plot, non-dimensionalization of chemostat | Sections 2.1.6-2.1.10 in ES |
| Week 5 | |||
| 9 | Tue 10/04 | Review of steady states for systems of ODEs, steady states of chemostat, conditions for steady states to exist in first quadrant, review of linearization, Hartman-Grobman Theorem | Sections 2.2.1, beginning 2.2.2 in ES Homework 4 (Note: now due on 10/13) |
| 10 | Thu 10/06 | Review of linear system phase portraits, trace-determinant plane to determine type of origin, began example of local phase portraits for nonlinear systems | Sections 2.2.2, 2.2.3 in ES (covered more in-depth though) |
| Week 6 | |||
| Remember: Exam 1 is on Tuesday, October 18 | |||
| 11 | Thu 10/11 | Finished local phase portrait example, analyzed phase portrait of chemostat, began nullclines | Sections 2.2.4, 2.4.1, 2.4.2, beginning 2.4.3 in ES Homework 5 (due on 10/20, but suggest to complete BEFORE Exam 1) |
| 12 | Tue 10/13 | Finished nullclines of chemostat, proof of global convergence to X 2 , macrophage dynamics nullcline example | Sections 2.2.3, 2.2.4 in ES |
| Week 7 | |||
| 13 | Tue 10/18 | Exam 1, in class | Exam 1 information |
| 14 | Thu 10/20 | Epidemiology: SIR and SIRS models, derivation of infection rate terms, reduction from 3 equations to 2, analysis of steady states and nullclines of reduced system, interpretation of R 0 , extensions to vaccines and other epidemiology models | Section 2.5 in ES Homework 6 (due 10/27) |
| Week 8 | |||
| 15 | Tue 10/25 | Exam 1 discussion, finish epidemiology. | See Lecture 14 above. |
| 16 | Thu 10/27 | Introduction to chemical kinetics and the law of mass action, and how to write a system of ODEs using mass action. Begin formalizing system of reactions as a chemical reaction network (CRN) | Sections 2.6.1, 2.6.2 in ES Homework 7 (due 11/08) |
| Week 9 | |||
| 17 | Tue 11/01 | Continue formalization of systems of chemical reactions as a CRN, stochiometry matrix and reaction vector, conservation laws and how to compute them to reduce the dimensionality of the system. | Sections 2.6.2 in ES |
| 18 | Thu 11/03 | Finished system reduction of phosphorylation/dephosporylation system using conservation laws, introduction to enzymatic reactions, differential equations representing basic enzyme-substrate dynamics | Sections 2.6.3, 2.6.4 in ES |
| Week 10 | |||
| 19 | Tue 11/08 | Quasi-steady state approximation for enzyme reactions (i.e. Michaelis-Menten kinetics), formulation, intuitive justification via nullclines. | Sections 2.6.5, 2.6.6 in ES Homework 8 (due 11/15) |
| 20 | Thu 11/10 | Methods of enzyme inhibition (competitive and allosteric), simple model of gene expression, sigmoidal kinetics, cooperativity and its relation to sigmoidal kinetics, hyperbolic vs. sigmoidal response. | Sections 2.6.9 - 2.6.12 in ES |
| Week 11 | |||
| Exam II announced: Tuesday, November 22 | |||
| 21 | Tue 11/15 | Fast-binding derivation of cooperativity, introduction to multi-stability and cooperative reactions, hyperbolic and sigmoidal responses in relation to decision making and binary responses | Sections 2.7.1, 2.7.2 in ES Homework 9 (due 12/06) |
| 22 | Thu 11/17 | Finish sigmoidal responses and dynamics, introduction to periodic orbits, motivation of biological importance of limit cycles, example of an explicit limit cycle using polar coordinates | Sections 2.7.2, 2.7.3 (please read), 2.8.1, 2.8.2 in ES |
| Week 12 | |||
| 23 | Tue 11/22 | Exam 2, in class | Exam 2 information |
| N/A | Thu 11/24 | Thanksgiving!! | Eat turkey (or meal of choice) |
| Week 13 | |||
| 24 | Tue 11/29 | Continue with periodic behavior and limit cycles, example of an explicit limit cycle, Poincare-Bendixson Theorem and the existence of periodic solutions, Van der Pol oscillator example | Sections 2.8.1-2.8.4 in ES |
| 25 | Thu 12/01 | Recap of Poincare-Bendixson Theorem, example usage for circle region D, Bendixson's criterion for the nonexistence of periodic orbits, introduction to partial differential equation models | Sections 2.8.4, 2.8.5, 3.1.1 in ES |
| Week 14 | |||
| 26 | Tue 12/06 | Conservation of mass equation, flux vector J, 3 main types of fluxes: advection (transport), chemotaxis, and diffusion, transport equation analysis (constant velocity and exponential growth only) |
Sections 3.1.1-3.1.6 in ES Homework 10 (due 12/13) |
| 27 | Thu 12/08 | Introduction to chemotaxis, PDEs resulting from a combination of fluxes (e.g. transport and chemotactic motion), chemotaxis qualitative analysis | Sections 3.1.7 and 3.2 (beginning) in ES Extra Credit homework 11 (due Friday 12/16) |
| Week 15 | |||
| 28 | Tue 12/13 | Introduction to diffusion equation, solution via separation of variables, determining separation constant from boundary conditions, determining constants from linearity (superposition) and initial conditions, no flux (i.e. closed endpoint) boundary conditions, general flux and relation to boundary conditions |
Sections 3.2, 3.2.3 - 3.2.5 in ES. See also 3.2.8 (didn't cover, but you should read it) |