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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 01/19 | Introduction to course, single difference equations, advantages/disadvantages of framework, exponential (Malthusian) equation | Section 1.1 in ES Review homework (due 01/26) |
| 2 | Thu 01/21 | 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 01/26 | Linearized stability of steady states, systems of difference equations | Finished Section 1.3 in ES, Systems example (please read) Homework 1 (due 02/02) |
| 4 | Thu 01/28 | 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 02/02 | Introduction to ODEs, exponential and logistic growth derivation |
Sections 2.1.1-2.1.4 in ES Homework 2 (due 02/09) |
| 6 | Thu 02/04 | Properties of logistic growth, alternate derivation based on nutrients, non-dimensionalization, introduction to the chemostat | Sections 2.1.3-2.1.5, beginning 2.1.6 in ES |
| Week 4 | |||
| 7 | Tue 02/09 | Finished chemostat derivation, Michaelis-Menten kinetics, Lineweaver-Burk plot, non-dimensionalization of chemostat |
Sections 2.1.6-2.1.10 in ES Homework 3 (due 02/16) |
| 8 | Thu 02/11 | 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 |
| Week 5 | |||
| 9 | Tue 02/16 | Review of linear system phase portraits, trace-determinant plane to determine type of origin, began example of local phase portraits for nonlinear system |
Sections 2.2.2, 2.2.3 in ES (covered more in-depth though) Homework 4 (due 02/23) |
| 10 | Thu 02/18 | 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 |
| Week 6 | |||
| Remember: Exam 1 is on Tuesday, March 1 | |||
| 11 | Tue 02/23 | Finished nullclines of chemostat, proof of global convergence to X 2 , macrophage dynamics nullcline example |
Sections 2.2.3, 2.2.4 in ES Homework 5 (due 03/01) |
| 12 | Thu 02/25 | Started epidemiology, SIR and SIRS models, derivation of infection rate terms, reduction from 3 equations to 2 | Section 2.5 in ES (beginning) |
| Week 7 | |||
| 13 | Tue 03/01 | Exam 1, in class | Exam 1 information |
| 14 | Thu 03/03 | Finished epidemiology: 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 (end) Homework 6 (due 03/10) |
| Week 8 | |||
| 15 | Tue 03/08 | Exam 1 discussion, introduction to chemical kinetics, motivation for law of mass action, writing ODEs for systems of chemical reactions | Section 2.6.1 in ES |
| 16 | Thu 03/10 | Formalizing system of reactions as a chemical reaction network (CRN), reaction vector and stoichiometry matrix, introduction to conservation laws and method find them |
Section 2.6.2 in ES (end) Homework 7 (due 03/24) |
| Week 9 | |||
| Spring break! | |||
| Week 10 | |||
| 17 | Tue 03/22 | Finished calculation of conservation laws from CRN formulation, introduction to enzymatic reactions | Sections 2.6.3, 2.6.4 in ES |
| 18 | Thu 03/24 | 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 04/05) |
| Week 11 | |||
| Exam II announced: Thursday, April 14 | |||
| 19 | Tue 03/29 | Methods of enzyme inhibition (competitive and allosteric), simple model of gene expression, began sigmoidal kinetics | Sections 2.6.9, 2.6.10, 2.6.11, 2.6.12 (begin) in ES |
| 20 | Thu 03/31 | Cooperativity and sigmoidal kinetics, hyperbolic vs. sigmoidal response, fast-binding derivation | Section 2.6.12 (finish) in ES |
| Week 12 | |||
| 21 | Tue 4/5 | Multi-stability and cooperative reactions, hyperbolic and sigmoidal responses in relation to decision making and binary responses, introduction to periodic behavior |
Sections 2.7.1, 2.7.2, 2.8 (beginning) in ES Homework 9 (due 04/12) |
| 22 | Thu 4/7 | 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 |
| Week 13 | |||
| 23 | Tue 04/12 | Recap of Poincare-Bendixson Theorem, example usage for circle region D, Bendixson's criterion for the nonexistence of periodic orbits | Sections 2.8.3 and 2.8.5 in ES |
| 24 | Thu 04/14 | Exam 2, in class | Exam 2 information |
| Week 14 | |||
| 25 | Tue 04/19 | Exam II recap, begin partial differential equation material, motivation for spatial variables, conservation of mass equation, flux vector J, 3 main types of fluxes: advection (transport), chemotaxis, and diffusion, begin transport equation |
Sections 3.1.1-3.1.6 (beginning of 3.1.6 only) in ES Homework 10 (due 04/26) |
| 26 | Thu 04/21 | Finish transport equation solution characterization (constant velocity and exponential growth only), introduction to chemotaxis, PDEs resulting from a combination of fluxes (e.g. transport and chemotactic motion) | Sections 3.1.6, 3.1.7 in ES |
| Week 15 | |||
| 27 | Tue 04/26 | Chemotaxis qualitative analysis, introduction to diffusion equation, solution via separation of variables |
Sections 3.2, 3.2.3, 3.2.4 (beginning) in ES Extra Credit homework 11 (due Monday 05/02) |
| 28 | Thu 04/28 | Finish diffusion equation, i.e. separable solutions 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 3.2.4 (finish), 3.2.5 in ES |