Math 336, Fall 2008: Lecture Outlines

  1. Lecture 1, September 2
    1. Definition of dynamical models for biological problems
    2. Principles of ODE modeling
    3. The exponential growth model
    4. The logistic equation: phenomenological derivation and mechanistic derivation.
    5. Dimensionless scaling of the logistic equation
  2. Lecture 2, September 4
    1. Qualtitative analysis of autonomous scalar differential equations.
    2. Application to the logistic equation
    3. Exact solution of the logistic equation
    4. Derivation of spruce budworm model
  3. Lecture 3, September 9
    1. Qualtitative analysis of the spruce budworm model, modeling outbreaks and occurence of hysteresis
    2. Derivation of chemostat model: part I
  4. Lecture 4, September 11
    1. Derivation of chemostat model: part II, dimensionless scaling of the chemostat model
    2. Systems of ordinary differential equations, review: Vector formulation of systems, linear systems, equilibrium points and solutions
    3. Equilibria of the chemostat model
  5. Lecture 5, September 16
    1. Stability of the origin for linear systems of ode's. (Edelstein-Keshet, 4.8)
    2. Stability for 2-dimensional systems in terms of the trace and determinant of the coefficient matrix. (Edelstein-Keshet, 4.9)
    3. Linearization of a nonlinear system of ode's about an equilibrium point.
      Stability of the linearized system implies stability of the equilibrium point of the nonlinear system. (Edelstein-Keshet, 4.7)
  6. Lecture 6, September 18
    1. Stability of the equilibria of the chemostat.
    2. Solving linear systems of 2 equations.
    3. A drug infusion model (Edelstein-Keshet, 4.11).
  7. Lecture 7, September 23
    1. Phase plane analysis: phase portraits and direction fields (Edelstein-Keshet, 5.2-5.4).
    2. Phase plane analyses; null clines (Edelstein-Keshet, 5.5-5.6).
    3. Application to (what else?) the chemostat model.
  8. Lecture 8, September 25
    1. Phase portraits of linear systems in 2 dimensions (EK, 5,7-5.8, and handouts in class).
    2. Phase portrait analysis of non-linear systems near an equilibrium point by linearization (EK, 5,9, and handouts in class).
    3. Full phase portrait analysis of the chemostat (Edelstein-Keshet, 5.10).
  9. Lecture 9, September 30
    1. Multiple species population models (EK, Chapter 6): general modeling framework.
    2. Predator-prey models and the Lotka-Volterra model (EK, 6.2).
    3. Models with competition; phase portrain analysis (EK, 6.3).
  10. Lecture 10, October 2
    1. Detailed analysis of phase portrait of a competition model (EK, 6.3, see also problem 15, page 259).
  11. Lecture 11, October 7
    1. Chemical kinetics, law of mass action and ode modeling of systems of chemical reactants.
    2. A model of catalysis by an enzyme and of nutrient uptake.
    3. Fast and slow variables and quasi-steady-state approximation.
    4. References: EK, 7.1-7.2; Sontag, Chapter 6.
  12. Lecture 12, October 9
    1. Quasi-steady state analysis of an enzyme model; Michaelis-Menten dynamics; EK, 7.2; Sontag, Chapter 6.