Math 336, Fall 2008: Lecture Outlines
- Lecture 1, September 2
- Definition of dynamical models for biological problems
- Principles of ODE modeling
- The exponential growth model
- The logistic equation: phenomenological derivation and
- Dimensionless scaling of the logistic equation
- Lecture 2, September 4
- Qualtitative analysis of autonomous scalar differential equations.
- Application to the logistic equation
- Exact solution of the logistic equation
- Derivation of spruce budworm model
- Lecture 3, September 9
- Qualtitative analysis of the spruce budworm model, modeling
outbreaks and occurence of hysteresis
- Derivation of chemostat model: part I
- Lecture 4, September 11
- Derivation of chemostat model: part II, dimensionless
scaling of the chemostat model
- Systems of ordinary differential equations, review:
Vector formulation of systems, linear systems, equilibrium
points and solutions
- Equilibria of the chemostat model
- Lecture 5, September 16
- Stability of the origin for linear systems of ode's.
- Stability for 2-dimensional systems in terms of
the trace and determinant of the coefficient matrix.
- 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)
- Lecture 6, September 18
Stability of the equilibria of the chemostat.
- Solving linear systems of 2 equations.
- A drug infusion model (Edelstein-Keshet, 4.11).
- Lecture 7, September 23
- Phase plane analysis: phase portraits and direction fields
- Phase plane analyses; null clines (Edelstein-Keshet, 5.5-5.6).
- Application to (what else?) the chemostat model.
- Lecture 8, September 25
- Phase portraits of linear systems in 2 dimensions
(EK, 5,7-5.8, and handouts in class).
- Phase portrait analysis of non-linear systems near
an equilibrium point by linearization
(EK, 5,9, and handouts in class).
- Full phase portrait analysis of the chemostat
- Lecture 9, September 30
- Multiple species population models (EK, Chapter 6): general
- Predator-prey models and the Lotka-Volterra model (EK, 6.2).
- Models with competition; phase portrain analysis (EK, 6.3).
- Lecture 10, October 2
Detailed analysis of phase portrait of a competition
(EK, 6.3, see also problem 15, page 259).
- Lecture 11, October 7
- Chemical kinetics, law of mass action and ode modeling
of systems of chemical reactants.
- A model of catalysis by an enzyme and of nutrient uptake.
- Fast and slow variables and quasi-steady-state
- References: EK, 7.1-7.2; Sontag, Chapter 6.
- Lecture 12, October 9
- Quasi-steady state analysis of an enzyme model;
Michaelis-Menten dynamics; EK, 7.2; Sontag, Chapter 6.