- 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 mechanistic derivation.
- 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. (Edelstein-Keshet, 4.8)
- Stability for 2-dimensional systems in terms of the trace and determinant of the coefficient matrix. (Edelstein-Keshet, 4.9)
- 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 (Edelstein-Keshet, 5.2-5.4).
- 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 (Edelstein-Keshet, 5.10).

- Lecture 9, September 30
- Multiple species population models (EK, Chapter 6): general modeling framework.
- 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 model (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 approximation.
- 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.