Goals for Math 2E03, introduction to modeling
Technical
- 'Fermi problems': order-of-magnitude calculations
- dimensional analysis/basic non-dimensionalization
- discrete-time deterministic models (single-variable):
- find equilibria
- determine stability graphically and analytically
- (R) simulate: basics, for loops
- discrete-time stochastic models
- basic probability rules
- moments and quantiles of random variables
- effects of stochasticity: absorbing boundaries, Jensen's inequality (?)
- distributions: uniform, normal, binomial, Poisson, possibly others
- (R) draw random deviates, experiments in probability
- (R) simulate, characterize mean and quantiles of behavior of dynamical systems over time
- analyze (expected mean and variance?)
- multidimensional linear (age/stage)-structured models
- construct Leslie/Lefkovitch matrix from description
- (R) use matrix multiplication to simulate
- eigenanalysis of long-term behavior [not reproductive value, sensitivity/elasticity]
- Markov chains
- (R) simulate
- eigenanalysis of stationary state, approach
- continuous-time deterministic models
- solve simple (separable) ODEs
- graphical and analytical stability (1D systems)
- (R) set up gradient functions and integrate numerically
- 2-variable systems: equilibria, characterize stability; phase plane analysis
- multi-variable systems: solve numerically. Linearized stability analysis (numerical)
Concepts
- separation of space/time scales, levels of hierarchy (Stommel diagram)
- calibration and validation of models [not details of parameterization/fitting]
- equilibria (fixed points, attractors)
- various senses of stability (eigenvalue, permanence, coefficient of variation)
- linearity/nonlinearity/"density dependence"/feedback; implications of nonlinear dynamics
- parsimony; "all models are wrong"
- predictive vs explanatory; mechanistic vs phenomenological
- where do parameters come from? parametric and structural sensitivity
- reproducible analysis/modeling
Skills
- Basic R programming
- Reading primary literature
Not included (except perhaps as special topics)
- Parameter estimation
- Phenomenological/statistical modeling (i.e. ch. 3 of Mooney & Swift)
- Individual-based models
- Continuous-time stochastic models
- Spatial models (diffusion, cellular automaton, IBM, etc.)
- Theoretical ecology/biology per se (although many examples will be drawn from the field)
- Tools for reproducible analysis (Sweave, odfWeave)