| Tentative schedule of topics for MTH 455, Spring 2005 | ||
| 1 | Population models | |
| Text | 1a | Haberman |
| Text | 1b | Ringland, Kopecky: Radioactive decay and SIR infection models |
| Text | 1c | Hofbauer, Sigmund |
| A Discrete One-Species Model (1st order linear difference eqn), geometric growth, Malthus | ||
| Struggle for existence, Density-dependent growth | ||
| Logistic map, cobweb diagrams, fixed points | ||
| linear stability analysis, topological conjugacy, Hartman-Grobman theorem | ||
| periodic points, deterministic chaos | ||
| Discrete One-Species Models with an Age Distribution, eigenvectors. | ||
| Continuous-time models, exponential growth, logistic equation | ||
| Stochastic Models. Probability concepts, random processes, realizations, mean and standard deviation. | ||
| Comparison of deterministic and stochastic models for radioactive decay and SIR infection model. | ||
| Density-Dependent Growth (logistic eqn), Phase Plane Solution of Logistic Equation, Explicit Solution of the Logistic Equation | ||
| Linear Constant Coefficient Difference and Differential Equations (incl polar rep of complex numbers) | ||
| Growth Models with Time Delays (discrete logistic eqn) | ||
| Destabilizing Influence of Delays, delay differential equations (destabilization in delayed SHO) | ||
| Two-Species Models | ||
| Phase Plane, Equilibrium, and Linearization | ||
| Systems of Two Constant Coefficient First Order Differential Equations (306 material) | ||
| Stability of Two-Species Equilibrium Populations (TD diagram) | ||
| Phase Plane of Linear Systems | ||
| Predator Prey Models | ||
| Derivation of the Lotka-Volterra Equations | ||
| Qualitative Solution of the Lotka-Volterra Equations | ||
| Average Populations of Predators and Prey | ||
| Man's Influence on Predator-Prey Systems | ||
| Limitations of Lotka-Volterra Equation | ||
| Two Competing Species | ||
| Genetics. Hardy-Weinberg Law. | ||
| 2 | Heat and diffusion. Diffusion Equation (in 1 spatial dimension) | |
| Text | 2a | Ringland: Heat and diffusion |
| Thermal equilibrium in rod. | ||
| Phase-lag in periodic planet-heating. | ||
| Fundamental plane-source solution, and applications (e.g. momentary contact with hot object) | ||
| 3 | Spontaneous Structure and Travelling Waves | |
| Text | 3a | Ringland: Mass action, Schlogl model, Brusselator |
| Text | 3b | Pearson: Complex structure from a simple model |
| Text | 3c | Wikipedia: Conway's Automaton |
| Text | 3d | Murray 6.5, 12.4: nerve impulses |
| Chemical reactions: the Principle of Mass Action | ||
| A<->B. The Schlogl Model (bistability) | ||
| Reaction-Diffusion Systems | ||
| Pattern-formation. The Brusselator: homogeneous steady state, spatial structures, spatio-temporal structures | ||
| Complex pattern-formation: The Pearson Experiment. (Euler's method, finite-difference approximation of derivatives.) | ||
| Conway's Cellular Automaton | ||
| Travelling waves. (Linearized Brusselator as example) | ||
| Hodgkin-Huxley Theory of Nerve Membranes: Fitzhugh-Nagumo Model | ||
| Waves in Excitable Media | ||
| 4 | Vibration: damped and forced oscillators | |
| Text | 4a | Ringland: Introduction |
| Text | 4b | Keener, Sneyd: Models of the cochlea |
| Coupled SHOs | ||
| Elastic bar | ||
| Cochlear membrane: pitch discrimination. |