MATH 550/CSCS 510 -- Winter 2001

WINTER TERM COURSE

MATH 550/PSCS 510: Introduction to Dynamical Systems for Biocomplexity

Instructor: Carl Simon (cpsimon@umich.edu) 647-9194

Time: Tu-Th 2-4

Place: TBA

SYLLABUS

Linear difference and differential equations on R1.
       Applications: population growth and finance.

Second order linear difference and differential equations on R1.
       Applications: spring, pendulum, Fibonacci systems.

Nonlinear differential equations on R1 and their phase diagrams.
       Applications: populations with carrying capacity, infection transmission.

Linear differential equations in R2. Solution via eigenvalues and phase diagrams.
       Applications: populations, combat models.

Nonlinear systems of differential equations.
       Applications: competing species systems, epidemiology.

First integrals and Lyapunov functions.
       Applications: Predator-prey systems, classical physics, HIV transmission.

Periodic orbits: Poincare-Bendixson Theorem, Bendixson-duLac Criterion, Hopf Bifurcation.
       Applications: More complex predator-prey models.

Linear difference equations in Rn. Solution by eigenvalues.
       Application: Age-structured population models.

Positive matrices; Perron-Frobenius Theorem.
       Application: Markov processes in biology and business
       Application: Leontieff input-output macroeconomic models.

Nonlinear difference equations in R1 and Rn. Steady states and their stability.
       Applications: Population interactions, Newton's Method.

Chaotic dynamics.
       Applications: Populations and economies.

Nonlinear methods of empirical analysis: distinguishing deterministic chaos from randomness.
       Application: economic and biological data sets.

Cellular Automata: definition, examples in R1 and R2.
       Application: population models over time and space, Game of Life.

Theory and simulation of one-dimensional cellular automata.
       Applications: plant and animal growth.

Zero-sum games; Nash equilibria; mixed strategies.
       Applications: market interactions, poker.

Non-zero sum games: 2x2 classification, Prisoner's dilemma (one-time and repeated).
       Applications: population and market interactions, economic competition.

Dynamics in non-zero sum games; replicator dynamics and Evolutionarily Stable Strategies.
       Applications: economics and evolution.

Introduction to linear partial differential equations (PDEs).
       Applications: populations parameterized by age or location, cellular automata.

Stochastic dynamic systems.
       Application: birth-death processes in population models,
       Application: PDEs for probability generating functions.

Introduction to Genetic Algorithms


PREREQUISITES: At least one solid course in calculus, familiarity with simple
probability. 


STUDENTS: Students in biology, economics, political science, natural resources who
have minimal math training and would like to learn some mathematical techniques
that are commonly used in building and studying models in their fields. This is
also the entry course for students in the certificate in complex systems.