Population Biology
BIOL 763/599 / SCCC 411B / MATH 599
Spring 1999
University of South Carolina

Professor Matt Miller
miller@math.sc.edu
Department of Mathematics
 Professor Dave Wethey
wethey@biol.sc.edu
Department of Biological Sciences

Feb 9-11: Systems (continued): equilibria, eigenvalues and eigenvectors, linearization, stability analysis

  • First the practical side. Be sure to walk through the Maple worksheets chemostat.mws, volterra.mws, matrix.mws and eigen.mws
  • Answer the questions in these worksheets, and ask us about anything you don't understand. The code you find here will serve as your templates and guides for your own work in the projects, so you want to get comfortable with it.
  • But don't ignore the theoretical side. Read the handout on vectors and transformations, as well as the basic materials on systems found in the various caclulus and DE texts.
  • For null-cline analysis see Hastings, chapters 7 and 8 (there's a bunch of other stuff woven in), or E-K sections 5.5, 5.10, 6.2, 6.3
  • For linearization see Hastings pp 125-126, and p 156; also E-K section 4.7
  • For local stability analysis (eigenvalue analysis) see Hastings pp 125-126, 156-162, or E-K p 137, section 4.9, pp 185-187, section 5.10. Since we will be using Maple as a "black box" to compute eigenvalues, you don't need to worry about the characteristic polynomial, the trace-determinant criteria, etc.
  • Better worth your time would be to try to get a firm grasp on what eigenvalue type corresponds to what behavior of trajectories near an equilibrium. For this purpose, step through the Maple worksheet localstab.mws