Notes for BMA 771

• Notes on the Lotka-Volterra competition model (non-dimensionalization, nullclines and stability analysis in the general case).
• Nullclines and stability. When are nullclines informative about equilibrium stability?
• Conversion from Cartesian to polar co-ordinates
• Notes on linear systems. (More in-depth than the coverage in class.)
• PPLANE website. You will need to download the PPLANE .jar file and have the java runtime environment installed on your computer. You may also have to do something with the security settings to allow unsigned apps to run. Brief notes on using PPLANE, borrowed from another course I teach.
• Vector and direction field plots in MATLAB using the “quiver” command
• Firefly model code: firefly.m
• Cusp surfaces: Maple worksheet for visualizing cusp surfaces (use Save As… and you might need to rename the file to end in .mws)
• Maple
  Maple is a symbolic algebra package that can do things such as solving systems of equations (algebraically or numerically), differentiate and integrate functions, solve differential equations (algebraically or numerically), make various plots—including slope fields, direction field plots and plots of implicitly-defined functions, and lots of other stuff.
  • Maple resources.
  • Maple introduction worksheet. Maple worksheet that I showed in class that includes a number of useful commands… copy/paste, edit to your needs! Note that you do have to hit enter at each line as you go through this. Also, some commands are available through packages that have to be added in (see the “with” commands for those).
  • Text file listing commands used.

  
  Note: because both of these are text files, you may have to use “Save As…” on the links to get a file, rather than seeing the contents in your browser.
  

• Numerical Integration of Differential Equations
  • Brief notes. Includes a link to an interesting page from MATLAB on its routines. (Apologies: the link in the PDF doesn’t work for some reason…)
  • Simulating the SIR epidemic model in MATLAB. Some notes on the SIR epidemic model and its simulation in MATLAB. The final three slides detail the ODE45 integration routine in MATLAB and its use for the SIR model. Text from the final slide could be copied and pasted into a MATLAB .m file and a simulation run from there.
• Potentials
  • Video on Potentials. Notes
  • Video on equilibria and their stability using geometric, algebraic and potentials approaches. Notes
• Separation of Variables
  • Video on Separable Differential Equations. Notes
  • Video: Method of Separation of Variables. Notes
  • Another example of separation of variables. Notes
  • Video: Solving exponential growth model. Notes
  • Notes from class
• Slope Fields
  • Video on Slope Fields Method. Notes

• Slope fields for one dimensional ODEs (includes Maple code and links to online applets)
• Analytic solutions for some special types of ODE.
• ODE terminology
• Introduction