Lotka-Volterra systems are one of the oldest and most fundamental mathematical renditions of ecological problems. They are typically illustrated with predator-prey relationships, but I was introduced to them dealing with problems of competition for space by plants in niche ecological communities, where they work as well.
I’m posting because I found this great site which has an applet that can be used to play with coefficients and demonstrate the variety of behaviors available for these systems. In my opinion, it’s also a great place to begin to understand the variety of behaviors that are possible from a set of relatively simple, coupled differential equations, just based upon initializing conditions. That’s a very powerful idea.
Incidently, this is also an important idea because of its relationship to trophic cascades.
See the Wikipedia entry for more on particulars of Lotka-Volterra systems.
Postscript 3rd January 2014: There’s a much better treatment of this now at the posting from today.