Consider the system: x' = x(1 - ax - y), y' = y(b - x - y) + h
Here a, b, and h are parameters. We assume that a, b > 0. If h > 0 then we are harvesting species y at a constant rate, while if h > 0, we add to the population y at a constant rate. The goal is to understand this system completely for all possible values of these parameters. As usual, we only consider the regime where x, y => 0. If y(t) <= 0 for any t > 0, then we consider this species to have become extinct.
1) First assume that h = 0. Give a complete synopsis of the behavior of this system by plotting the different behaviors you find in the a, b parameter plane.
2) Identify the points or curves in the ab-plane where bifurcations occur when h = 0 and describe them.
3) Now let h > 0. Describe the ab-parameter plane for various (fixed) h-values.
4) Repeat the previous exploration for h > 0.
5) Describe the full 3-D parameter space using pictures, flip books, 3D models, movies, or whatever you find most appropriate.
Sounds simple, doesn't it? Well... let's just say this this pretty much ripped me a new digestive tract... And no, I won't post my answers to them. At least not yet. There's still a gaggle of undergrads that are taking this final at Caltech. I don't want to violate the Honor Code
I really should be working on the rest of my finals... More finals ranting on the way as my stress level reaches that unstable fixed point again!
Next installment: New uses for Foxtrot and Quickstep music and anything I can dig up about geography... maybe...
1 comment:
Might be a tad late since I'm a bit slow on comments but I used Mathematica and Excel to do this. Basically I used Mathematica for the 3-D parameter space. In retrospect, a movie or flip book would've done much better because I had a nightmare of a time trying to do 3-D graphing, but then again, at the time my knowledge of Mathematica isn't that strong.
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