A (not so good) predator-prey problem

Here's another gem from Strogatz. A classic model of predator prey interaction favored by text-book writers (and dismissed by biologists) is the Lotka-Volterra predator-prey model which we can think of as rabbits vs. coyotes (or coyotes vs. ACME). If we let R(t) be the number of rabbits and C(t) be the number of Coyotes per 100 acres, then we can write this model as

 

where and are growth rates and and are specific, constant values of population density of coyotes and rabbits.

• Discuss the biological meaning of each equation and comment on any unrealistic assumptions (hint: think of each equation like a simple growth rate equation e.g. (Equation 2) but now the growth rate r is a function of coyotes or rabbits). This is tricky but worth the discussion. You should also do it at the end of the problem.
• Show that the two points (R=0,C=0) and ( ) are fixed points in this system.
• Extra credit: Prove that those points are the only fixed points (hint: draw the null-clines).
• Sketch some direction arrows for the phase-plane plot and predict the behavior of the two fixed points.
• Choose , and (or any other values you please) and use STELLA to test your intuition and show that the model predicts cycles in the populations of both species, for almost all initial conditions. (which initial conditions do not produce cycles). What happens if you start with 1 rabbit and 10 coyotes per 100 acres? Does this make sense to you? Why or Why not?
• Discuss Strogatz's point that ``This model is popular with many textbook writers because it's simple, but some are beguiled into taking it too seriously. Mathematical biologists dismiss the Lotka-Volterra model because it is not structurally stable, and because real predator-prey cycles typically have a characteristic amplitude. In other words, realistic models should predict a single closed orbit, or perhaps finitely many, but not a continuous family of neutrally stable cycles. See the discussions in May (1972) [4], Edelstein-Keshet (1988) [5], or Murray (1989) [6]