While x and y may change over time, C(t) does not. Thus, C is a conserved quantity and its existence gives a conservation law. A trajectory that begins at a point (x, y) at time t = 0 gives a value of C that remains unchanged at future times. Each value of the constant C gives a trajectory for the autonomous system, and these trajectories close up, rather than spiraling inward or outward. The rabbit and fox populations oscillate through repeated cycles along a fixed trajectory. Figure 9.37 shows several trajectories for the predator-prey system.

Using a procedure similar to that in the text for the competitivehunter model, show that each trajectory is traversed in a counterclockwise direction as time t increases.

Fox population a b 0 d Rabbit population X FIGURE 9.37 Some trajectories along which C is conserved.