Question 7.4.2: Using a Direction Field to Sketch a Phase Portrait Sketch a ...
Using a Direction Field to Sketch a Phase Portrait
Sketch a direction field of \frac{d y}{d x}=\frac{-0.1 y+0.1 x y}{0.2 x-0.1 x^2-0.4 x y} \text {, } and use the resulting phase portrait to determine the stability of the three equilibrium points (0, 0), (2, 0) and(1, 0.25).
The direction field generated by our CAS (see Figure 7.20) is not especially helpful, largely because it does not show sufficient detail near the equilibrium points. To more clearly see the behavior of solutions near each equilibrium solution, we zoom in on each equilibrium point in turn and plot a number of solution curves, as shown in Figures 7.21a, 7.21b and 7.21c.
Since the arrows in Figure 7.21a point away from (0, 0), we refer to (0, 0) as an unstable equilibrium. That is, solutions that start out close to (0, 0) move away from that point as t \rightarrow \infty . Similarly, most of the arrows in Figure 7.21b point away from (2, 0) and so, we conclude that (2, 0) is also unstable. Finally, in Figure 7.21c, the arrows spiral in toward the point (1, 0.25), indicating that solutions that start out near (1, 0.25) tend toward that point as t \rightarrow \infty, making this is a stable equilibrium. From this, we conclude that the naturally balanced state is for the two species to coexist, with 4 times as many prey as predators.
