Basins of Attraction Problem the specify one of the competition scenarios in the previous...

Basins of Attraction Problem the specify one of the competition scenarios in the previous set of problems. For each stable equilibrium point in the given model, find and color the set of points in the plane whose associated solution curves eventually approach that equilibrium point. An example is given in Fig. 12.

Figure 12 For the system x′ = −x − y, y′ = x − y2, the basin of attraction for the equilibrium at (0, 0) is shaded.

Such a set of points is called the basin of attraction for that equilibrium. Relate the coloring of these basins to your description of the long-term fate of the species given in the corresponding problem.

Problem 2

Predicting System Behavior Consider the systems in Problem.

(a) Determine and plot the equilibrium points and nullclines for the systems.

(b) Show the direction of the vector field between the null- clines, as illustrated in Example 2 and Fig. 4.

(c) Sketch some solution curves starting near, but not on, the equilibrium point(s).

(d) Label each, equilibrium as stable or unstable depending on the behavior of solutions that start nearby, and describe the long-term behavior of all the solutions.

dx/dt = 1 − x − y

dy/dt = x – y2