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 18

Competition Analyze the models for competition between two species given in Problem, using the following outline. NOTE: We require x and y to be nonnegative because this is a population model.

(a) Find and plot the equilibrium points and nullclines. Determine the directions of the vector fields between the nullclines.

(b) Decide whether the equilibrium points are unstable (repelling at least some nearby solutions) or stable (repelling no nearby solutions).

(c) Sketch portions of solution curves near equilibrium points; then complete the phase portrait of the system.

(d) Decide whether the two species described by the model can coexist. What conditions are required for coexistence when it is possible?