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 21, with a=b=c=d=1

Problem 21

Simpler Competition Consider a situation of two populations competing according to the simpler model

where a, b, c, and d are positive constants. Find and sketch the equilibrium points and the nullclines, and determine directions of the vector field between the nullclines. Then determine the stability of the equilibrium points, and sketch the phase portrait of the system.