Problems 4 through 7 deal with the competition system

in which c₁c₂ = 9 > 8 = b₁b₂, so the effect of competition should exceed that of inhibition. Problems 4 through 7 imply that the four critical points (0,0), (0, 21), (15,0), and (6, 12) of the system in (2) resemble those shown in Fig. 9.3.9-a nodal source at the origin, a nodal sink on each coordinate axis, and a saddle point interior to the first quadrant. In each of these problems use a graphing calculator or computer system to construct a phase plane portrait for the linearization at the indicated critical point. Finally, construct a first-quadrant phase plane portrait for the nonlinear system in (2). Do your local and global portraits look consistent?

Show that the coefficient matrix of the linearization x' = 60x, y' = 42y of (2) at (0,0) has positive eigenvalues λ₁ = 60 and  λ₂ = 42. Hence (0,0) is a nodal source for (2).

Transcribed Image Text:

dx dt dy dt 60x - 4x² - 3xy, = 42y - 2y? - 3x y, (2)