Prove that neither of the following systems have limit cycles using the given multipliers:

(a) ˙ x = x(4 + 5x + 2y), ˙ y = y(−2 + 7x + 3y), ψ = 1 xy2 .

(b) ˙ x = x(β − δx − γy), ˙ y = y(b − dy − cx), ψ = 1 xy .

In case (b), prove that there are no limit cycles in the first quadrant only.

These differential equations were used as a general model for competing species in Chapter 3.