How to implement Dulac's criterion?

Question

For this question, I'm having trouble getting started. Can anyone please help me out?

Consider the system

$ x' = x(A-ax+by)$

$ y' = y(B-cy+dx)-x^2y$

Where A,B,a,b,c,d are constants and a,c >0. Show that it cannot be periodic in the first quadrant.

Hint: Use Dulbac's criterion $\delta(x,y) = \frac{1}{xy}$

Answer 1

First show that the first quadrant is invariant. Then multiply by $\delta$ and compute the divergence. To show invariance, note that a solution that touches the axis must stay there forever (compute $x',y'$ for $x=0$ and $y=0$ respectively)

Answer 2

Proposing the test function $\phi(x,y) = x^m y^n$ and proceeding according to the Bendixson-Dulac technique

$$ (\phi(x,y)\dot x)_x + (\phi(x,y)\dot y)_y = \phi(x,y)\left((-a (n+2) x+A (n+1)+b (n+1) y+B (m+1)-c (m+2) y+(m+1) x (d-x))\right) $$

now choosing $m=n=-1, a=\epsilon_1^2, c=\epsilon_2^2$ we have

$$ (\phi(x,y)\dot x)_x + (\phi(x,y)\dot y)_y = -\frac{x\epsilon_1^2+y\epsilon_2^2}{xy}\ne 0 $$

for the first quadrant interior so no limit cycles there.