Positive-quadrant Dulac multiplier for quadratic activation ($\nabla \cdot (u^{-2}(f,g))=(b-1)/u^2$)

For the diffusionless quadratic activator-inhibitor model, choose the Dulac function $u^{-2}$. Then $u^{-2}f=1/v-b/u$ and $u^{-2}g=1-v/u^2$, giving divergence $(b-1)/u^2$. For $0<b<1$ it is strictly negative throughout the simply connected positive quadrant. Green's theorem rules out a periodic orbit: its boundary flux would be zero because the rescaled vector field is tangent to the orbit, while its interior divergence integral would be negative.