 Solution

There is no stable positive coexistence state: the only one is a saddle for every $\rho ,\alpha >0$ when $\beta >1$. Moreover, the Bendixson-Dulac criterion with Dulac function $B(x,y)=1/(xy)$ applies in the positive quadrant, sinceThus no periodic orbit can support long-term coexistence. Except on the saddle's stable manifold, one species is excluded: species 2 becomes extinct for initial data in the basin of $(1,0)$, ecologically when species 1 is sufficiently abundant and species 2 sufficiently scarce. On the other side, species 1 becomes extinct and the unregulated second species grows.