 Solution

Since $a,b>0$, the coexistence value $P_{\ast }=(\log r)/a$ is positive exactly when $r>1$. Under that condition, $H_{\ast }>0$ as well. Therefore the required condition isFor $0<r<1$, the formal nonzero candidate has negative parasitoid density and is not biologically admissible; $r=1$ gives only the host-only line of equilibria.