= Solution
At an <equilibrium>, $H=rHe^{-aP}$ and $P=bH(1-e^{-aP})$. Taking $H=0$ forces $P=0$, giving extinction. If $H\ne0$, the first equation gives $e^{-aP}=1/r$ and hence
$$
\boxed{P_*=\frac{\log r}{a},\qquad H_*=
\frac{r\log r}{ab(r-1)}}\qquad(r\ne1).
$$
These are the only isolated candidates obtained from the equations. There is an exceptional parameter case: at $r=1$, $P=0$ and every $H\ge0$ is an <equilibrium>. Thus the assertion of exactly two <equilibria> needs $r\ne1$, and only the positive candidate belongs to the biological state space when $r>1$.
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