 Solution

The transition rates of this birth-death process are $q_{n,n+1}=\lambda n$ and $q_{n,n-1}=n(\gamma +\beta n)$. Thus, with $p_{-1}=0$,Multiplying by $n$, summing, and shifting indices givesAt a finite-moment steady state with positive mean and $\lambda >\gamma$,Since $\mathbb{E}(N^2)\ge (\mathbb{E}N{)}^2$, this impliesIf $\lambda \le \gamma$, both terms in the mean equation are nonpositive, and stationarity forces $\mathbb{E}N=0$. Thus the population is extinct almost surely in any such steady state.