 Pair immigration with linear deaths

A population jumps from $n$ to $n+2$ at a constant immigration rate $\lambda$ and from $n$ to $n-1$ at rate $\beta n$. Applying its Markov jump-process generator to $n$ and $n^2$ gives the displayed mean and variance equations. For $\beta >0$ and finite initial second moment, the stationary mean and variance are $2\lambda /\beta$ and $3\lambda /\beta$. Interpreting the pair-arrival rate as proportional to population instead would produce a different branching process.