= Pair immigration with linear deaths
{title2=$m'=2\lambda-\beta m$, $V'=4\lambda+\beta m-2\beta V$}
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.
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