 Constant-birth pair-annihilation process

For jumps $n\to n+1$ at rate $\lambda$ and $n\to n-2$ at rate $\beta n^2$, the diffusion approximation hasThe stable deterministic population is $n_{\ast }=\sqrt{\lambda /(2\beta )}$. Its linear-noise stationary approximation is normal with mean $n_{\ast }$ and variance $3n_{\ast }/4$.