Consider a stochastic birth-death process in a population of size $n(t)$, where deaths occur in pairs for $n \geqslant 2$. The probability per unit time of a birth, $n \rightarrow n+1$ for $n \geqslant 0$, is $b$, that of a pair of deaths, $n \rightarrow n-2$ for $n \geqslant 2$, is $d n$, and that of the death of a lonely singleton, $1 \rightarrow 0$, is $D$.

(a) Write down the master equation for $p_{n}(t)$, the probability of a population of size $n$ at time $t$, distinguishing between the cases $n \geqslant 2, n=0$ and $n=1$.

(b) For a function $f(n), n \geqslant 0$, show carefully that

$$\frac{d}{d t}\langle f(n)\rangle=b \sum_{n=0}^{\infty}\left(f_{n+1}-f_{n}\right) p_{n}-d \sum_{n=2}^{\infty}\left(f_{n}-f_{n-2}\right) n p_{n}-D\left(f_{1}-f_{0}\right) p_{1}$$

where $f_{n}=f(n)$.

(c) Deduce the evolution equation for the mean $\mu(t)=\langle n\rangle$, and simplify it for the case $D=2 d$.

(d) For the same value of $D$, show that

$$\frac{d}{d t}\left\langle n^{2}\right\rangle=b(2 \mu+1)-4 d\left(\left\langle n^{2}\right\rangle-\mu\right)-2 d p_{1}$$

Deduce that the variance $\sigma^{2}$ in the stationary state for $b, d>0$ satisfies

$$\frac{3 b}{4 d}-\frac{1}{2}<\sigma^{2}<\frac{3 b}{4 d}$$