Consider a population subject to the following birth-death process. When the number of individuals in the population is $n$, the probability of an increase from $n$ to $n+1$ in unit time is $\beta n+\gamma$ and the probability of a decrease from $n$ to $n-1$ is $\alpha n(n-1)$, where $\alpha, \beta$ and $\gamma$ are constants.

Show that the master equation for $P(n, t)$, the probability that at time $t$ the population has $n$ members, is

$$\frac{\partial P}{\partial t}=\alpha n(n+1) P(n+1, t)-\alpha n(n-1) P(n, t)+(\beta n-\beta+\gamma) P(n-1, t)-(\beta n+\gamma) P(n, t) .$$

Show that $\langle n\rangle$, the mean number of individuals in the population, satisfies

$$\frac{d\langle n\rangle}{d t}=-\alpha\left\langle n^{2}\right\rangle+(\alpha+\beta)\langle n\rangle+\gamma$$

Deduce that, in a steady state,

$$\langle n\rangle=\frac{\alpha+\beta}{2 \alpha} \pm \sqrt{\frac{(\alpha+\beta)^{2}}{4 \alpha^{2}}+\frac{\gamma}{\alpha}-(\Delta n)^{2}}$$

where $\Delta n$ is the standard deviation of $n$. When is the minus sign admissable?

Show how a Fokker-Planck equation of the form

$$\frac{\partial P}{\partial t}=\frac{\partial}{\partial n}[g(n) P(n, t)]+\frac{1}{2} \frac{\partial^{2}}{\partial n^{2}}[h(n) P(n, t)]$$

may be derived under conditions to be explained, where the functions $g(n)$ and $h(n)$ should be evaluated.

In the case $\alpha \ll \gamma$ and $\beta=0$, find the leading-order approximation to $n_{*}$ such that $g\left(n_{*}\right)=0$. Defining the new variable $x=n-n_{*}$, where $g\left(n_{*}\right)=0$, approximate $g(n)$ by $g^{\prime}\left(n_{*}\right) x$ and $h(n)$ by $h\left(n_{*}\right)$. Solve $(*)$ for $P(x)$ in the steady-state limit and deduce leading-order estimates for $\langle n\rangle$ and $(\Delta n)^{2}$.