Consider a birth-death process in which the birth rate per individual is $\lambda$ and the death rate per individual in a population of size $n$ is $\beta n$.

Let $P(n, t)$ be the probability that the population has size $n$ at time $t$. Write down the master equation for the system, giving an expression for $\partial P(n, t) / \partial t$.

Show that

$$\frac{d}{d t}\langle n\rangle=\lambda\langle n\rangle-\beta\left\langle n^{2}\right\rangle$$

where $\langle.\rangle$ denotes the mean.

Deduce that in a steady state $\langle n\rangle \leqslant \lambda / \beta$.