(a) A stochastic birth-death process has a master equation given by

$$\frac{d p_{n}}{d t}=\lambda\left(p_{n-1}-p_{n}\right)+\beta\left[(n+1) p_{n+1}-n p_{n}\right]$$

where $p_{n}(t)$ is the probability that there are $n$ individuals in the population at time $t$ for $n=0,1,2, \ldots$ and $p_{n}=0$ for $n<0$.

(i) Give a brief interpretation of $\lambda$ and $\beta$.

(ii) Derive an equation for $\frac{\partial \phi}{\partial t}$, where $\phi$ is the generating function

$$\phi(s, t)=\sum_{n=0}^{\infty} s^{n} p_{n}(t)$$

(iii) Assuming that the generating function $\phi$ takes the form

$$\phi(s, t)=e^{(s-1) f(t)}$$

find $f(t)$ and hence show that, as $t \rightarrow \infty$, both the mean $\langle n\rangle$ and variance $\sigma^{2}$ of the population size tend to constant values, which you should determine.

(b) Now suppose an extra process is included: $k$ individuals are added to the population at rate $\epsilon(n)$.

(i) Write down the new master equation, and explain why, for $k>1$, the approach used in part (a) will fail.

(ii) By working with the master equation directly, find a differential equation for the rate of change of the mean population size $\langle n\rangle$.

(iii) Now take $\epsilon(n)=a n+b$ for positive constants $a$ and $b$. Show that for $\beta>a k$ the mean population size tends to a constant, which you should determine. Briefly describe what happens for $\beta<a k$.