The master equation describing the evolution of the probability $P(n, t)$ that a population has $n$ members at time $t$ takes the form

$$\frac{\partial P(n, t)}{\partial t}=b(n-1) P(n-1, t)-[b(n)+d(n)] P(n, t)+d(n+1) P(n+1, t)$$

where the functions $b(n)$ and $d(n)$ are both positive for all $n$.

From (1) derive the corresponding Fokker-Planck equation in the form

$$\frac{\partial P(x, t)}{\partial t}=-\frac{\partial}{\partial x}\left\{a_{1}(x) P(x, t)\right\}+\frac{1}{2} \frac{\partial^{2}}{\partial x^{2}}\left\{a_{2}(x) P(x, t)\right\}$$

making clear any assumptions that you make and giving explicit forms for $a_{1}(x)$ and $a_{2}(x)$.

Assume that (2) has a steady state solution $P_{s}(x)$ and that $a_{1}(x)$ is a decreasing function of $x$ with a single zero at $x_{0}$. Under what circumstances may $P_{s}(x)$ be approximated by a Gaussian centred at $x_{0}$ and what is the corresponding estimate of the variance?