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In the birth-death master equation, The upward jump has rate $\lambda$ and the downward jump rate $\beta x$, so the first two jump moments per unit time are $A(x)=\lambda -\beta x$ and $B(x)=\lambda +\beta x$. The second-order Kramers-Moyal expansion gives the diffusion approximation of a birth-death processFor the associated Fokker-Planck equation, integrating by parts with vanishing moment boundary terms gives $d⟨f⟩/dt=⟨A{f}^{\prime }+\frac{1}{2}B{f}^{\prime \prime }⟩$. Taking $f=x,x^2$ yieldsFor example, if $m_0=⟨x(0)⟩$, then $m(t)=\lambda /\beta +(m_0-\lambda /\beta )e^{-\beta t}$, while the variance satisfies $v^{\prime }=-2\beta v+\lambda +\beta m$. The two boxed moment equations also follow exactly from the discrete Markov jump-process generator, since quadratic functions have no higher jump terms. A reflecting diffusion confined to $x\ge 0$ can have extra boundary contributions; the formal approximation near extinction is not an exact replacement for the discrete process.

With $p_{-1}=0$, the birth-death master equation for birth rate $b_n=Bn$ and death rate $d_n=Dn$ is

Multiply the birth-death master equation by $n$ and sum. Index shifts giveFor the second moment, a birth changes $n^2$ by $2n+1$ and a death changes it by $-2n+1$, soSince ${\sigma }^2=\mathbb{E}[N^2]-{\mu }^2$,Together these are the moment equations of a linear birth-death process.

The birth-death master equation must include jumps of size two on death. For $n\ge 2$,At the boundary states it is

This is a birth-death process with birth and death ratesIts transition diagram has the two arrowsThe gain into state $n$ comes from a death in state $n+1$ or a birth in state $n-1$, while the loss is the sum of both rates out of state $n$. The birth-death master equation is consequently

Multiplying by $n$, summing over the nonnegative integers, and shifting the summation indices shows that each birth contributes $+1$ and each death contributes $-1$. Thus the first-moment equation of a birth-death process givesAt a stationary state, use the variance identity $⟨n^2⟩=⟨n{⟩}^2+(\Delta n{)}^2$. Solving the resulting quadratic equation givesThe minus root is inadmissible whenever it is negative, namely when $(\Delta n{)}^2<\gamma /\alpha$. Equality would give zero mean, which is also incompatible with a positive immigration rate $\gamma$, so for $\gamma >0$ the minus branch requires $(\Delta n{)}^2>\gamma /\alpha$ even to be a possible mean.

For a continuum approximation, write $b(n)=\gamma +\beta n$ and $d(n)=\alpha n(n-1)$. Applying the Kramers-Moyal expansion to the two gain terms and retaining derivatives through second order gives the Fokker-Planck equationwhere the negative drift and infinitesimal jump variance areThis truncation requires the typical population and the scale on which $P$ varies to be much larger than the unit jump size, with the rates varying smoothly across that scale.

The positive zero of $g$ isso in particular $n_{\ast }\sim \beta /\alpha$ as $\alpha /\beta \to 0$. Put $x=n-n_{\ast }$. The linear noise approximation useswhereand, because $b(n_{\ast })=d(n_{\ast })$,In a stationary state with zero Fokker-Planck probability current,Its normalized solution is the normal distributionIt follows that

These estimates agree with the exact stationary first-moment relation on its plus branch to leading order: inserting $(\Delta n{)}^2\sim \beta /\alpha$ gives $⟨n⟩\sim \beta /\alpha$. Moreover,Thus the stationary mass lies far from the boundary $n=0$, is narrow relative to its mean, yet changes across many lattice sites. These are precisely the large-population and slow-variation conditions needed for the diffusion approximation.

This birth-death process has birth rates $b_n=\alpha +\beta n$ and death rates $d_n=\gamma n(n-1)$. With $P_{-1}=0$, its birth-death master equation isThe first two terms are inflow by a birth from $n-1$ and a death from $n+1$; the bracketed term is the total outflow from $n$.

Multiply the birth-death master equation by $n$ and sum. Equivalently, a birth changes $n$ by $+1$ and a death by $-1$, soThe equation is not closed in the mean because the quadratic death rate introduces the second factorial moment.