Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 13A (Mathematical Biology)

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.

For a Poisson random variable of mean $\lambda$,Substitution into part (b) gives the closed Riccati equationIts positive equilibrium is stable on the nonnegative half-line and equalsThus under the Poisson approximation,

The diffusion approximation of a birth-death process uses the first two jump moments. For these unit upward and downward jumps they giveThe factor $1/2$ in $D$ is required because the equation is written with $+{\partial }_x^2(DP)$, rather than $\frac{1}{2}{\partial }_x^2(BP)$.

Assuming the boundary terms vanish, integration by parts givesConsequentlywhich is precisely the discrete first-moment equation in part (b).