Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 13E (Mathematical Biology)

For jump rate $W(n,r)$ from $n$ to $n+r$, probability enters state $n$ from $n-r$ and leaves it from $n$. The resulting kramers-Moyal master equation is

When jumps are small relative to a large population and moments exist, the Kramers-Moyal expansion through second order givesThis Fokker-Planck equation is a conservation law with vanishing boundary flux; $u$ is the drift and $D$ the diffusivity. Equal constant rates for jumps $r=\pm 1$ give pure diffusion.

Integration by parts gives $d⟨x⟩/dt=⟨u⟩$ and $d\mathrm{var}x/dt=2⟨D⟩+2\mathrm{Cov}(x,u)$.

Here $u=-2\sinh (\lambda x)$ and $D=\cosh (\lambda x)$. Zero stationary flux gives $(DP{)}^{\prime }=uP$, so $P\propto D^{-1}\exp \int u/D,dx\propto \cosh (\lambda x{)}^{-(1+2/\lambda )}$, a symmetric unimodal distribution.

For constants $u=a-b$ and $D=(a+b)/2$, $z=x-ut$ removes drift and gives ${\tilde{P}}_t=D{\tilde{P}}_{zz}$.

At the earliest negative minimum, $\tilde{P}=0$, ${\tilde{P}}_z=0$, ${\tilde{P}}_{zz}\ge 0$, so the diffusion equation gives ${\tilde{P}}_t\ge 0$, contradicting first passage below zero. This is the maximum principle.

The third Kramers-Moyal expansion coefficient is $F=\frac{1}{6}{\sum }_r{r}^{3}W=(a-b)/6$. At a first minimum the third derivative has no fixed sign, so the maximum-principle argument fails; finite truncations beyond second order need not preserve positivity.