Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 13B (Mathematical Biology)

Removing the death term with $u(x,t)=e^{\mu t}n(x,t)$ gives the heat equation $u_t=D{u}_{xx}$ with initial data $N\delta (x)$. The heat kernel therefore yieldsFor each fixed $x$, the exponential mortality factor and diffusive prefactor show that $n(x,t)\to 0$ as $t\to \infty$. Its spatial integral is $N{e}^{-\mu t}$, the expected number of surviving larvae.

Because $e(x,0)=0$, the egg density is the accumulated depositionIt is convenient to use the Fourier transform. The larval equation givesApplying the stated inverse-transform integral with $\alpha =\sqrt{\mu /D}$ gives the remnant density from diffusion with mortality and deposition

Each larva undergoes diffusion over its exponentially distributed lifetime of mean $1/\mu$ and deposits eggs at rate $\lambda$ wherever it is found. The characteristic diffusion length before death is $\sqrt{D/\mu }$, which explains the exponential spatial decay. Integrating the remnant over space givesThis is also immediate physically: each of the $N$ larvae lives for mean time $1/\mu$ and therefore lays an expected $\lambda /\mu$ eggs. The eggs remain as a stationary record because they neither die nor diffuse.