Each larva undergoes <diffusion equation>[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 gives
$$
\int_{-\infty}^{\infty}e(x,\infty)\,dx=\frac{N\lambda}{\mu}.
$$
This 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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 13B / iii / Solution

{title2=$\sqrt{D/\mu}$}
{wiki}

For diffusivity $D$ and removal rate $\mu$, the diffusion length $\sqrt{D/\mu}$ is the characteristic distance travelled before removal.


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 Incoming links: Diffusion length