= Solution
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.
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