 Solution

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.