Dispersal with mortality and immobile deposition

For diffusing organisms of density $n$ that die at rate $\mu$ and deposit an immobile material of density $e$ at rate $\lambda$,An initial point population $N\delta (x)$ leaves the remnant density from diffusion with mortality and deposition

The remnant profile is the time integral of a decaying heat kernel. Its length scale is $\sqrt{D/\mu }$ and its total mass is $N\lambda /\mu$, the initial population times the mean deposition over an exponential lifetime.

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