Dispersal with mortality and immobile deposition
= 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 $\lambda$,
$$
n_t=Dn_{xx}-\mu n,
\qquad
e_t=\lambda n.
$$
An initial point population $N\delta(x)$ leaves the <remnant density from diffusion with mortality and deposition>
$$
e(x,\infty)=\frac{N\lambda}{2\sqrt{D\mu}}e^{-|x|\sqrt{\mu/D}}.
$$