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>
$$
e(x,\infty)=\frac{N\lambda}{2\sqrt{D\mu}}e^{-|x|\sqrt{\mu/D}}.
$$

Dispersal with mortality and immobile deposition

Because $e(x,0)=0$, the egg density is the accumulated deposition
$$
e(x,t)=\lambda\int_0^t n(x,s)\,ds.
$$
It is convenient to use the <Fourier transform>. The larval equation gives
$$
\widehat n(k,t)=N e^{-(Dk^2+\mu)t},
\qquad
\widehat e(k,\infty)=\frac{N\lambda}{Dk^2+\mu}.
$$
Applying the stated inverse-transform integral with $\alpha=\sqrt{\mu/D}$ gives the <remnant density from diffusion with mortality and deposition>
$$
\boxed{e(x,\infty)=\frac{N\lambda}{2\sqrt{D\mu}}
\exp\left(-|x|\sqrt{\frac\mu D}\right).}
$$

Solution

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

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.

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