The larva of a parasitic worm disperses in one dimension while laying eggs at rate $\lambda>0$. The larvae die at rate $\mu$ and have diffusivity $D$, so that their density, $n(x, t)$, obeys

$$\frac{\partial n}{\partial t}=D \frac{\partial^{2} n}{\partial x^{2}}-\mu n, \quad(D>0, \mu>0)$$

The eggs do not diffuse, so that their density, $e(x, t)$, obeys

$$\frac{\partial e}{\partial t}=\lambda n$$

At $t=0$ there are no eggs and $N$ larvae concentrated at $x=0$, so that $n(x, 0)=N \delta(x)$.

(i) Determine $n(x, t)$ for $t>0$. Show that $n(x, t) \rightarrow 0$ as $t \rightarrow \infty$.

(ii) Determine the limit of $e(x, t)$ as $t \rightarrow \infty$.

(iii) Provide a physical explanation for the remnant density of the eggs identified in part (ii).

[You may quote without proof the results

$$\begin{aligned} \int_{-\infty}^{\infty} \exp \left(-x^{2}\right) d x &=\sqrt{\pi} \\ \int_{-\infty}^{\infty} \frac{\exp (i k x)}{k^{2}+\alpha^{2}} d k &=\pi \exp (-\alpha|x|) / \alpha, \quad \alpha>0 \end{aligned}$$