Solution
= Solution
Removing the death term with $u(x,t)=e^{\mu t}n(x,t)$ gives the <heat equation> $u_t=Du_{xx}$ with initial data $N\delta(x)$. The <heat kernel> therefore yields
$$
n(x,t)=\frac{N}{\sqrt{4\pi Dt}}
\exp\left(-\frac{x^2}{4Dt}-\mu t\right).
$$
For each fixed $x$, the exponential mortality factor and diffusive prefactor show that $n(x,t)\to0$ as $t\to\infty$. Its spatial integral is $Ne^{-\mu t}$, the expected number of surviving larvae.