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

Removing the death term with $u(x,t)=e^{\mu t}n(x,t)$ gives the heat equation $u_t=D{u}_{xx}$ with initial data $N\delta (x)$. The heat kernel therefore yieldsFor each fixed $x$, the exponential mortality factor and diffusive prefactor show that $n(x,t)\to 0$ as $t\to \infty$. Its spatial integral is $N{e}^{-\mu t}$, the expected number of surviving larvae.