A nonlinear model of insect dispersal with exponential death rate takes the form (for insect population $n(x, t)$ )

$$\tag{*} \frac{\partial n}{\partial t}=-\mu n+\frac{\partial}{\partial x}\left(n \frac{\partial n}{\partial x}\right)$$

At time $t=0$ the total insect population is $Q$, and all the insects are at the origin. A solution is sought in the form

$$\tag{†} n=\frac{e^{-\mu t}}{\lambda(t)} f(\eta) ; \quad \eta=\frac{x}{\lambda(t)}, \quad \lambda(0)=0$$

(a) Verify that $\int_{-\infty}^{\infty} f d \eta=Q$, provided $f$ decays sufficiently rapidly as $|x| \rightarrow \infty$.

(b) Show, by substituting the form of $n$ given in equation $(†)$ into equation $(*)$, that $(*)$ is satisfied, for nonzero $f$, when

$$\frac{d \lambda}{d t}=\lambda^{-2} e^{-\mu t} \quad \text { and } \quad \frac{d f}{d \eta}=-\eta$$

Hence find the complete solution and show that the insect population is always confined to a finite region that never exceeds the range

$$|x| \leqslant\left(\frac{9 Q}{2 \mu}\right)^{1 / 3}$$