= Solution
Put $\xi=r/(r_0\lambda)$ and $n=n_0\lambda^{-2}F(\xi)$. Direct <differentiation> gives
$$
n_t=-n_0\lambda^{-3}\dot\lambda(2F+\xi F'),\qquad
\frac{D_0}{r}\partial_r\!\left[r(n/n_0)^2n_r\right]
=\frac{D_0n_0}{r_0^2\lambda^8}\frac1\xi(\xi F^2F')'.
$$
Thus the <partial differential equation> reduces to an <ordinary differential equation> precisely when $\lambda^5\dot\lambda=CD_0/r_0^2$. The profile then satisfies $(\xi F^2F')'=-C(\xi^2F)'$. Regularity and zero radial <flux> at the origin make the integration constant zero. Wherever $F>0$, $FF'=-C\xi$, so $F^2=1-C\xi^2$ using $F(0)=1$. Requiring the front at $\xi=1$ selects $C=1$. Point release requires $\lambda(0)=0$, giving
$$
\boxed{\lambda(t)=\left(\frac{6D_0t}{r_0^2}\right)^{1/6},\qquad F(\xi)=\begin{cases}\sqrt{1-\xi^2},&0\leq\xi<1,\\0,&\xi\geq1.\end{cases}}
$$
Although $F'$ diverges at the front, $F^2F'$ tends to zero, so this compactly supported profile satisfies the equation in the <weak solution> sense.
The <compact radial cubic-diffusion profile> conserves the total population:
$$
Q=2\pi\int_0^\infty rn(r,t)\,dr
=2\pi n_0r_0^2\int_0^1\xi\sqrt{1-\xi^2}\,d\xi
=\frac{2\pi n_0r_0^2}{3}.
$$
Consequently \b[$\boxed{r_0=\sqrt{3Q/(2\pi n_0)}}$.] The front radius grows as $t^{1/6}$, while the central density decays as $t^{-1/3}$. The conserved mass concentrates at the origin as $t\downarrow0$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/ii/paper-2-spread.png]
{title=Compact insect-density profiles spreading outward while decreasing in height}
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