Consider a model of insect dispersal in two dimensions given by

$$\frac{\partial C}{\partial t}=\frac{1}{r} \frac{\partial}{\partial r}\left(r D C \frac{\partial C}{\partial r}\right)$$

where $r$ is a radial coordinate, $t$ is time, $C(r, t)$ is the density of insects and $D$ is a constant coefficient such that $D C$ is a diffusivity.

Show that under suitable assumptions

$$2 \pi \int_{0}^{\infty} r C d r=N$$

where $N$ is constant, and interpret this condition.

Suppose that after a long time the form of $C$ depends only on $r, t, D$ and $N$ (and is thus independent of any detailed form of the initial condition). Show that there is a solution of the form

$$C(r, t)=\left(\frac{N}{D t}\right)^{1 / 2} g\left(\frac{r}{(N D t)^{1 / 4}}\right) \text {, }$$

and deduce that the function $g(\xi)$ satisfies

$$\frac{d}{d \xi}\left(\xi g \frac{d g}{d \xi}+\frac{1}{4} \xi^{2} g\right)=0$$

Show that this equation has a continuous solution with $g>0$ for $\xi<\xi_{0}$ and $g=0$ for $\xi \geqslant \xi_{0}$, and determine $\xi_{0}$. Hence determine the area within which $C(r, t)>0$ as a function of $t$.