The radially symmetric spread of an insect population density $n(r, t)$ in the plane is described by the equation

$$\frac{\partial n}{\partial t}=\frac{D_{0}}{r} \frac{\partial}{\partial r}\left[r\left(\frac{n}{n_{0}}\right)^{2} \frac{\partial n}{\partial r}\right]$$

Suppose $Q$ insects are released at $r=0$ at $t=0$. We wish to find a similarity solution to $(*)$ in the form

$$n(r, t)=\frac{n_{0}}{\lambda^{2}(t)} F\left(\frac{r}{r_{0} \lambda(t)}\right)$$

Show first that the PDE $(*)$ reduces to an ODE for $F$ if $\lambda(t)$ obeys the equation

$$\lambda^{5} \frac{d \lambda}{d t}=C \frac{D_{0}}{r_{0}^{2}}$$

where $C$ is an arbitrary constant (that may be set to unity), and then obtain $\lambda(t)$ and $F$ such that $F(0)=1$ and $F(\xi)=0$ for $\xi \geqslant 1$. Determine $r_{0}$ in terms of $n_{0}$ and $Q$. Sketch the function $n(r, t)$ at various times to indicate its qualitative behaviour.