A proposed model of insect dispersal is given by the equation

$$\frac{\partial n}{\partial t}=D \frac{\partial}{\partial x}\left[\left(\frac{n_{0}}{n}\right) \frac{\partial n}{\partial x}\right]$$

where $n(x, t)$ is the density of insects and $D$ and $n_{0}$ are constants.

Interpret the term on the right-hand side.

Explain why a solution of the form

$$n(x, t)=n_{0}(D t)^{-\beta} g\left(x /(D t)^{\beta}\right),$$

where $\beta$ is a positive constant, can potentially represent the dispersal of a fixed number $n_{0}$ of insects initially localised at the origin.

Show that the equation (1) can be satisfied by a solution of the form (2) if $\beta=1$ and find the corresponding function $g$.