A model of insect dispersal and growth in one spatial dimension is given by

$$\frac{\partial N}{\partial t}=D \frac{\partial}{\partial x}\left(N^{2} \frac{\partial N}{\partial x}\right)+\alpha N, \quad N(x, 0)=N_{0} \delta(x),$$

where $\alpha, D$ and $N_{0}$ are constants, $D>0$, and $\alpha$ may be positive or negative.

By setting $N(x, t)=R(x, \tau) e^{\alpha t}$, where $\tau(t)$ is some time-like variable satisfying $\tau(0)=0$, show that a suitable choice of $\tau$ yields

$$R_{\tau}=\left(R^{2} R_{x}\right)_{x}, \quad R(x, 0)=N_{0} \delta(x)$$

where subscript denotes differentiation with respect to $x$ or $\tau$.

Consider a similarity solution of the form $R(x, \tau)=F(\xi) / \tau^{\frac{1}{4}}$ where $\xi=x / \tau^{\frac{1}{4}}$. Show that $F$ must satisfy

$$-\frac{1}{4}(F \xi)^{\prime}=\left(F^{2} F^{\prime}\right)^{\prime} \quad \text { and } \quad \int_{-\infty}^{+\infty} F(\xi) d \xi=N_{0}$$

[You may use the fact that these are solved by

$$F(\xi)= \begin{cases}\frac{1}{2} \sqrt{\xi_{0}^{2}-\xi^{2}} & \text { for }|\xi|<\xi_{0} \\ 0 & \text { otherwise }\end{cases}$$

where $\left.\xi_{0}=\sqrt{4 N_{0} / \pi} .\right]$

For $\alpha<0$, what is the maximum distance from the origin that insects ever reach? Give your answer in terms of $D, \alpha$ and $N_{0}$.