The spatial density $n(x, t)$ of a population at location $x$ and time $t$ satisfies

$$\frac{\partial n}{\partial t}=f(n)+D \frac{\partial^{2} n}{\partial x^{2}}$$

where $f(n)=-n(n-r)(n-1), 0<r<1$ and $D>0$.

(a) Give a biological example of the sort of phenomenon that this equation describes.

(b) Show that there are three spatially homogeneous and stationary solutions to $(*)$, of which two are linearly stable to homogeneous perturbations and one is linearly unstable.

(c) For $r=\frac{1}{2}$, find the stationary solution to $(*)$ subject to the conditions

$$\lim _{x \rightarrow-\infty} n(x)=1, \quad \lim _{x \rightarrow \infty} n(x)=0 \quad \text { and } \quad n(0)=\frac{1}{2}$$

(d) Write down the differential equation that is satisfied by a travelling-wave solution to $(*)$ of the form $n(x, t)=u(x-c t)$. Let $n_{0}(x)$ be the solution from part (c). Verify that $n_{0}(x-c t)$ satisfies this differential equation for $r \neq \frac{1}{2}$, provided the speed $c$ is chosen appropriately. [Hint: Consider the change to the equation from part (c).]

(e) State how the sign of $c$ depends on $r$, and give a brief qualitative explanation for why this should be the case.