A model of wound healing in one spatial dimension is given by

$$\frac{\partial S}{\partial t}=r S(1-S)+D \frac{\partial^{2} S}{\partial x^{2}}$$

where $S(x, t)$ gives the density of healthy tissue at spatial position $x$ at time $t$ and $r$ and $D$ are positive constants.

By setting $S(x, t)=f(\xi)$ where $\xi=x-c t$, seek a steady travelling wave solution where $f(\xi)$ tends to one for large negative $\xi$ and tends to zero for large positive $\xi$. By linearising around the leading edge, where $f \approx 1$, find the possible wave speeds $c$ of the system. Assuming that the full nonlinear system will settle to the slowest possible speed, express the wave speed as a function of $D$ and $r$.

Consider now a situation where the tissue is destroyed in some window of length $W$, i.e. $S(x, 0)=0$ for $0<x<W$ for some constant $W>0$ and $S(x, 0)$ is equal to one elsewhere. Explain what will happen for subsequent times, illustrating your answer with sketches of $S(x, t)$. Determine approximately how long it will take for this wound to heal (in the sense that $S$ is close to one everywhere).