Consider the nonlinear equation describing the invasion of a population $u(x, t)$

$$u_{t}=m u_{x x}+f(u)$$

with $m>0, f(u)=-u(u-r)(u-1)$ and $0<r<1$ a constant.

(a) Considering time-dependent spatially homogeneous solutions, show that there are two stable and one unstable uniform steady states.

(b) In the case $r=\frac{1}{2}$, find the stationary 'front' which has

$$u \rightarrow 1 \text { as } x \rightarrow-\infty \quad \text { and } \quad u \rightarrow 0 \text { as } x \rightarrow \infty$$

[Hint: $f(u)=F^{\prime}(u)$ where $F(u)=-\frac{1}{4} u^{2}(1-u)^{2}+\frac{1}{6}\left(r-\frac{1}{2}\right) u^{2}(2 u-3)$.]

(c) Now consider travelling-wave solutions to (1) of the form $u(x, t)=U(z)$ where $z=x-v t$. Show that $U$ satisfies an equation of the form

$$m \ddot{U}+v \dot{U}=-V^{\prime}(U),$$

where $\left({ }^{\cdot}\right) \equiv \frac{d}{d z}(\quad)$ and ()$^{\prime} \equiv \frac{d}{d U}(\quad)$.

Sketch the form of $V(U)$ for $r=\frac{1}{2}, r>\frac{1}{2}$ and $r<\frac{1}{2}$. Using conditions (2), show that

$$v \int_{-\infty}^{\infty} \dot{U}^{2} d z=F(1)-F(0) .$$

Deduce how the sign of the travelling-wave velocity $v$ depends on $r$.