(a) The populations of two competing species satisfy

$$\begin{aligned} \frac{d N_{1}}{d t} &=N_{1}\left[b_{1}-\lambda\left(N_{1}+N_{2}\right)\right] \\ \frac{d N_{2}}{d t} &=N_{2}\left[b_{2}-\lambda\left(N_{1}+N_{2}\right)\right] \end{aligned}$$

where $b_{1}>b_{2}>0$ and $\lambda>0$. Sketch the phase diagram (limiting attention to $\left.N_{1}, N_{2} \geqslant 0\right)$.

The relative abundance of species 1 is defined by $U=N_{1} /\left(N_{1}+N_{2}\right)$. Show that

$$\frac{d U}{d t}=A U(1-U)$$

where $A$ is a constant that should be determined.

(b) Consider the spatial system

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

and consider a travelling-wave solution of the form $u(x, t)=f(x-c t)$ representing one species $(u=1)$ invading territory previously occupied by another species $(u=0)$. By linearising near the front of the invasion, show that the wave speed is given by $c=2 \sqrt{D}$.

[You may assume that the solution to the full nonlinear system will settle to the slowest possible linear wave speed.]