We have the Fisher-KPP equation: $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + ru(1-u)$
We can reduce this to a second order ODE: $cu_{\xi} = u_{\xi\xi}+u(1-u)$ where $\xi = x+ct$. We can rewrite this as a system of 2 first-order equations,
$u' = V$
$V' = cV-u(1-u)$
Clearly the fixed points are $(U,V) = (0,0), (1,0)$.
Question:
How can we show that a seperatrix exists, which connects the 2 equilibrium points. Our hint is that for $c\geq2$ no trajectories can enter the triangle,T given by:
$0\leq u\leq1$ and $0\leq V\leq \frac{cu}{2}$.
We sort of verified this graphically, by plotting the phase plane, but I am not sure that that is enough to show the existence of the speratrix.