= Phase-plane construction of a Fisher travelling front
{title2=$c\geq2\sqrt{k}$}
For $u_t=u_{xx}+u(k-u)$ with $k>0$, a decreasing <travelling wave> connecting $k$ to $0$ satisfies $w'=p$, $p'=-cp-w(k-w)$. If $c\geq2\sqrt{k}$, choose $a>0$ with $a(c-a)\geq k$. The triangle $0\leq w\leq k$, $-aw\leq p\leq0$ is forward invariant: on its sloping side, $(p+aw)'=w[a(c-a)-k+w]\geq0$. The descending <unstable manifold> of the <saddle equilibrium> $(k,0)$ stays in this triangle and tends to $(0,0)$, producing a positive <heteroclinic orbit>. Below the threshold the origin is a <stable spiral> for positive $c$, forcing sign changes; at $c=0$ conserved energy also excludes the connection.
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