  In Part II of this series of papers, we consider an initial-boundary value
problem for the Kolmogorov--Petrovskii--Piscounov (KPP) type equation with a
discontinuous cut-off in the reaction function at concentration $u=u_c$. For
fixed cut-off value $u_c \in (0,1)$, we apply the method of matched asymptotic
coordinate expansions to obtain the complete large-time asymptotic form of the
solution which exhibits the formation of a permanent form travelling wave
structure. In particular, this approach allows the correction to the wave speed
and the rate of convergence of the solution onto the permanent form travelling
wave to be determined via a detailed analysis of the asymptotic structures in
small-time and, subsequently, in large-space. The asymptotic results are
confirmed against numerical results obtained for the particular case of a
cut-off Fisher reaction function.
