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In Part II of this series of papers, we consider an initial-boundary valueproblem for the Kolmogorov--Petrovskii--Piscounov (KPP) type equation with adiscontinuous cut-off in the reaction function at concentration $u=u_c$. Forfixed cut-off value $u_c \in (0,1)$, we apply the method of matched asymptoticcoordinate expansions to obtain the complete large-time asymptotic form of thesolution which exhibits the formation of a permanent form travelling wavestructure. In particular, this approach allows the correction to the wave speedand the rate of convergence of the solution onto the permanent form travellingwave to be determined via a detailed analysis of the asymptotic structures insmall-time and, subsequently, in large-space. The asymptotic results areconfirmed against numerical results obtained for the particular case of acut-off Fisher reaction function.

The Evolution of Travelling Waves in a KPP Reaction-Diffusion Model with cut-off Reaction Rate. II. Evolution of Travelling Waves