Rigorous results for the speed of Kolmogorov--Petrovskii--Piscounov fronts with a cutoff

We study the effect of a cut-off on the speed of pulled fronts of the one dimensional reaction diffusion equation. We prove rigorous upper and lower bounds on the speed in terms of the cut-off parameter epsilon. From these bounds we estimate the range of validity of the Brunet--Derrida formula for a general class of reaction terms.