Fronts with a growth cutoff but with speed higher than the linear spreading speed

Fronts, propagating into an unstable state phi=0, whose asymptotic speed v(as) is equal to the linear spreading speed v* of infinitesimal perturbations about that state (so-called pulled fronts), are very sensitive to changes in the growth rate f(phi) for phi<<1. It was recently found that with a small cutoff, f(phi)=0 for phi< epsilon, v(as) converges to v* very slowly from below, as ln(-2) epsilon. Here we show that with such a cutoff and a small enhancement of the growth rate for small phi behind it, one can have v(as)>v*, even in the limit epsilon -->0. The effect is confirmed in a stochastic lattice model simulation where the growth rules for a few particles per site are accordingly modified.