Fronts with a Growth Cutoff but Speed Higher than $v^*$

TL;DR

This paper demonstrates that by introducing a small growth enhancement behind a cutoff in the growth rate, the asymptotic front speed can exceed the linear spreading speed, even as the cutoff approaches zero, confirmed through simulations.

Contribution

It reveals that a small growth rate enhancement behind a cutoff can cause fronts to propagate faster than the linear spreading speed, challenging previous assumptions.

Findings

Asymptotic speed can surpass $v^*$ with growth enhancement.

Speed exceeds $v^*$ even as cutoff $ o 0$.

Simulation confirms theoretical prediction.

Abstract

Fronts, propagating into an unstable state $\phi=0$, whose asymptotic speed $v_{\text{as}}$ 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 $\phi \ll 1$. It was recently found that with a small cutoff, $f(\phi)=0$ for $\phi < \epsilon$, $v_{\text{as}}$ converges to $v^*$ very slowly from below, as $\ln^{-2} \epsilon$. Here we show that with such a cutoff {\em and} a small enhancement of the growth rate for small $\phi$ behind it, one can have $v_{\text{as}} > v^*$, {\em even} in the limit $\epsilon \to 0$. The effect is confirmed in a stochastic lattice model simulation where the growth rules for a few particles per site are accordingly modified.