Fig. 4.

Front wandering identifies two classes of pushed waves. (A) Fluctuations in the front position can be described by simple diffusion for both pulled and pushed waves. (B) The front diffusion is caused by the number fluctuations, so the effective diffusion constant, $D_{\mathrm{f}}$, decreases with the carrying capacity, $N$. For pulled waves, $D_{\mathrm{f}}\sim {\ln }^{-3}N$ (33), while, for pushed waves, $D_{\mathrm{f}}$ can decrease much faster as $N^{-1}$ (38). We quantify the scaling of $D_{\mathrm{f}}$ with $N$ by the exponent ${\alpha }_{\mathrm{D}}$ equal to the slope on the log–log plot shown. The overlap of the two red lines highlights the fact that, even though ${\alpha }_{\mathrm{D}}$ should equal $0$ for pulled waves, the limited range of $N$ results in a different value of ${\alpha }_{\mathrm{D}}\approx -0.33$. (C) The dependence of the scaling exponent on cooperativity identifies two distinct classes of pushed waves.