FKPP fronts in quenched random media

Abstract

We study numerically the evolution of one-dimensional FKPP fronts initiated from steep initial conditions in the presence of a quenched random growth rate. Compared to both the homogeneous case (with velocity $v_0$) and deterministic disorder, quenched randomness increases the average propagation speed. We show that the velocity shift relative to the homogeneous case scales linearly with the disorder variance $\ddot{I}{ƒ}^2$, with a universal prefactor -- independent of the specific distribution of the disorder -- such that $v=v_0+a\ddot{I}{ƒ}^2$, with $a\approx 0.02432\pm 0.00002$. Moreover, the front position exhibits diffusive fluctuations across disorder realizations. The corresponding effective diffusion coefficient scales quadratically with $\ddot{I}ƒ$, $D=\frac{b^2\ddot{I}{ƒ}^2}{2}$, with $b\approx 0.223\pm 0.002$. These results suggest a universal statistical response of FKPP fronts to quenched heterogeneity.

7 pages, 7 figures