Generalized principal eigenvalues of elliptic operators and spreading speeds of Fisher-KPP equations in two-scale almost periodic media
Xing Liang, Linfeng Xu, Tao Zhou
TL;DR
The paper analyzes how two-scale almost periodic heterogeneity affects the generalized principal eigenvalues and spreading speeds in Fisher–KPP equations. By recasting the problem through homogenization and effective Hamiltonians, it derives rigorous limits as the fast-scale parameter $L$ tends to 0 or ∞, and establishes convergence rates that depend on quantitative measures of almost periodicity. The results reveal how mean-zero perturbations in growth or slow oscillating drift can accelerate or decelerate propagation, and provide explicit characterizations in 1D alongside general multidimensional phenomena. These findings advance understanding of invasion dynamics in highly heterogeneous media and connect spectral theory with Hamilton–Jacobi homogenization in almost periodic settings.
Abstract
This paper is concerned with the asymptotic behavior of the generalized principal eigenvalues of elliptic operators and spreading speeds of Fisher-KPP equations in two-scale almost periodic media where one scale is fixed and another one approaches zero or infinity. We transform the problem into the homogenization of certain effective Hamiltonian and then establish the asymptotic limits and the convergence rates. Based on the analysis of the asymptotic behavior of effective Hamiltonians, we investigate how the heterogeneity of the advection and growth rates affect on the propagation in the case where the media has very rapid or slow spatial oscillation: We show a normal scale perturbation of the growth rate with mean zero can accelerate the propagation in the media with rapid or slow oscillation; and an advection with slow oscillation and mean zero can decelerate the propagation in 1-D case.