Generalized principal eigenvalues of elliptic operators and spreading speeds of Fisher-KPP equations in two-scale almost periodic media

TL;DR

This paper studies how the heterogeneity and oscillation scales of media influence the spreading speeds of Fisher-KPP equations, using homogenization of effective Hamiltonians to analyze asymptotic behaviors.

Contribution

It introduces a novel analysis of the asymptotic limits of principal eigenvalues and spreading speeds in two-scale almost periodic media, linking media heterogeneity to propagation dynamics.

Findings

Rapid or slow oscillations with mean zero can accelerate propagation.

Slow oscillation advection with mean zero can decelerate propagation.

Established convergence rates for the asymptotic limits of eigenvalues.

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.