Spreading speeds of KPP-type lattice systems in heterogeneous media

TL;DR

This paper analyzes the spreading speeds of solutions to a KPP-type lattice system in heterogeneous media, developing new estimates and characterizing speeds via eigenvalues, with exact speeds derived for random and almost periodic coefficients.

Contribution

It introduces novel discrete Harnack estimates and homogenization techniques to determine spreading speeds in heterogeneous lattice systems, including cases with random and almost periodic coefficients.

Findings

Established bounds for spreading speeds using generalized eigenvalues.

Derived exact spreading speeds for random stationary ergodic and almost periodic coefficients.

Proved symmetry of spreading speeds in positive and negative directions under certain conditions.

Abstract

In this paper, we investigate spreading properties of the solutions of the Kolmogorov-Petrovsky-Piskunov-type, (to be simple,KPP-type) lattice system \begin{equation}\label{firstequation}\overset{.}u_{i}(t) =d^{\prime}_{i}(u_{i+1}(t)-u_{i}(t))+d_{i}(u_{i-1}(t)-u_{i}(t))+f(i,u_{i}).\end{equation} we develop some new discrete Harnack-type estimates and homogenization techniques for this lattice system to construct two speeds $\overline\omega \leq \underline \omega$ such that $\displaystyle{\lim_{t\rightarrow+\infty}}\sup \limits_{i\geq\omega t}|u_i(t)|=0$ for any $\omega>\overline{\omega}$, and $\displaystyle{\lim_{t\rightarrow+\infty}}\sup \limits_{0\leq i\leq\omega t}|u_i(t)-1|=0$ for any $\omega<\underline{\omega}$. These speeds are characterized by two generalized principal eigenvalues of the linearized systems. In particular, we derive the exact spreading speed when the coefficients are random stationary ergodic or almost periodic (where $\underline\omega = \overline\omega$). Finally, in the case where $f_{s}^{\prime}(i,0)$ is almost periodic in $i$ and the diffusion rate $d_i'=d_i$ is independent of $i$, we show that the spreading speeds in the positive and negative directions are identical even if $ f(i,u_{i})$ is not invariant with respect to the reflection.