Spreading speeds of KPP-type lattice systems in heterogeneous media

Abstract

In this paper, we investigated spreading properties of the solutions of the Kolmogorov–Petrovsky–Piskunov–type (KPP-type) lattice system [Formula: see text] Motivated by the work in [H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction–diffusion equations, J. Math. Phys. 53(11) (2012) 115619, 23 pp.], we develop some new discrete Harnack-type estimates and homogenization techniques for the lattice system [Formula: see text] to construct two speeds [Formula: see text] such that [Formula: see text] for any [Formula: see text], and [Formula: see text] for any [Formula: see text]. These speeds are characterized by two generalized principal eigenvalues of the linearized systems of [Formula: see text]. In particular, we derive the exact spreading speed when the coefficients are random stationary ergodic or almost periodic (where [Formula: see text]). Finally, in the case where [Formula: see text] is almost periodic in [Formula: see text] and the diffusion rate [Formula: see text] is independent of [Formula: see text], we show that the spreading speeds in the positive and negative directions are identical even if [Formula: see text] is not invariant with respect to the reflection.