Analysis of the periodically fragmented environment model: II—biological invasions and pulsating travelling fronts

Abstract

This paper is concerned with propagation phenomena for reaction–diffusion equations of the type: u t − ∇ ⋅ ( A ( x ) ∇ u ) = f ( x , u ) , x ∈ R N , where A is a given periodic diffusion matrix field, and f is a given nonlinearity which is periodic in the x-variables. This article is the sequel to [H. Berestycki, F. Hamel, L. Roques, Analysis of the periodically fragmented environment model: I—influence of periodic heterogeneous environment on species persistence, Preprint]. The existence of pulsating fronts describing the biological invasion of the uniform 0 state by a heterogeneous state is proved here. A variational characterization of the minimal speed of such pulsating fronts is proved and the dependency of this speed on the heterogeneity of the medium is also analyzed.