Analysis of the periodically fragmented environment model : II - Biological invasions and pulsating travelling fronts.
Résumé
This paper is concerned with propagation phenomena for reaction-diffusion equations of the type $$u_t-\nabla\cdot(A(x)\nabla u)=f(x,u),\ x\in\mathbb{R}^N$$ 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, J. Math. Biol. (2005)]. 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.