Travelling waves, spreading and extinction for Fisher–KPP propagation driven by a line with fast diffusion. (May 2016)

• Record Type: Journal Article
• Authors: Berestycki, Henri; Roquejoffre, Jean-Michel; Rossi, Luca
• Is Part Of: Nonlinear analysis. Volume 137(2016)
• Page Start: 171
• Page End: 189
• Publication Date: 2016-05
• Subjects: Reaction–diffusion system -- Travelling wave -- Speed of propagation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.01.023
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In an earlier work (Berestycki et al., 2013), we introduced a parabolic system to describe biological invasions in the presence of a line (the "road") with a specific diffusion, included in a domain (the "field") subject to Fisher–KPP propagation. We determined the asymptotic spreading speed in the direction of the road, w ∗, in terms of the various parameters. The new result we establish here is the existence of travelling fronts for this system for any speed c ⩾ w ∗ . In addition, we show that no front can travel with a speed c < w ∗ . We further extend the results to a new model where there is Fisher–KPP growth on the road, embedded in an elsewhere unfavourable plane. This is relevant to discuss certain invasions that have been observed along roads. We establish sharp conditions for extinction/invasion and describe the invasion dynamics.