The spreading frontiers in partially degenerate reaction–diffusion systems. (July 2015)

• Record Type: Journal Article
• Authors: Wang, Jie; Cao, Jia-Feng
• Is Part Of: Nonlinear analysis. Volume 122(2015)
• Page Start: 215
• Page End: 238
• Publication Date: 2015-07
• Subjects: 35K57 -- 35B40 -- 92D25; Spreading frontiers -- Partially degenerate reaction–diffusion systems -- Free boundary; 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.2015.04.003
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we study the spreading frontiers in the partially degenerate reaction–diffusion systems with a mobile and a stationary component, which are described by a class free boundary condition. The center of our investigation is to understand the effect of the dispersal rate D of the mobile component, the expansion capacity μ (the ratio of the expansion speed of the free boundary and the gradient of the mobile component at the spreading frontiers) and the initial numbers u 0 and v 0 on the long-run dynamics of the spreading frontiers. It is shown that a spreading–vanishing dichotomy holds, and the sharp criteria for the spreading and vanishing by choosing D, μ, u 0 and v 0 as variable factors are also obtained. In particular, our results still reveal that slow dispersal rate unconditionally favors the frontiers to spread, but the fast one, however, leads to a conditional vanishing of the frontiers. Moreover, as applications, we consider a man–environment–man epidemic model in physiology and a reaction–diffusion model with a quiescent stage in population dynamics.