2015/04/04 by Yonggang Zhao, Mingxin Wang, Zhao, Yonggang +1 · 1 citation
Mathematics · Medicine · #35B40 #35K20 #35R35 #92B05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Stochastic processes and statistical mechanics #math.AP #msc:35B40 #msc:35K20 #msc:35R35 #msc:92B05
paper · pdf · doi:10.48550/arxiv.1504.00998
32 pages
openalex publication_date 2015/04/04 · arxiv created 2015/08/13 · arxiv updated 2015/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate a reaction-diffusion-advection equation of the form ut-uxx+βux=f(u) (t>0, 0<x<h(t)) with mixed boundary condition at x=0 and a free boundary condition at x=h(t). Such a model may be applied to describe the dynamical process of a new or invasive species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundary representing the expanding front. The goal of this paper is to understand the effect of advection environment and no flux across the left boundary on the dynamics of this species. When |β|<c0, we first derive the spreading-vanishing dichotomy and sharp threshold for spreading and vanishing. Then provide a much sharper estimate for the spreading speed of h(t) and the uniform convergence of u(t,x) when spreading happens. For the case |β|≥ c0, some results concerning spreading, virtual spreading, vanishing and virtual vanishing are obtained. Where c0 is the minimal speed of traveling waves of the differential equation.