The diffusive farmers and hunter-gatherers model with a free boundary in a heterogeneous environment

2021 ◽  
pp. 125317
Author(s):  
Wonhyung Choi ◽  
Inkyung Ahn ◽  
Changwook Yoon
Keyword(s):  
Free Boundary ◽  
Heterogeneous Environment ◽  
Hunter Gatherers

An evolutional free-boundary problem of a reaction–diffusion–advection system

2017 ◽  
Vol 147 (3) ◽  
pp. 615-648 ◽  
Author(s):  
Ling Zhou ◽  
Shan Zhang ◽  
Zuhan Liu
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Ecological Model ◽  
Reaction Diffusion ◽  
Spreading Speed ◽  
Heterogeneous Environment ◽  
Long Run ◽  
Strong Competition ◽  
Asymptotic Spreading

In this paper we consider a system of reaction–diffusion–advection equations with a free boundary, which arises in a competition ecological model in heterogeneous environment. The evolution of the free-boundary problem is discussed, which is an extension of the results of Du and Lin (Discrete Contin. Dynam. Syst. B19 (2014), 3105–3132). Precisely, when u is an inferior competitor, we prove that (u, v) → (0, V) as t→∞. When u is a superior competitor, we prove that a spreading–vanishing dichotomy holds, namely, as t→∞, either h(t)→∞ and (u, v) → (U, 0), or limt→∞h(t) < ∞ and (u, v) → (0, V). Moreover, in a weak competition case, we prove that two competing species coexist in the long run, while in a strong competition case, two species spatially segregate as the competition rates become large. Furthermore, when spreading occurs, we obtain some rough estimates of the asymptotic spreading speed.

The diffusive model for West Nile virus with advection and expanding fronts in a heterogeneous environment

2020 ◽  
Vol 13 (07) ◽  
pp. 2050057
Author(s):  
Zhengdi Zhang ◽  
Abdelrazig K. Tarboush
Keyword(s):  
West Nile Virus ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Risk Index ◽  
Sufficient Conditions ◽  
Heterogeneous Environment ◽  
West Nile

In this paper, we investigate a reaction–diffusion–advection model with expanding fronts, which models the spatial transmission of West Nile virus (WNv) in a heterogeneous environment. A free boundary problem is formulated and the global existence and uniqueness of the solution is presented. In addition to a classical basic reproduction number, the spatial-temporal basic reproduction number for the model with null Dirichlet boundary condition is introduced and the risk index associated with the virus in spatial setting is defined, and their properties are discussed. Sufficient conditions for the WNv to vanish or spread are given, and the asymptotic behavior of the solution to the free boundary problem when the spreading occurs is established. Our results show that the initial number of infected populations and the expanding capability of the expanding fronts exhibit important impacts on the extinction or persistence of the virus.

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