Spreading in a Shifting Environment Modeled by the Diffusive Logistic Equation with a Free Boundary

Du, Yihong; Wei, Lei; Zhou, Ling

doi:10.1007/s10884-017-9614-2

Author(s)

Du, Yihong

Wei, Lei

Zhou, Ling

Publication Date

2018-12

Abstract

We investigate the influence of a shifting environment on the spreading of an invasive species through a model given by the diffusive logistic equation with a free boundary. When the environment is homogeneous and favourable, this model was first studied in Du and Lin (SIAM J Math Anal 42:377–405, 2010), where a spreading–vanishing dichotomy was established for the long-time dynamics of the species, and when spreading happens, it was shown that the species invades the new territory at some uniquely determined asymptotic speed c0 > 0. Here we consider the situation that part of such an environment becomes unfavourable, and the unfavourable range of the environment moves into the favourable part with speed c > 0. We prove that when c ≥ c0, the species always dies out in the long-run, but when 0 < c < c0, the long-time behavior of the species is determined by a trichotomy described by (a) vanishing, (b) borderline spreading, or (c) spreading. If the initial population is written in the form u0(x) = σϕ(x) with ϕ fixed and σ > 0 a parameter, then there exists σ0 > 0 such that vanishing happens when σ ∈ (0,σ0), borderline spreading happens when σ = σ0, and spreading happens when σ > σ0.

Citation

Journal of Dynamics and Differential Equations, 30(4), p. 1389-1426

ISSN

1572-9222

1040-7294

Link

https://hdl.handle.net/1959.11/26537

Language

en

Publisher

Springer New York LLC

Title

Spreading in a Shifting Environment Modeled by the Diffusive Logistic Equation with a Free Boundary

Type of document

Journal Article

Entity Type

Publication

Author(s)	Du, Yihong Wei, Lei Zhou, Ling
Publication Date	2018-12
Abstract	We investigate the influence of a shifting environment on the spreading of an invasive species through a model given by the diffusive logistic equation with a free boundary. When the environment is homogeneous and favourable, this model was first studied in Du and Lin (SIAM J Math Anal 42:377–405, 2010), where a spreading–vanishing dichotomy was established for the long-time dynamics of the species, and when spreading happens, it was shown that the species invades the new territory at some uniquely determined asymptotic speed c<sub>0</sub> > 0. Here we consider the situation that part of such an environment becomes unfavourable, and the unfavourable range of the environment moves into the favourable part with speed c > 0. We prove that when c ≥ c<sub>0</sub>, the species always dies out in the long-run, but when 0 < c < c<sub>0</sub>, the long-time behavior of the species is determined by a trichotomy described by (a) vanishing, (b) borderline spreading, or (c) spreading. If the initial population is written in the form u<sub>0</sub>(x) = σϕ(x) with ϕ fixed and σ > 0 a parameter, then there exists σ<sub>0</sub> > 0 such that vanishing happens when σ ∈ (0,σ<sub>0</sub>), borderline spreading happens when σ = σ<sub>0</sub>, and spreading happens when σ > σ<sub>0</sub>.
Citation	Journal of Dynamics and Differential Equations, 30(4), p. 1389-1426
ISSN	1572-9222 1040-7294
Link	https://hdl.handle.net/1959.11/26537
Language	en
Publisher	Springer New York LLC
Title	Spreading in a Shifting Environment Modeled by the Diffusive Logistic Equation with a Free Boundary
Type of document	Journal Article
Entity Type	Publication

Spreading in a Shifting Environment Modeled by the Diffusive Logistic Equation with a Free Boundary

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