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

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
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

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