Diffusive logistic equations with indefinite weights: Population models in disrupted environments

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Abstract

The dynamics of a population inhabiting a strongly heterogeneous environment are modelled by diffusive logistic equations of the form ut= d Δm + [m(x)-cu]u in Q X (0, ∞), where u represents the population density, c, d>0 are constants describing the limiting effects of crowding and the diffusion rate of the population, respectively, and m(x) describes the local growth rate of the population. If the environment Ωis bounded and is surrounded by uninhabitable regions, then u = 0 on Ω x (0,∞). The growth rate m(x) is positive on favourable habitats and negative on unfavourable ones. The object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled. The models are shown to possess a unique, stable, positive steady state (implying persistence for the population) provided 1/ d> λ1 +-(m), where λ1 +(m) is the principle positive eigenvalue for the problem —Δϕ — λm(x)ϕ in Ω, ϕ = 0 on 3Ω. Analysis of how λt +(m) depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat. In the limit as the diffusion rate d ↓ 0, the solutions tend toward the positive part of m(x)/c, and if m is discontinuous develop interior transition layers. The analysis uses bifurcation and continuation methods, the variational characterisation of eigenvalues, upper and lower solution techniques, and singular perturbation theory.

Original languageEnglish (US)
Pages (from-to)293-318
Number of pages26
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume112
Issue number3-4
DOIs
StatePublished - 1989

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Indefinite Weight
Logistic Equation
Population Model
Bifurcation Method
Interior Layer
Eigenvalue
Singular Perturbation Theory
Transition Layer
Upper and Lower Solutions
Continuation Method
Heterogeneous Environment
Persistence
Arrangement
Limiting
Tend

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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title = "Diffusive logistic equations with indefinite weights: Population models in disrupted environments",
abstract = "The dynamics of a population inhabiting a strongly heterogeneous environment are modelled by diffusive logistic equations of the form ut= d Δm + [m(x)-cu]u in Q X (0, ∞), where u represents the population density, c, d>0 are constants describing the limiting effects of crowding and the diffusion rate of the population, respectively, and m(x) describes the local growth rate of the population. If the environment Ωis bounded and is surrounded by uninhabitable regions, then u = 0 on Ω x (0,∞). The growth rate m(x) is positive on favourable habitats and negative on unfavourable ones. The object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled. The models are shown to possess a unique, stable, positive steady state (implying persistence for the population) provided 1/ d> λ1 +-(m), where λ1 +(m) is the principle positive eigenvalue for the problem —Δϕ — λm(x)ϕ in Ω, ϕ = 0 on 3Ω. Analysis of how λt +(m) depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat. In the limit as the diffusion rate d ↓ 0, the solutions tend toward the positive part of m(x)/c, and if m is discontinuous develop interior transition layers. The analysis uses bifurcation and continuation methods, the variational characterisation of eigenvalues, upper and lower solution techniques, and singular perturbation theory.",
author = "Robert Cantrell and George Cosner",
year = "1989",
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language = "English (US)",
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pages = "293--318",
journal = "Proceedings of the Royal Society of Edinburgh Section A: Mathematics",
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T1 - Diffusive logistic equations with indefinite weights

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N2 - The dynamics of a population inhabiting a strongly heterogeneous environment are modelled by diffusive logistic equations of the form ut= d Δm + [m(x)-cu]u in Q X (0, ∞), where u represents the population density, c, d>0 are constants describing the limiting effects of crowding and the diffusion rate of the population, respectively, and m(x) describes the local growth rate of the population. If the environment Ωis bounded and is surrounded by uninhabitable regions, then u = 0 on Ω x (0,∞). The growth rate m(x) is positive on favourable habitats and negative on unfavourable ones. The object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled. The models are shown to possess a unique, stable, positive steady state (implying persistence for the population) provided 1/ d> λ1 +-(m), where λ1 +(m) is the principle positive eigenvalue for the problem —Δϕ — λm(x)ϕ in Ω, ϕ = 0 on 3Ω. Analysis of how λt +(m) depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat. In the limit as the diffusion rate d ↓ 0, the solutions tend toward the positive part of m(x)/c, and if m is discontinuous develop interior transition layers. The analysis uses bifurcation and continuation methods, the variational characterisation of eigenvalues, upper and lower solution techniques, and singular perturbation theory.

AB - The dynamics of a population inhabiting a strongly heterogeneous environment are modelled by diffusive logistic equations of the form ut= d Δm + [m(x)-cu]u in Q X (0, ∞), where u represents the population density, c, d>0 are constants describing the limiting effects of crowding and the diffusion rate of the population, respectively, and m(x) describes the local growth rate of the population. If the environment Ωis bounded and is surrounded by uninhabitable regions, then u = 0 on Ω x (0,∞). The growth rate m(x) is positive on favourable habitats and negative on unfavourable ones. The object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled. The models are shown to possess a unique, stable, positive steady state (implying persistence for the population) provided 1/ d> λ1 +-(m), where λ1 +(m) is the principle positive eigenvalue for the problem —Δϕ — λm(x)ϕ in Ω, ϕ = 0 on 3Ω. Analysis of how λt +(m) depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat. In the limit as the diffusion rate d ↓ 0, the solutions tend toward the positive part of m(x)/c, and if m is discontinuous develop interior transition layers. The analysis uses bifurcation and continuation methods, the variational characterisation of eigenvalues, upper and lower solution techniques, and singular perturbation theory.

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