TY - JOUR

T1 - Diffusive logistic equations with indefinite weights

T2 - Population models in disrupted environments

AU - Cantrell, Robert Stephen

AU - Cosner, Chris

PY - 1989/1/1

Y1 - 1989/1/1

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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U2 - 10.1017/S030821050001876X

DO - 10.1017/S030821050001876X

M3 - Article

AN - SCOPUS:0000883114

VL - 112

SP - 293

EP - 318

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 3-4

ER -