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 language | English (US) |
|---|---|
| Pages (from-to) | 293-318 |
| Number of pages | 26 |
| Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |
| Volume | 112 |
| Issue number | 3-4 |
| DOIs | |
| State | Published - 1989 |
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Cite this
Diffusive logistic equations with indefinite weights : Population models in disrupted environments. / Cantrell, Robert; Cosner, George.
In: Proceedings of the Royal Society of Edinburgh Section A: Mathematics, Vol. 112, No. 3-4, 1989, p. 293-318.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Diffusive logistic equations with indefinite weights
T2 - Population models in disrupted environments
AU - Cantrell, Robert
AU - Cosner, George
PY - 1989
Y1 - 1989
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.
UR - http://www.scopus.com/inward/record.url?scp=0000883114&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0000883114&partnerID=8YFLogxK
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 -