caa a22 4500 386382549 CHVBK 20180307112047.0 cr unu---uuuuu 161130s1989 xx s 000 0 eng 10.1017/S030821050001876X doi S030821050001876X pii (NATIONALLICENCE)cambridge-10.1017/S030821050001876X Diffusive logistic equations with indefinite weights: population models in disrupted environments [Elektronische Daten] The dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form ut = d Δu + [m(x) — cu]u in Ω × (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 ∞ is bounded and is surrounded by uninhabitable regions, then u = 0 on ∂∞× (0, ∞). The growth rate m(x) is positive on favourablehabitats 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 l/d> where is the principle positive eigenvalue for the problem — Δϕ=λm(x)ϕ in Χ,ϕ=0 on ∂Ω. Analysis of how 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. Copyright © Royal Society of Edinburgh 1989 Cantrell Robert Stephen Department of Mathematics and Computer Science, The University of Miami, Coral Gables, Florida 33124, U.S.A. Cosner Chris Department of Mathematics and Computer Science, The University of Miami, Coral Gables, Florida 33124, U.S.A. Proceedings of the Royal Society of Edinburgh: Section A Mathematics Royal Society of Edinburgh Scotland Foundation 112/3-4(1989), 293-318 0308-2105 112:3-4<293 1989 112 PRM https://doi.org/10.1017/S030821050001876X text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S030821050001876X text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Cantrell Robert Stephen Department of Mathematics and Computer Science, The University of Miami, Coral Gables, Florida 33124, U.S.A NATIONALLICENCE 700 1- Cosner Chris Department of Mathematics and Computer Science, The University of Miami, Coral Gables, Florida 33124, U.S.A NATIONALLICENCE 773 0- Proceedings of the Royal Society of Edinburgh: Section A Mathematics Royal Society of Edinburgh Scotland Foundation 112/3-4(1989), 293-318 0308-2105 112:3-4<293 1989 112 PRM CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge