%0 Journal Article %1 cantrell1989diffusive %A Cantrell, Robert Stephen %A Cosner, Chris %D 1989 %J Proceedings of the Royal Society of Edinburgh, Section: A Mathematics %K diffusive_logistic_equation eigenvalue_problem fishers_diffusion probability_of_survival spatial_structure %N 3-4 %P 293--318 %R 10.1017/S030821050001876X %T Diffusive logistic equations with indefinite weights: population models in disrupted environments %U http://journals.cambridge.org/article_S030821050001876X %V 112 %X The dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form ut = d Δu + m(x) — cuu 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> S030821050001876X_inline1 where S030821050001876X_inline1 is the principle positive eigenvalue for the problem — Δϕ=λm(x)ϕ in Χ,ϕ=0 on ∂Ω. Analysis of how S030821050001876X_inline1 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.

@article{cantrell1989diffusive, abstract = {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> S030821050001876X_inline1 where S030821050001876X_inline1 is the principle positive eigenvalue for the problem — Δϕ=λm(x)ϕ in Χ,ϕ=0 on ∂Ω. Analysis of how S030821050001876X_inline1 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.}, added-at = {2013-03-13T13:06:17.000+0100}, author = {Cantrell, Robert Stephen and Cosner, Chris}, biburl = {https://www.bibsonomy.org/bibtex/27aec3f932daf769754d94acad835966b/peter.ralph}, doi = {10.1017/S030821050001876X}, interhash = {7c182ef717165a7961058a18e39572a1}, intrahash = {7aec3f932daf769754d94acad835966b}, issn = {1473-7124}, journal = {Proceedings of the Royal Society of Edinburgh, Section: A Mathematics}, keywords = {diffusive_logistic_equation eigenvalue_problem fishers_diffusion probability_of_survival spatial_structure}, month = {0}, number = {3-4}, numpages = {26}, pages = {293--318}, timestamp = {2014-07-01T02:42:33.000+0200}, title = {Diffusive logistic equations with indefinite weights: population models in disrupted environments}, url = {http://journals.cambridge.org/article_S030821050001876X}, volume = 112, year = 1989 }