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

The dynamics of a population inhabiting a strongly heterogeneous environment are modelled by diffusive logistic equations of the form u^t= 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 λ^t+(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.