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Traditionally, ecologists interested in habitat edges have focused on edge-related gradients in patterns of species richness or abiotic variables. Here, however, we take a different perspective, attempting to synthesize recent empirical results concerning the effects of habitat edges on population dynamics with contemporary theoretical developments to outline the ways in which species interactions, and the dynamics of the communities in which they are embedded, can be changed by habitat edges. We find a striking convergence between empirical notions of a patch's core area and analytical results from partial differential equation models. A review of both empirical and theoretical studies suggests four general classes of mechanisms through which habitat edges can have similar impacts on dissimilar types of species interactions. Specifically, we focus on edges' roles as dispersal barriers or filters, edges' influences on mortality, edges' involvement in spatial subsidies (in which dispersers' intrapatch impacts are maintained by their activities in other habitats), and edges' roles as generators of novel interactions. For each class of edge-mediated effects, we provide examples of how one can use spatial modeling to address the relevant questions on these topics, which together form a key link between community dynamics and landscape structure.

SynopsisThe 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.

With the recent resurgence of vector-borne diseases due to urbanization and development there is an urgent need to understand the dynamics of vector-borne diseases in rapidly changing urban environments. For example, many empirical studies have produced the disturbing finding that diseases continue to persist in modern city centers with zero or low rates of transmission. We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian models that mimic human commuting behavior and (ii) Eulerian models that mimic human migration. We determine the basic reproduction number R0 for both modeling approaches. We show that for both approaches that if the disease free equilibrium is stable (R0 < 1) then it is globally stable and if the disease free equilibrium is unstable (R0 > 1) then there exists a unique positive (endemic) equilibrium that is globally stable among positive solutions. Finally, we prove in general that Lagrangian and Eulerian modeling approaches are not equivalent. The modeling approaches presented provide a framework to explore spatial vector-borne disease dynamics and control in heterogeneous environments. As an example, we consider two patches in which the disease dies out in both patches when there is no movement between them. Numerical simulations demonstrate that the disease becomes endemic in both patches when humans move between the two patches.