Robert Stephen Cantrell

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

The dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form u t = 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=λ m ( 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.

The effects of spatial heterogeneity in population dynamics

The dynamics of a population inhabiting a heterogeneous environment are modelled by a diffusive logistic equation with spatially varying growth rate. The overall suitability of an environment is characterized by the principal eigenvalue of the corresponding linearized equation. The dependence of the eigenvalue on the spatial arrangement of regions of favorable and unfavorable habitat and on boundary conditions is analyzed in a number of cases.

Diffusive logistic equations with indefinite weights: population models in disrupted environments II

The dynamics of a population inhabiting a strongly heterogeneous environment are modeled by diffusive logistic equations of the form

Permanence in ecological systems with spatial heterogeneity

u_1 = \nabla \cdot (d(x,u) + \nabla u) - {\bf b}(x) \cdot \nabla u + m(x)u - cu^2

The effects of human movement on the persistence of vector-borne diseases

in

Evolution of dispersal and the ideal free distribution

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The ideal free distribution as an evolutionarily stable strategy

, where u represents the population density,

Should a park be an island

d(x,u)

Practical persistence in ecological models via comparison methods

the (possibly) density dependent diffusion rate,

How Predator Incursions Affect Critical Patch Size: The Role of the Functional Response*

{\bf b(x)}