Abstract
We analyze the positive solutions to \t\t\t{−Δv=λv(1−v);Ω0,∂v∂η+γλv=0;∂Ω0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} - \\Delta v = \\lambda v(1-v); & \\Omega_{0}, \\\\ \\frac{\\partial v}{\\partial\\eta} + \\gamma\\sqrt{\\lambda} v =0 ; & \\partial\\Omega_{0}, \\end{cases} $$\\end{document} where Omega_{0}=(0,1) or is a bounded domain in mathbb{R}^{n}, n =2,3, with smooth boundary and |Omega_{0}|=1, and λ, γ are positive parameters. Such steady state equations arise in population dynamics encapsulating assumptions regarding the patch/matrix interfaces such as patch preference and movement behavior. In this paper, we will discuss the exact bifurcation diagram and stability properties for such a steady state model.
Highlights
Habitat fragmentation creates landscape-level spatial heterogeneity which influences the population dynamics of the resident species
Fragmentation often leads to declines in abundance of the species as the fragmented landscape becomes more susceptible to edge effects between the remnant habitat patches and the lower quality human-modified “matrix” surrounding these focal patches [1,2,3]
Studies of movement behavior in response to different habitat edge conditions clearly demonstrate that the composition of the matrix can influence emigration rates, patterns of movement, and withinpatch distributions of a species (e.g., [4,5,6])
Summary
Habitat fragmentation creates landscape-level spatial heterogeneity which influences the population dynamics of the resident species. We will briefly summarize the modeling framework and boundary condition derivation given in [12] and present our main results. Organisms modify their movement behavior at the patch/matrix interface and would have a probability α of remaining in or leaving different from 50%.
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