An asymptotic analysis of the persistence threshold for the diffusive logistic model in spatial environments with localized patches

An indefinite weight eigenvalue problem characterizing the threshold condition for extinction of a population based on the single-species diffusive logistic model in a spatially heterogeneous environment is analyzed in a bounded two-dimensional domain with no-flux boundary conditions. In this eigenvalue problem, the spatial heterogeneity of the environment is reflected in the growth rate function, which is assumed to be concentrated in $n$ small circular disks, or portions of small circular disks, that are contained inside the domain. The constant bulk or background growth rate is assumed to be spatially uniform. The disks, or patches, represent either strongly favorable or strongly unfavorable local habitats. For this class of piecewise constant bang-bang growth rate function, an asymptotic expansion for the persistence threshold λ₁, representing the positive principal eigenvalue for this indefinite weight eigenvalue problem, is calculated in the limit of small patch radii by using the method of matched asymptotic expansions. By analytically optimizing the coefficient of the leading-order term in the asymptotic expansion of λ₁, general qualitative principles regarding the effect of habitat fragmentation are derived. In certain degenerate situations, it is shown that the optimum spatial arrangement of the favorable habit is determined by a higher-order coefficient in the asymptotic expansion of the persistence threshold.