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

Abstract

An indefinite weight eigenvalue problem characterizing the thresholdcondition for extinction of a population based on the single-speciesdiffusive logistic model in a spatially heterogeneous environment isanalyzed in a bounded two-dimensional domain with no-flux boundaryconditions. In this eigenvalue problem, the spatial heterogeneity ofthe environment is reflected in the growth rate function, which isassumed to be concentrated in $n$ small circular disks, or portions ofsmall circular disks, that are contained inside the domain. Theconstant bulk or background growth rate is assumed to be spatiallyuniform. The disks, or patches, represent either strongly favorable orstrongly unfavorable local habitats. For this class of piecewise constantbang-bang growth rate function, an asymptotic expansion for thepersistence threshold λ1, representing the positive principaleigenvalue for this indefinite weight eigenvalue problem, iscalculated in the limit of small patch radii by using the method ofmatched asymptotic expansions. By analytically optimizing thecoefficient of the leading-order term in the asymptotic expansion ofλ1, general qualitative principles regarding the effect ofhabitat fragmentation are derived. In certain degenerate situations,it is shown that the optimum spatial arrangement of the favorablehabit is determined by a higher-order coefficient in the asymptoticexpansion of the persistence threshold.