A reaction-diffusion model for population dynamics in patchy landscapes.

We address the linear eigenvalue problem associated with reaction-diffusion models for patchy linear domains with any finite number of unique patches. Interface conditions between adjacent patches may be chosen independently to accurately reflect organism behavior. We show that there exists a real principal eigenvalue with a corresponding non-negative eigenfunction that is non-zero interior to the domain. We demonstrate that our model is a generalization of several existing two-patch models. We consider simple three- and four-patch landscapes representing fragmented reserves to illustrate how our model can be used to analyze population persistence, spatial distribution, and migration dynamics. Fragmented landscapes with increased heterogeneity capture migration dynamics not found in landscapes with only two patch types, such as favorable habitat patches that experience net immigration. Conditions for the existence and non-existence of zero-flux conditions interior to three-patch systems are shown, which provide insight to the location of sources and direction(s) of migration. • Principal eigenvalues are studied for patchy domain problems, and corresponding eigenfunctions are positive on the interior. • An implicit equation is used to obtain numerical solutions to eigenvalue problems on previously intractable patchy domains. • Conditions are derived for the existence of extrema for principal eigenfunctions. • Results are used to simulate different landscapes with fragmented reserves. • Predictions are made for migration across landscapes, including when favorable patches might receive net immigration. [ABSTRACT FROM AUTHOR]