Metapopulation structure and persistence in random fragmented landscapes

Background/Question/Methods

In metapopulation theory, local patches of suitable habitat are connected by dispersal. The balance between dispersal and extinction rates is key for metapopulation persistence, as originally shown by the classic work of Levins (1969). Hanski and Ovaskainen (2000) extended Levins' model to colonization rates which depend on the distance between patches. They studied the matrix M whose (i, j)th entry expresses the colonization rate of patch j for individuals in patch i, and found a simple persistence rule: metapopulations persist when the dominant eigenvalue of M is larger than the extinction rate. Though this result provides a framework for studying the persistence of a metapopulation given a particular landscape, it cannot address the question of which features of the landscape are the key drivers of persistence in general. To answer this question, we propose a null model drawing from the mathematics of Random Geometric Graphs and Euclidean Random Matrices. In a nutshell: patches are uniformly distributed in the landscape, and connected with a dispersal rate that is a function of between-patch distance.

Results/Conclusions

Using this null model as our framework, we arrive at three important conclusions. First, the dominant eigenvalue of M is accurately approximated by the "effective number of neighbors" each patch has. This number depends on the density of patches in the landscape, the dimensionality (e.g. 1D for dunes on the shores of Lake Michigan, 2/3D for fragmented forests), and the dispersal kernel. Using this effective measure for the number of neighbors, we can write a simple criterion for metapopulation persistence for arbitrary dimensions, kernel, and density. Second, we also show that patches with high probability of occurrence are invariably spatially localized, forming clusters that are surrounded by patches with lower occurrence probabilities. Third, a random grid enhances the formation of such clusters compared to a regularly spaced one, and thus increases overall persistence probability. This has important consequences for conservation and landscape design: regularly spaced habitats turn out to be considerably less efficient than randomly distributed ones.