Hierarchical spatial segregation of two Mediterranean vole species: the role of patch‑network structure and matrix composition

According to ecological theory, the coexistence of competitors in patchy environments may be facilitated by hierarchical spatial segregation along axes of environmental variation, but empirical evidence is limited. Cabrera and water voles show a metapopulation-like structure in Mediterranean farmland, where they are known to segregate along space, habitat, and time axes within habitat patches. Here, we assess whether segregation also occurs among and within landscapes, and how this is influenced by patch-network and matrix composition. We surveyed 75 landscapes, each covering 78 ha, where we mapped all habitat patches potentially suitable for Cabrera and water voles, and the area effectively occupied by each species (extent of occupancy). The relatively large water vole tended to be the sole occupant of landscapes with high habitat amount but relatively low patch density (i.e., with a few large patches), and with a predominantly agricultural matrix, whereas landscapes with high patch density (i.e.,many small patches) and low agricultural cover, tended to be occupied exclusively by the small Cabrera vole. The two species tended to co-occur in landscapes with intermediate patch-network and matrix characteristics, though their extents of occurrence were negatively correlated after controlling for environmental effects. In combination with our previous studies on the Cabrera-water vole system, these findings illustrated empirically the occurrence of hierarchical spatial segregation, ranging from withinpatches to among-landscapes. Overall, our study suggests that recognizing the hierarchical nature of spatial segregation patterns and their major environmental drivers should enhance our understanding of species coexistence in patchy environments.

On lattices from combinatorial game theory modularity and a representation theorem: Finite case

We show that a self-generated set of combinatorial games, S, may not be hereditarily closed but, strong self-generation and hereditary closure are equivalent in the universe of short games. In [13], the question “Is there a set which will give an on-distributive but modular lattice?” appears. A useful necessary condition for the existence of a finite non-distributive modular L(S) is proved. We show the existence of S such that L(S) is modular and not distributive, exhibiting the first known example. More, we prove a Representation Theorem with Games that allows the generation of all finite lattices in game context. Finally, a computational tool for drawing lattices of games is presented.