Marked decline in forest-dependent small mammals following habitat loss and fragmentation in an Amazonian deforestation frontier

Agricultural frontier expansion into the Amazon over the last four decades has created million hectares of fragmented forests. While many species undergo local extinctions within remaining forest patches, this may be compensated by native species from neighbouring open-habitat areas potentially invading these patches, particularly as forest habitats become increasingly degraded. Here, we examine the effects of habitat loss, fragmentation and degradation on small mammal assemblages in a southern Amazonian deforestation frontier, while accounting for species-specific degree of forest-dependency. We surveyed small mammals at three continuous forest sites and 19 forest patches of different sizes and degrees of isolation. We further sampled matrix habitats adjacent to forest patches, which allowed us to classify each species according to forest-dependency and generate a community-averaged forest-dependency index for each site. Based on 21,568 trap-nights, we recorded 970 small mammals representing 20 species: 12 forest-dependents, 5 matrix-tolerants and 3 open-habitat specialists. Across the gradient of forest patch size, small mammal assemblages failed to show the typical species-area relationship, but this relationship held true when either species abundance or composition was considered. Species composition was further mediated by community-averaged forest-dependency, so that smaller forest patches were occupied by a lower proportion of forest-dependent rodents and marsupials. Both species richness and abundance increased in less isolated fragments surrounded by structurally simplified matrix habitats (e.g. active or abandoned cattle pastures). While shorter distances between forest patches may favour small mammal abundances, forest area and matrix complexity dictated which species could persist within forest fragments according to their degree of forest-dependency. Small mammal local extinctions in small forest patches within Amazonian deforestation frontiers are therefore likely offset by the incursion of open-habitat species. To preclude the dominance of those species, and consequent losses of native species and associated ecosystem functions, management actions should limit or reduce areas dedicated to pasture, additionally maintaining more structurally complex matrix habitats across fragmented landscapes

Foliations, solvability and global injectivity

Let F: R^n -> R^n be a C^\infty map such that DF(x) is invertible for all x in R^n. We know that F is a local diffeomorphism but, in general, it is not globally injective. We discuss the relations between some additional hypothesis that guarantee the global injectivity of F. Further, based on one of these hypotheses, we establish a necessary condition for the existence of F: R^n -> R^n such that det DF = h, where h: R^n -> [0,\infty) is a given C^\infty function