**2.2 Criteria for defining populations as native, introduced, or range expansion**

I defined all population living within the diagonal of dry areas in South America (Caatinga, Cerrado, Chaco and Pampas domains [4, 6]) as native even if resulting from accidental or purposeful reintroductions if and only if within native range [33–35]. I made exceptions for individuals living in restingas (coastal savannahs over sand dunes) were the species did not occur historically. These populations on restingas were labeled as introduced. I used the same criteria for other populations outside of the dry area's diagonal, unless the population could result from natural emigration. Populations resulting from emigration towards forest domains—arising from forest degradation—were labeled as range expansion. Populations within ecotone areas were labeled as native, since several open-field species occur within grassland enclaves in a matrix of forest (e.g. marsh deer *Blastocerus dichotomus*, pampas deer, and greater rheas themselves [36–38]).

## **2.3 Mapping and vegetation cover**

I collected vegetation cover maps for depicting canopy closeness from MOD44B. v006, representing areas currently covered by tall wood vegetation, not appropriated for rheas. The MOD44B.v006 image layer in the MODIS Vegetation Continuous Fields product provides a continuous, quantitative representation of global tree cover (greater rhea non-habitat) at a 250 m spatial resolution [39]. I used vegetation ecoregions from WWF categorization [40], chosen because it defines ecoregions as relatively large landscapes, each containing a distinct assemblages of species, with boundaries similar to the original extent of natural communities—prior to modern land-use change.
