For a similar comparison, wild horses do surprisingly well in some desert regions of the Southwest US. It's a surprising treat to see a healthy herd of horses meandering through Joshua Trees. They can apparently do pretty well on sparse grasses and shrubs.

Those horse populations do well enough that the BLM regularly kills some off to limit the numbers. Ostensibly it's to protect the environment, but given the number of legal and illegal cattle that's allowed to graze on those same public lands I think the culling is mostly at the behest of ranchers.