Introduction of horses changed the landscape. They damage the roots when grazing. Bison and cows don't. Many of these desert scrub areas may have had much more grass cover pre-columbus.
According to Allan Savory and some other respectable scientists the problem arose because the plants evolved to support huge buffalo populations and depended on them to clean and fertilize the ground. As their population declined, there was no more mega-herds migrating annually through the prairies and smaller groups of animals had different grazing patterns. Plants couldn't adapt to the new situation quick enough and the balance got broken. Less cover meant more evaporation, drier soil, erosion and rain washing off the soil rich in organic matter, and it all spiraled down to the land turning into semi-deserts. Allan Savory thinks (and he proved this method successful in practice) that the solution is to have large flocks of grazing animals moving around periodically, which stimulates grass growth and grass protects the soil and helps other plants to colonize the area.
There is some evidence that the huge Buffalo herds did not exist until Indians died off after Columbus "discovered" America. Pre-Columbian indigenous societies hunted the Bison enough to keep their population in check.
