Just speculating here but fire and overgrazing strike me as similar in effect. Might an arid grassland tend to remain an arid grassland with adequate rain?
No, fire burns off the grass tops and leaves ash in the soil, which is beneficial in the long term. Grazing, especially by sheep, gets down to the root systems and keeps the grass from coming back after a rain. Then the soil is open to hardier desert plants.
This was a rare occasion when Hanson Baldwin got one wrong. His column calling Normandy an Anzio-style stalemate was published right at the time of the breakout.