The Plants adapted to the climate, too

Plants of the savanna are highly specialised to grow in this environment of long periods of drought. They have long top roots that can reach the deep water table, thick bork to resist annual fires, trucks taht can store water and leaves that drop of during the winter to conserve water.

The grasses leave adaptations that discourage anumals from grazing on them. Some grasses are too sharp or biter to taste for some animals but not others to eat.

The side benefit of this is that every species of animal has something to eat. Different species will also eat different part of the grass. Many grasses grow from the bottom up, so that the grouth tissue doesn't damaged by grasses.