Both warm-season grasses, zoysia and St. Augustine grasses relish warm soil and air temperatures. With different blade textures and winter cold tolerances, these two grasses are not often deliberately grown together -- one species will be the weed and the other desirable.
Zoysia grass, with thin, soft leaves, grows more slowly than St. Augustine grass. Zoysia also tolerates drier soils as well as much chillier winter conditions; it turns tan in color when dormant. St. Augustine grass grows quickly with stiff, coarse blades. It falters in dry soil and dies back in subfreezing cold.
If the environmental conditions favor one grass species over another, "choking out" may occur. For zoysia grass to overtake and smother St. Augustine grass, the soil must become drier to favor the zoysia's slower growth and cause the St. Augustine to wither. In light shade or subfreezing winter weather, zoysia will persist even when dormant, but the St. Augustine will be partially to fully killed.
Zoysia grass is typically mowed short, as low as 1 inch, far too short for St. Augustine grass to still display green leaves. Zoysia doesn't overtake St. Augustine unless the latter withers, weakens from pests and diseases and dies.