When agriculture was started, humans no longer relied on hunting and gathering to survive, but could make their own food. Societies relying on agriculture can be nomadic in a sense that they move their crops around in order for the ground to remain fertile. However, nomadism decreased rather than increased with the introduction of agriculture, because hunters and gatherers (people before) needed to move all the time in search for food, so they were far more nomadic. Also your second statement doesn't seem correct cause food production started (!) with the introduction of agriculture. It did eventually lead to the development of civilizations, which (also seen today) finally led to the decrease of birthrates, but i wouldn't know about the rapidity.
