The idea of farming took shape in one region only and then spread to the rest of the world over thousands of years?

I doubt it. Most of the world was undiscovered by the other. Once the explorers did discover the other regions, they brought with them the seeds of the fruits and vegetables of the area and spread it around Potatoes and corn and tobacco were cultivated by the Aztecs I believe. PineApple from Hawaii. Kiwi fruit from New Zealand.