I studied anthropology as my college minor and the consensus in my classes seemed to be that farming was never really a new discovery that was 'invented.' Rather it seems likely that ancient people had known about the possibility of raising crops for a while but had simply forgone that in favor of less labor intensive hunting and foraging. Later, as the large game that early humans relied on began to die out, we resorted to the much more labor intensive lifestyle of farming.

Some points of evidence include:

1) The advent of farming coincides roughly with the Late Pleistocene Extinction event which saw many of the large game mammals (human prey) die out

2) Bone samples from the era show that the advent of farming coincides with a decline in human height and greater enamel defects showing malnutrition. This is strong evidence that farming was a step down for our early ancestors not a step up.

3) Tribes like the San Bushman and early Australian aborigines knew full well about farming but chose to forgo it because in their region, food was plentiful enough to be hunted and gathered. This is evidence that farming is not necessarily the result of new innovation (people don't instantly and automatically switch to farming just because they know about it).

see:

http://discovermagazine.com/1987/may/02-the-worst-mistake-in...

Based on this it seems like farming was not invented separately in separate places but rather that most early humans were aware of the possibility of farming and that switched to it as conditions deteriorated as a result of climate change at the end of the last ice age.