It's unlikely that rich countries would experience famine as severely as poor ones and consequently they would probably still demand meat. Grain that could feed people would still feed livestock.
A draw down of animal stocks increases meat supply in the short term. As grain gets more expensive, farmers sell animals for meat rather than keeping them to reproduce.
But “As grain gets more expensive” middle eastern countries (that rely almost entirely on import for their grain source) would start facing grain shortage (due to balance of payment issues) or at least severe deprivation of the poorer part of their population.
The Tunisian, Egyptian, Syrian and Libyan revolutions didn't occur at the same moment out of coincidence…