I'm a bit confused by the prevalence of the "so how do they cook it?" question in such discussions when I point out that one can live sustainably, if not well, on very little. It's as though many have never gone camping and cooked over a fire. Gas/electric stoves and metal cookware are recent developments which humans got by without for a long time - and many still do. For starters, shish-kabobs come to mind for cooking veggies on a stick.
In a cramped/austerity situation, little actually needs cooking, given that much can be either eaten raw, or just dried and eaten straight or reconstituted as needed. What needs cooking can be over a fire using the dried inedible remainder of the plants. Stir-fry Solar ovens are not hard to build. High-tech solutions, at the other extreme, could provide very efficient heat sources. (We need not abandon high- and future-tech for high-population-density scenarios; to the contrary, there would be great motivation to advance technology to match demands for efficiency.)
Desertification correlating with population growth? Will have to look into it more. Causation is not obvious (which influences which how & more?). Seems people _are_ figuring out how to feed a population with little resources, translating to sustainability with small per-capita footprint.
BTW: Gladwell's "Outliers" spends a lot of time researching how ancient/remote/low-tech/cramped/austerity populations managed to feed themselves with few farming resources. For one, Japanese rice farming was a tour-de-force of squeezing as many calories out of a very limited space per capita. Worth a read.