Several things in Illyriad are based off of population (when you can start your next city for instance), and my population is definitely growing, but I don't know what influences it. Does population grow with time, excess food, or activity (building, producing, etc)?
It is directly based on the level of building you have built and that buildings food consumption on a 1:1 basis. However, some buildings increase faster in food consumption than others.
If you run out of food you can only upgrade resource structures (e.g. farms); but can't queue production of items, troops, do research or build non-resource buildings so its worth keeping an eye on your stockpile.
Population/food consumed also is related to your gold income. At 25% tax your income is 4 x your population.