Farm animals have a lifecycle just like humans. They get born, mature into adults, have children and die of old-age. But in contrary to humans they procreate asexually. A single animal in a pasture is enough to start a livestock industry.
No pasture will produce any meat until its population cap is reached. The maximum cap depends on the size of the pasture, but you can artificially reduce it with a slider in the menu of the pasture. When the cap of the pasture is exceeded, the herdsmen will butcher any excess animals for food. Animals which die of old-age do not provide any resources.
Of all animals chicken have the highest rate of growth. They more than double their number each year. But they also have the lowest yield of food per butchered animal: a mere 6 units.
Cattle and sheep procreate much slower. Sheep double about every 30 months, cattle slightly lower with every 36. But butchering a cow or sheep yields a whole 200 units of food, enough to feed two citizens for a year.
Looking at the production statistics window of a fully-stocked, maximum-size, maximum-workers pastries of chicken, cattle and sheep for a year and adding up the production of each season gave the following results:
Meat Other products
25 Sheep 4580 384 wool
20 Cattle 3880 110 leather
66 Chicken 580 3054 eggs
Sheep are the preferred farm animal according to these numbers. They produce the most meat and the most clothing-material as by-product.
Cattle are quite meh compared to sheep. They produce less food and less clothing material. The only reason to have them is when you want some leather for warm coats. When it comes to leather-production, a cattle farm is more personnel-efficient than a hunters lodge, and when you count the whole forest of the hunters lodge as used space, also more space-efficient. Create three cattle-farms for each sheep-farm and you should have an equal supply of leather and wool for your tailors.
Chicken are the least efficient animals to have around. When you add up their egg- and meat production, they are still slightly less productive than cattle, and they have no other by-product you can use. But they have the advantage that they are much quicker to set up due to their high reproduction rate and they already produce food during the setup phase.