Considering the descriptions of the exploding populations of pigeons, rats, and roaches in the depopulated (by humans, that is) city, it is time to examine methods of harvesting these bounties.
First, pigeons.
Canon has it that huge flocks of hundreds, even thousands of pigeons have flocked together and often find nesting areas in the large empty galleries whose windows have been blown out. These flocks are described as darkening the skies when they are startled into flight.
Solution: scout out the roosting places. Construct large strong nets to be suspended over the openings to the roosting areas--fishing line, nylon cording, anything that is nearly invisible but strong. After the nets are quietly rigged and the roost has accumulated a large population, throw in a couple of firecrackers or their homemade equivalent to spook the birds into fleeing their shelter--right into the dropped netting, which captures large numbers of them with minimal damage. At this phase the decision becomes, "How many to eat, how many to smoke/preserve, how many to keep for breeding/eggs, and how many to trade?" The Duke might send mating pairs as gifts to entice independent (but hungry) enclaves into joining him. Oh, yeah--pigeon manure is among the most nutrient-rich manures available. So the the Duke, trying to wring out every calorie from his intensive gardening efforts, will have working crews scraping up and saving every jot and tittle of pigeon poop, then wetting the floors with clean water and sopping up the dissolved nutrients ("pigeon tea") to add to the soil. Imagine the rumors _that_ will generate about the mental state of the Downtown ruler.
And don't forget the feathers and down for pillows or insulating clothing or quilts
Next, rats. One of the narrative bits has a chilling encounter with swarming, chittering, squealing masses of rats in a sewer. Another passage describes rats en masse trying to escape a flooding tunnel. Well, if they are so eager to run through small passages and pipes, why not build some diversion ductwork to pipe off a stream of eager-to-escape rats, with the outflow end depositing the rodents into a gnaw-proof container? Think a 3-dimensional fish seine. Again, a push at the far end of things will motivate the rats into fleeing down the wide end of the funnel(s) that will bring them to the harvesters. And the meat isn't the only benefit from the wee beasties-- you get furs, small though they be, and bones, which can be dried and ground up for soil amendments.
Last,Roaches. Even fried in olive oil with garlic, roaches would not be the first choice in protein sources. In fact, they approach infinity minus 1 in numerical order of what I'd choose. (I can just see Jay Leno with a bag of crispy-fried roaches in Nacho Cheese flavor--"Crunch all you want--we'll breed more!"
) However they ARE full of protein, so why waste them? Just feed them to your pigeons, chickens, rats, or other insectivore food sources.