I feel your pain! I live in a pasture where rabbits and rodents of all descriptions flourish. I used to lay in bed an night listening to them chewing my home up from underneath. We had to replace spark plug wires on two vehicles just days apart because the rats chewed them to pieces over night. Over ONE night! Rabbits chewed holes in every water hose and every faucet post on the place the first winter we were there. They also chewed holes in the PVC water pipes under the house and made nests in my stored hay.
I trapped a lot of them but the field rats are big guys and they were carrying my traps off. Rabbits, yes folks, cotton tail rabbits, also carry the traps off. Turns out they like cheese and peanut butter just as much as rats and mice. Even when I staked the traps down, they would disappear. Live traps work but are a pain because I then have to dispatch the critter caught and I refuse to use poison of any sort. It is a lot of trouble when I was trapping 10 or more rats and rabbits every night. Without any deterrent, they just kept coming to a steady food source in my hay and chicken feed. No way to keep them out of the chicken feeder and still feed the chickens and no way to keep them out of the wheat hay or hay grazer we often bought for winter.
Cats are my long term solution. I feed them very well and they still hunt constantly. I don't have any rodents, rabbits, snakes, bugs, and very few sparrows within 50 yards of my home due to the cats. I started with 5 kittens my mother found at the cemetery when she went to "visit" my dad. That was 15 years ago! I only have some fixed, mostly get the males neutered, so they have kittens every now and then. They are not pets but not quite feral and I keep them healthy and their numbers under control. The mortality rate is pretty high, nothing I can do about coyotes, owls, hawks, the occasional stray dog or mules/donkeys, but they are very effective at what they do. I live far enough away from any public roadway to avoid them being run over. They don't run off very often, I think they like the food!
I have house cats that do not go outside except in their specially designed cage they access form the window. They have no contact with the outside cats since the cage is in the front yard and the dogs keep the outside cats out of the yard. They have no rodent control duties since the outside cats don't let anything get to the house! I have to buy them toys to play with.
The outside cats, I can't call them barn cats since I don't really have a barn, live under the house for the most part. I feed them 4Health (a Tractor Supply brand) dry and canned food. I give them two small cans of wet food every morning and about 10 cups of dry. I feed an additional 4 cups of dry in the evening. They are fat, soft and shiny! I give them canned food in the morning so I can get a look at them, handle any kittens a little bit and keep them coming up so they don't get hurt under one of the vehicles as we come and go for work and such. They hang around the back porch until I come out and feed them so they are not under the vehicles. We, unfortunately killed a lot of cats before I figured out how to do this. I have about 15 cats right now which is about the right number for the amount of territory they defend for me. One un-neutered male, four adult females but two are unable to complete a prenancy for some reason and two 1/2 grown litters of 4 and 6 kittens. I need to catch kittens and get them fixed or re-homed soon. I try to keep two or three females that can breed and only one tom. The one I have now and the one previous are very good with the kittens and do not tolerate any stray males hanging around. They leave the male kittens alone if I get them neutered young but those guys are not good hunters, the females are voracious hunters even when well fed! I try to give the males away once I get them neutered since they tend to be free-loaders.
They give me a chance to train any of my dogs to leave the cats alone. Good training for dogs I often take to town or camping with me. Between the cats and the chickens, my dogs learn Leave it! pretty quickly.
Sorry for the novel. I get a lot of flack from folks for letting the cats breed but the local shelter will not let me adopt cats if I don't promise to keep them indoors. I always feel like I have to explain my set up and practices to keep from being labeled an abuser.