Most cats cannot catch rats. I currently have three cats and only one of them catches rats. The one who does is a very large male. All three are pretty good for getting mice. My trio controls rodent populations on about 10 acres. It is easy to find a good mouser, difficult to find a good ratter.
A terrier would be good control of rats, but it would hunt your birds too. If you are a dog person this might be a good idea, but it could not be trusted with chickens or ducks off leash. They have a very high prey drive.
Now for more than you ever wanted to know about cats...
My cats all come from generations of farm bred mousers, they are much more active than most peoples house cats. Some cats do not know how to hunt or are not very good at it. For a good cat that will like to hunt, find a kitten or a young cat from a neighbor with farm cats or a feral cat rescue. These will have come from parents that have earned their keep hunting on a farm or in the case of a feral cat have fed themselves by hunting.
Farms who have cats that breed usually will give kittens away for free, feral cat rescues will vaccinate and alter cats then
sell them for less than the cost of the vet care they have received.
You might get some people suggesting that you do not need to
feed a mouser, or feed it less than it needs, as it will find its own food. This will not work, a healthy well fed cat has the
energy to be a better hunter, plus if it is not getting food from you it will move out and find someone who will feed it.
I prefer my mousers altered and vaccinated. Sick cats do not make good mousers. Altered cats need about 25-33% less food than unaltered cats due to changes in metabolism rate, so you need to feed them less. Unaltered males are obnoxious, they spray
urine to mark territory, then leave to find females. A unaltered female will have kittens and spend all her time looking after them instead of hunting.
Adult chickens and ducks are way too big and tough for a cat to hunt.
When picking out a kitten as a likely mouser watch them play for a while. Take the most active one you can find, it will remain playful as an adult, and to a cat killing rodents is playing.
In my
experience cats are not useful for catching mice until they are at least one year old. They don't catch anything bigger until several months of hunting experience.
If there is anything else you want to know about cats just ask me.