Cats may make you feel good by bringing you a dead or dying animal now and then
(usually not the target species) but they will not significantly affect the
population of a rodent. Rats and mice can outbreed the ability of a cat to
catch them and they can stay away from the cat by hiding or travelling in
places too small for the cat to follow. Maybe that is why people think they
have so much success. Out of sight, out of mind.
You really need to seal up the means of entry and trap out the ones that are
inside. Anything else is just ****ing on the fire.