Snap traps baited with peanut butter, cat food, sunflower seeds or your birds food. Place it along the path with the non baited side blocked off a bit so the rat must approach the bait side, and can't accidentally set it off from behind.
A wily and tenacious dog will hunt rats.
Poison if you can keep it isolated from your chickens. Make sure to safely dispose of dead rats into the trash.
Rat-X pet safe poison doesn't work well from my experience.
Keep feed away at night, sweep up spilled feed, make sure that you're not feeding the rats.
If you have one rat you have more than one rat. Take care.